AND

JAMES SERRIN

Abstract. In this paper we first present the classical maximum principle due to E. Hopf, together with an extended commentary and discussion of Hopf’s paper. We emphasize the comparison technique invented by Hopf to prove this principle, which has since become a main mathematical tool for the study of second order elliptic partial differential equations and has generated an enormous number of important applications. While Hopf’s principle is generally understood to apply to linear equations, it is in fact also crucial in nonlinear theories, such as those under consideration here. In particular, we shall treat and discuss recent generalizations of the strong maximum principle, and also the compact support principle, for the case of singular quasilinear elliptic differential inequalities, under generally weak assumptions on the quasilinear operators and the nonlinearities involved. Our principal interest is in necessary and sufficient conditions for the validity of both principles; in exposing and simplifying earlier proofs of corresponding results; and in extending the conclusions to wider classes of singular operators than previously considered. The results have unexpected ramifications for other problems, as will develop from the exposition, e.g. (i) two point boundary value problems for singular quasilinear ordinary differential equations (Sections 3, 4); (ii) the exterior Dirichlet boundary value problem (Section 5); (iii) the existence of dead cores and compact support solutions, i.e. dead cores at infinity (Section 7); (iv) Euler–Lagrange inequalities on a Riemannian manifold (Section 9); (v) comparison and uniqueness theorems for solutions of singular quasilinear differential inequalities (Section 10). The case of p-regular elliptic inequalities is briefly considered in Section 11.

Contents 1. 2. 3. 4. 5. 6. 7. 8.

Introduction The Hopf Maximum Principle Some preliminary lemmas A singular two-point boundary value problem Radial solutions of an exterior Dirichlet problem Proofs of Theorems 1.1 and 1.2 Dead cores More general quasilinear inequalities 8.1. The strong maximum principle 8.2. The compact support principle 9. Riemannian weighted norms 10. Comparison and uniqueness theorems for singular divergence form operators 10.1. Comparison results 10.2. Uniqueness of the Dirichlet problem 11. p–regular equations 12. Special cases

2 7 10 14 20 24 26 28 29 32 35 39 39 42 43 45

1991 Mathematics Subject Classification. Primary, 35J15, Secondary, 35J70. Key words and phrases. Quasilinear singular elliptic inequalities, Strong maximum and compact support principles. 1

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P. PUCCI AND J. SERRIN

12.1. The linear case 12.2. The degenerate Laplacian case References

45 46 46

1. Introduction The strong maximum principle of Eberhard Hopf is a classical and bedrock result of the theory of second order elliptic partial differential equations. It goes back to the maximum principle for harmonic functions, already known to Gauss in 1839 on the basis of the mean value theorem. On the other hand, it carries forward to maximum principles for singular quasilinear elliptic differential inequalities, a theory initiated particularly by V´azquez and Diaz in the 1980’s, but with earlier intimations in the work of Benilan, Brezis and Crandall. Our purpose here is to provide a clear explanation of this type of result, from its beginnings, to show its relation with and differences from the classical theory of Hopf, and to develop the ramifications of these ideas in rather unexpected byways. In particular, there are intimate connections with a number of fundamental questions of elliptic partial differential equations, more specifically in the noteworthy directions: (i) two point boundary value problems for singular quasilinear ordinary differential equations (Sections 3, 4); (ii) the exterior Dirichlet boundary value problem (Section 5); (iii) the existence of dead cores and compact support solutions, i.e. dead cores at infinity (Section 7); (iv) Euler–Lagrange inequalities on a Riemannian manifold (Section 9); (v) comparison and uniqueness theorems for solutions of singular quasilinear differential inequalities (Section 10). These areas and their relevant connections will be developed throughout the course of the article, see especially Sections 3, 4, 5, 7, 9 and 10. We shall particularly emphasize and maintain the nonlinear nature of the operators involved, in contrast to the naive view sometimes expressed that Hopf’s original result applies principally to linear operators. After an initial discussion of the maximum principle of Eberhard Hopf, Section 2, we shall turn our attention in the following sections especially to the strong maximum principle and the compact support principle for quasilinear differential inequalities. To introduce these questions in the most natural way, it is convenient first to describe a canonical type of inequality to which the discussion applies, and to clarify the structure of these model inequalities by means of special examples. Thus we consider in the first instance the strong maximum principle and the compact support principle for quasilinear elliptic differential inequalities, under generally weak assumptions on the quasilinear operators in question, in the canonical divergence structure (1.1) and (1.2)

div{A(|Du|)Du} − f (u) ≤ 0,

u ≥ 0,

div{A(|Du|)Du} − f (u) ≥ 0,

u ≥ 0,

Rn ,

in a domain (connected open set) Ω in n ≥ 2. Here Du denotes the vector gradient of the given function u = u(x), x ∈ Rn . We assume throughout the paper, unless otherwise stated explicitly, the following conditions on the operator A = A(ρ) and the nonlinearity f = f (u), (A1) A ∈ C(0, ∞), 7 ρA(ρ) is strictly increasing in (0, ∞) and ρA(ρ) → 0 as ρ → 0; (A2) ρ → (F1) f ∈ C[0, ∞),

STRONG MAXIMUM PRINCIPLE

3

(F2) f (0) = 0 and f is non–decreasing on some interval (0, δ), δ > 0. Condition (A2) is a minimal requirement for ellipticity of (1.1)–(1.2). Furthermore, it allows singular and degenerate behavior of the operator A at ρ = 0, that is at critical points of u. We emphasize that no assumptions of differentiability are made on either A or f when dealing with the canonical models (1.1) and (1.2). The operator div{A(|Du|)Du} will be called the A–Laplace operator, so as to place it in the context of well–known elliptic theory. By a classical solution (or a classical distribution solution) of (1.1) or (1.2) in Ω we mean a non–negative function u ∈ C 1 (Ω) which satisfies (1.1) or (1.2) in the distribution sense. With the notation Φ(ρ) = ρA(ρ) when ρ > 0, and Φ(0) = 0, we introduce the function Z ρ (1.3) Φ(s)ds, ρ ≥ 0. H(ρ) = ρΦ(ρ) − 0

This function is easily seen to be strictly increasing, as follows from the inequality Z ρ1 ρ1 Φ(ρ1 ) − ρ0 Φ(ρ0 ) > (ρ1 − ρ0 )Φ(ρ1 ) > Φ(s)ds ρ0

when ρ1 > ρ0 ≥ 0. Alternatively, monotonicity follows from the representation Z Φ(ρ) (1.4) ρ ≥ 0, H(ρ) = Φ−1 (ω)dω, 0

Rρ this being a consequence of the Stieltjes formula H(ρ) = 0 s dΦ(s). For the Laplace operator, that is when (1.1) takes the classical form ∆u − f (u) ≤ 0,

u ≥ 0,

we have A(ρ) ≡ 1 and H(ρ) = 12 ρ2 . Similarly, for the degenerate p–Laplace operator, here denoted by ∆p , p > 1, we have A(ρ) = p ρp−2 and H(ρ) = (p − 1)ρp /p, p while for the mean 2 curvature operator, one has A(ρ) = 1/ 1 + ρ and H(ρ) = 1 − 1/ 1 + ρ2 . In the last example, note the anomalous behavior Φ(∞) = H(∞) = 1, a possibility which occasionally requires extra care in the statement and treatment of results. It is also worth observing that (1.1), when equality holds, is precisely the Euler–Lagrange equation for the variational integral Z u Z F (u) = (1.5) I[u] = {G(|Du|) + F (u)}dx, f (s)ds, Ω

0

where G and A are related by A(ρ) = G 0 (ρ)/ρ, ρ > 0. In this case H(ρ) = ρG 0 (ρ) − G(ρ), the pre–Legendre transform of G. Further comments and other examples of operators satisfying (A1), (A2) are given in [30]. By the strong maximum principle for (1.1) we mean the statement that if u is a classical solution of (1.1) with u(x0 ) = 0 for some x0 ∈ Ω, then u ≡ 0 in Ω. We can now state the main results of [27], which are proved in Section 6 using a very much simplified method based on the results of Sections 3, 4 and 5. Theorem 1.1. (Strong maximum principle). In order for the strong maximum principle to hold for (1.1) it is necessary and sufficient either that f (s) ≡ 0 for s ∈ [0, µ), µ > 0, or that f (s) > 0 for s ∈ (0, δ) and Z δ ds (1.6) = ∞. −1 (F (s)) H 0 As is well known, the strong maximum principle is extremely useful when studying the qualitative behavior of solutions of differential equations and inequalities. The choice of the base level zero for the statement of the principle is of course a matter only of convenience, as is whether we deal with minimum or maximum values at the base point x0 .

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The background and literature for Theorem 1.1 is fairly complicated and deserves a number of comments: The necessity of (1.6) for the case of the Laplace operator is due to Benilan, Brezis and Crandall [4], while for the p–Laplacian it is due to V´azquez [41]. In these cases we observe that (1.6) reduces respectively to Z δ Z δ ds ds p = ∞ and = ∞. 1/p [F (s)] F (s) 0 0

For general operators satisfying (A1), (A2), necessity is due to Diaz ([11], Theorem 1.4), see also ([30], Corollary 1). Sufficiency for the case of the Laplace operator and also for the p–Laplacian is again due to V´azquez [41], see also [11] and [38]. For general operators satisfying (A1), (A2), sufficiency was proved in Theorem 1 of [30] under an additional technical assumption, and in Theorem 1 of [27] without the technical assumption. For the vectorial case see [16]. The case R when f ≡ 0 was studied by Cellina [5] for non–negative minimizers of the integral Ω G(|Du|)dx. An alternative abstract approach to the strong maximum principle appears in [6]. The regular case. If A(ρ) is continuous on [0, ∞), limρ→0 A(ρ) = α > 0, and f (u) ≤ Const. u, (u ≥ 0), then clearly Φ(ρ) ≈ αρ and H(ρ) ≈ αρ2 /2 for small ρ, while also F (u) ≤ Const. u2 ; thus obviously the strong maximum principle is valid. In fact, far stronger results are known in this direction [36]: Let u and v be classical distribution solutions of the differential inequalities ˆ u, Du) ≤ 0 ˆ u, Du) − B(x, divA(x, ˆ v, Dv) ≥ 0, ˆ v, Dv) − B(x, divA(x, ˆ z, ξ) are continuously differˆ z, ξ) and the scalar B(x, in Ω, where the vector function A(x, ˆ is positive definite for all values of entiable in the variables z, ξ, and the matrix [∂ A/∂ξ] its variables. If u ≥ v in Ω, then either u ≡ v or u > v in Ω. ˆ We shall not pursue this direction further, since our interest is essentially in functions A ˆ and B which are singular or degenerate, respectively when Du = 0 and when u = 0. A rigorous treatment of the full sufficiency result of Theorem 1.1, avoiding use of the technical assumption (2.5) of [30], is not entirely obvious, involving as it does the solution of differential inequalities whose structure includes driving and amplifying terms which reinforce each other. The proof here uses only standard calculus, and the elementary Leray– Schauder theorem (see [18], Theorem 11.6), but requires neither monotone operator theory (as [41], [11]–[14]), nor Orlicz–Sobolev space theory (as [23]), nor viscosity solution theory (as [21]), nor probabilistic methods. The proofs have further applications as well, for example to dead core theory, see Section 7 and uniqueness for the Dirichlet problem, see Section 10. In the next result we consider the situation when the integral in (1.6) is convergent. Here the appropriate hypotheses are that u satisfies the converse inequality (1.2) and also “vanishes” at ∞, rather than at some finite point x0 ∈ Ω. More precisely, by the compact support principle for (1.2) we mean the statement that if u is a classical solution of (1.2) in an exterior domain Ω, with u(x) → 0 as |x| → ∞, then u has compact support in Ω. Theorem 1.2. (Compact support principle). Let f (u) > 0 for 0 < u < δ. Then in order for the compact support principle to hold for (1.2) in an exterior domain Ω, it is

STRONG MAXIMUM PRINCIPLE

5

necessary and sufficient that Z

(1.7)

δ

0

ds H −1 (F (s))

< ∞.

As in the case of the strong maximum principle it is worth commenting on the background and literature for Theorem 1.2. Necessity was first shown in Corollary 2 of [30] under the additional technical assumption (2.5) of [30], and in [27], with a proof which is in totality not at all easy. The proof given here is simpler and at the same time provides an existence theorem for radial solutions of exterior Dirichlet problems, see Theorem 5.1. The sufficiency of (1.7) is Theorem 2 of [30], but see also [31] and the remarks following the statement of Theorem 2 in [30]. For radially symmetric solutions of (1.2) sufficiency was proved in [17] under the weaker assumption that F (s) > 0 for s ∈ (0, δ), see Proposition 1.3.1 of [17]. If Theorem 1.2 were an exact analogue of Theorem 1.1, the conclusion of the compact support principle would be that u ≡ 0 in Ω, but this would be incorrect since (1.2) admits non–trivial compact support solutions under assumption (1.7), see [17] and Theorem 7.5 below. The existence of compact support solutions for quasilinear equations was studied extensively in the 80’s, as well as other properties of the set where the solution u vanishes, for example the case of dead cores. In chemical models, when u represents the density of a reactant, the vanishing of a solution then delineates a region where no reactant is present (see [1], [12]). A short discussion of dead cores for (1.1), with equality sign, is given in Section 7, see Theorems 7.2 and 7.3. The results described above can be extended to a wider class of differential inequalities by replacing div {A(|Du|)Du} by the more general operator Di {aij (x, u)A(|Du|)Dj u} and f (u) by B(x, u, Du), where [aij (x, u)] is a continuously differentiable positive–definite symmetric matrix on Ω × R0+ and where B is continuous and satisfies a (typical) condition of the form (1.8)

−Const. Φ(|ξ|) + g(u) ≤ B(x, u, ξ) ≤ Const. Φ(|ξ|) + f (u)

for x ∈ Ω, u ≥ 0 and all ξ ∈ Rn with |ξ| sufficiently small, and with f and g satisfying (F1) and (F2); see Theorem 8.1 and 8.5, and their corollaries, these being the second main goal of the paper; see also Section 9. An important prototype is the equation ∆p u − |Du|q − f (u) = 0,

(1.9)

p > 1, q > 0.

Since Φ(ρ) = ρp−1 for this case, condition (1.8) applies with f = g and requires q ≥ p − 1; that is, the strong maximum principle holds for (1.9) when q ≥ p − 1 and either f ≡ 0 in [0, µ], µ > 0, or f obeys (1.6) – see Corollary 8.3. On the other hand, when q ∈ (0, p − 1) the strong maximum principle can fail, even when f ≡ 0, e.g. the C 1 function u(x) = C|x|k satisfies ∆p u − |Du|q = 0,

(1.10) where p−q k= , s

(p − 1)n − (n − 1)q 1/s 1 , =k C s

s=p−1−q >0

(for p = 2, this example is due to Barles, Diaz and Diaz [3]). It is of further interest in connection with this example that the compact support principle can fail even if (1.8) is satisfied, namely when q > p − 1! Indeed, the function u(x) = L|x|−l satisfies (1.10) in

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P. PUCCI AND J. SERRIN

ΩR = Rn \ BR , with l = (p − q)/t > 0, provided that 1 (n − 1)q − (p − 1)n 1/t n(p − 1) , L= , q> n−1 l t

t = q − p + 1.

For the special case when A ≡ 1 and f (u) = uq , q > 0, the strong maximum principle holds for non–negative C 1 distribution solutions of ∆u−uq ≤ 0 if and only if q ≥ 1, while the compact support principle holds for non–negative C 1 distribution solutions of ∆u − uq ≥ 0 if and only if 0 < q < 1. Actually by the main results of [17], or by Section 7 below, there exist C 2 non–negative radially symmetric compact support solutions of ∆u − uq = 0 when 0 < q < 1. Note that when q = 0 our analysis cannot be applied. Let c ∈ R. The strong maximum principle holds for non–negative C 1 distribution solutions of ∆u − c ≤ 0 only if c ≤ 0. Indeed the equation ∆u − 2n = 0 in any domain Ω of Rn containing the origin admits the non–trivial solution u(x) = |x|2 , but u(0) = 0. We also note that the equation ∆u − c = 0, with c 6= 0, admits no compact support solutions no matter what of the sign of c, as follows from the Hopf boundary point lemma. The same remarks apply to the p–Laplacian analogue ∆p u − uq = 0, p > 1 and q > 0, for which the compact support principle holds for non–negative C 1 distribution solutions if and only if 0 < q < p − 1, while the strong maximum principle holds if and only if q ≥ p − 1. As we shall note in Section 2, dedicated to the original work of E. Hopf (see also [37]), the Maximum Principle implies the Comparison Principle, Theorem 2.4. On the other hand, for singular equations, even if they are smooth, the situation is more delicate. Consider for example ∆4 u + |Du|2 = 0,

(1.11)

n = 2,

which, when expanded to the form F(Du, D2 u) = 0 is smooth (even analytic), elliptic when Du 6= 0, and degenerate,1 that is, ∂F/∂(D2 u) = 0 when Du = 0. The Strong Maximum Principle continues to hold (see Theorem 8.1), while on the other hand (1.11) admits two unequal solutions u ≡ 0 and u(x) = 18 (R2 − |x|2 ) in BR , both with the same boundary values. The paper is structured as follows. In Section 2 we present the classical Hopf Maximum Principle together with some comments of independent interest. Section 3 is devoted to some preliminary lemmas, and Section 4 to existence and uniqueness for related two point boundary value problems for quasilinear ordinary differential equations. Section 5 deals with the existence and uniqueness of classical radial solutions of the exterior Dirichlet problem for (1.1), or (1.2), with equality sign, namely for the case of equations. The results are important in the proof of the compact support principle, but are also of independent interest. In Section 6 we prove the main Theorems 1.1 and 1.2 for the canonical models (1.1) and (1.2). In Section 7 the existence of dead cores for (1.1), with equality sign, is proved, and also the existence of compact support solutions of (1.1) in exterior domains. In Sections 8.1 and 8.2 we consider the case of fully quasilinear inequalities (1.12)

Di {aij (x, u)A(|Du|)Dj u} − B(x, u, Du) ≤ 0 (≥ 0),

u≥0

1In particular, in this case

F(Du, D2 u) = |Du|2 ∆u + 2

2 X i,j=1

2 Di uDj uDij u + |Du|2 .

STRONG MAXIMUM PRINCIPLE

7

(where the obvious summation convention is used). Section 9 extends these considerations to the quasilinear inequality (1.13)

p

Di {aij (x, u)A(|Du|g )Dj u} − B(x, u, Du) ≤ 0,

where |Du|g = g ij (x, u)Di uDj u is a gradient norm of Riemannian type, a case of importance when one treats variational problems on a manifold; in this regard we emphasize particularly Theorem 9.3. Section 10 contains a series of general comparison principles for singular elliptic inequalities of divergence type. These results, which extend well–known theorems of Gilbarg and Trudinger, are important not only in proving our main conclusions for the strong maximum principle, but naturally are useful well beyond this application. In particular, they imply various uniqueness results for the Dirichlet problem, see e.g. Theorems 10.8 and 10.10, which appear to be new in the generality given. Section 11 contains a brief discussion of the strong maximum principle for p–regular inequalities, alternative to the previous considerations. Finally, in Section 12 we treat several special cases where the main proof of Proposition 4.1 reduces to a simpler form. As a byproduct of this discussion we obtain a rational comparison function for some special inequalities, alternative to the classical exponential function of E. Hopf. 2. The Hopf Maximum Principle Before giving the main results already stated, we present the classical principle due to E. Hopf in [20], together with an extended commentary and discussion of Hopf’s original paper by J. Serrin [37]. The maximum principle for harmonic and subharmonic functions was known to Gauss on the basis of the mean value theorem (1839); an extension to elliptic inequalities however remained open until the twentieth century. Bernstein (1904), Picard (1905), Lichtenstein (1912, 1924) then obtained various results by difficult means, as well as use of regularity conditions for the coefficients of the highest order terms. It was Hopf’s genius to see that a “g˝anzlich elementare Begr˝ unden” could be given. The comparison technique he invented for this purpose is essentially so transparent that it has generated an enormous number of important applications in many further directions. Here is Hopf’s theorem in its main form: Let u = u(x), x = (x1 , . . . , xn ), be a C 2 function which satisfies the differential inequality X X ∂u ∂2u Lu ≡ aij bi + ≥0 ∂xi ∂xj ∂xi i,j

i

in a domain Ω, where the (symmetric) matrix aij = aij (x) is locally uniformly positive definite in Ω and the coefficients aij , bi = bi (x) are locally bounded. If u takes a maximum value M in Ω, then u ≡ M in Ω.

Hopf’s proof (Section I of [20]), now a classic of the subject, is reproduced in the monographs [26] and [18], and in many other texts as well, particularly the second volume of [7]. The hypothesis that u is of class C 2 is essential for the theorem, though not always strictly noted in presentations of the result. For maximum principles when u is not of class C 2 , and even possibly only measurable, see e.g. Littman [22]; for the case of C 1 distribution solutions, see the later results of the present paper, as discussed in the introduction. Hopf next observes (Section II of [20]) that one can allow the coefficients to depend on the solution u itself, provided that when they are evaluated along a solution the resulting functions aij (x), bi (x) satisfy the conditions of the main theorem. This allows him to deal explicitly with nonlinear as well as linear equations. In the same section he then notices two important corollaries (S˝atze 2, 3) dealing with the differential inequality Lu + cu ≥ 0. First, for the case c = c(x) ≤ 0 and a positive

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maximum, and second, when there is an extremum M = 0 with c being bounded but not necessarily non–positive. The latter possibility is not mentioned in [18]. Moreover, Courant and Hilbert in their formulation of Satz 2 in [7] do not include the crucial restriction to a positive maximum. Because Hopf’s formulation of these results is somewhat obscure, the main conclusions are worth restating here, which we do in terms of the operator L. Theorem 2.1. Let u be a C 2 function satisfying the differential inequality (2.1)

Lu + cu ≥ 0

(≤ 0)

in a domain Ω, where the coefficients of L satisfy the previous conditions, and c = c(x) is a non-positive function on Ω. If u takes a positive maximum (negative minimum) value M in Ω, then u ≡ M .

Theorem 2.2. Let the hypotheses of Theorem 2.1 hold, except that one now assumes only that the function c is locally bounded. If u takes on a vanishing maximum (minimum) value M = 0 in Ω, then u ≡ 0.

The real depth of Hopf’s nonlinear analysis shows up only in Section III of [20], though the presentation is seriously obscured by the restriction to exact equations, as well as to the case where one of the solutions in question is assumed to vanish identically (“engere Voraussetzungen” according to Hopf). Accordingly we shall again restate the results, in slightly greater generality and in more usual notation. Theorem 2.3. (Touching Lemma). Let u, v be C 2 (Ω) solutions of the nonlinear differential inequalities F(x, u, Du, D2 u) ≥ 0,

F(x, v, Dv, D2 v) ≤ 0,

where F is of class C 1 in the variables u, Du, D2 u (notation obvious). Suppose also that the matrix ∂F 2 2 Qij ≡ 2 u) (x, u, Du, θD u + (1 − θ)D v) ∂(Dij is positive definite in Ω for all θ ∈ [0, 1]. If u ≤ v in Ω and u = v at some point x0 in Ω, then u ≡ v in Ω. The terms u, Du in Q can be replaced by v, Dv. Proof. Essentially following Hopf’s proof of Satz 30 of [20], we write 0 ≥ F(x, v, Dv, D2 v) − F(x, u, Du, D2 u) = F(x, u, Du, D2 v) − F(x, u, Du, D2 u)

+ F(x, u, Dv, D2 v) − F(x, u, Du, D2 v) + F(x, v, Dv, D2 v) − F (x, u, Dv, D2 v) X X 2 = aij Dij bi Di (v − u) + c(v − u) = L(v − u) + c(v − u), (v − u) +

where, for some values θ, θ1 , θ2 ∈ [0, 1] we have aij =

∂F 2 2 2 u) (x, u, Du, θD v + (1 − θ)D u) = Qij ∂(Dij

∂F (x, v, θ1 Dv + (1 − θ1 )Du, D2 u) ∂Di u ∂F c= (x, θ2 v + (1 − θ2 )u, Dv, D2 v). ∂u Clearly aij , bi , c are locally bounded, and equally by continuity the coefficient matrix aij is locally uniformly positive definite on Ω. Since by assumption v − u ≥ 0 and (v − u)(x0 ) = 0, it now follows from Theorem 2.2 that v ≡ u in Ω. bi =

STRONG MAXIMUM PRINCIPLE

9

To obtain the final conclusion of the theorem, one proceeds in the same way, though starting from the alternative decomposition 0 ≥ F(x, v, Dv, D2 v) − F (x, u, Du, D2 u) = F(x, v, Dv, D2 v) − F(x, v, Dv, D2 u)

+ F(x, v, Dv, D2 u) − F(x, v, Du, D2 u) + F(x, v, Du, D2 u) − F(x, u, Du, D2 u).

The next result (essentially Satz 20 of [20] in a more general context and formulation) is stated as a comparison result, rather than a maximum principle, this being the underlying content of Hopf’s result. Theorem 2.4. (Comparison Lemma). Let u, v be C 2 (Ω) ∩ C(Ω) solutions of the nonlinear differential inequalities given in Theorem 2.3. Suppose that the matrix Q = Qij is positive definite in Ω and that ∂F (x, w, Dv, D2 v) ≤ 0 Ψ= ∂u for all functions w ≥ v (or simply for all functions w on Ω). If u ≤ v on ∂Ω, then u ≤ v in Ω. The terms u, Du in Q can be replaced by v, Dv if at the same time the terms Dv, D2 v in Ψ are replaced by Du, D2 u and the condition w ≥ v is replaced by w ≤ u. Proof. Suppose for contradiction that the conclusion v − u ≥ 0 in Ω fails. Then there will be a subdomain Ω0 of Ω in which v −u ≤ 0 but is not identically constant, and in which also v − u takes on a negative minimum M . As in the proof of Theorem 2.3 one obtains L(v − u) + c(v − u) ≤ 0, while by hypothesis c ≤ 0 in Ω0 . Hence by Theorem 2.1 we get v − u ≡ M in Ω0 , a contradiction. The final conclusion is obtained from the alternative decomposition in the proof of Theorem 2.3. Using other decompositions, one can obtain various related results, e.g. Theorem 31 of Chapter 2 of [26]. A direct consequence of Theorem 2.4 is a uniqueness theorem for the Dirichlet problem for the nonlinear equation F(x, u, Du, D2 u) = 0, a fact mentioned by Hopf in the final paragraph of [20], though not explicitly formulated by him. Since the result is important, and a precise formulation is in fact not immediate from Hopf’s analysis, it is worth stating a definite result here. Theorem 2.5. Let u and v be C 2 solutions of the nonlinear equation F(x, u, Du, D2 u) = 0

in a domain Ω, with u = v on ∂Ω. Suppose Q is positive definite in Ω for all θ ∈ [0, 1], and Ψ ≤ 0 in Ω for all functions w. Then u ≡ v. This is an immediate corollary of Theorem 2.4, the main result being used to establish that u ≤ v and the final part of the theorem to get v ≤ u. Here it is crucial that Ψ ≤ 0 for all functions w. It is surprising that the matrix Q in the hypothesis of Theorem 2.5 is, insofar as its second and third arguments are concerned, to be evaluated solely on the functions u and Du, without any symmetric reference to v and Dv. Indeed specializing Theorem 2.5 to quasilinear equations, we find that for the equation 2 Aij (x, Du)Dij u − B(x, u, Du) = 0

a sufficient condition for uniqueness is that the matrix Qij = Aij (x, Du) needs to be positive definite (i.e. the equation needs to be elliptic) only when evaluated for either one (!) of the solutions u or v, provided that B(x, u, ξ) is a non–decreasing function of u for arbitrary

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P. PUCCI AND J. SERRIN

arguments x, ξ. This last result (essentially due to Hopf, though not explicitly mentioned or stated by him) seems to have appeared first in [18], first edition, Chapter 8. The result applies at once to the quasilinear operator X ∂u ∂u ∂ 2 u F = (1 + |Du|2 )∆u − ∂xi ∂xj ∂xi ∂xj

(mean curvature), since clearly

Qij = (1 + |Du|2 )Iij − Di uDj u

is positive definite for all values of its arguments. Here of course there is no need to use the full strength of Theorem 2.5. On the other hand, if we consider the Dirichlet problem X ∂u ∂u ∂ 2 u =0 (1 + |Du|2 )∆u − 2 ∂xi ∂xj ∂xi ∂xj

in Ω, with u = 0 on ∂Ω, then the matrix Q is not positive definite for arbitrary arguments D2 u. Nevertheless Q = I for the function u ≡ 0, whence it follows that this function is the unique solution of the stated Dirichlet problem. A second and more subtle example is the elementary Monge–Amp`ere equation in R2 2 2 ∂ u ∂2u ∂2u − = g(x, y). 2 2 ∂x ∂y ∂x∂y Here one checks that

Qij ξi ξj =

∂2u 2 ∂2u 2 ∂2u + ξ ξ − 2 ξ . ξ 2 1 1 ∂y 2 ∂x∂y ∂x2 2

The discriminant of Q is then det Q = det Hu =

∂2u ∂2u − ∂y 2 ∂x2

∂2u ∂x∂y

2

,

which is precisely g = g(x, y) when evaluated at a solution u. Suppose in particular that g > 0. It is easy to see then, that any solution u is either everywhere strictly convex or everywhere strictly concave. From this, one can check without difficulty that if u and v are two convex solutions then 2 (θu + (1 − θ)v). Q is positive definite for the arguments Dij Hence the Dirichlet problem for the elementary Monge–Amp`ere equation above has at most one convex solution. On the other hand, if u and v are concave solutions, then −u and −v are convex solutions and so, similarly, the Dirichlet problem can have at most one concave solution; altogether then the problem can have at most two solutions. This result is a special case of a theorem of Rellich [32]; see [7], page 324. Other related maximum and comparison principles are discussed in the Notes to Chapter 2 of [26], and in Chapter 10 of [18], to which the reader is strongly referred; see also the references cited on page 314 of [42]. A viscosity based maximum principle for singular fully nonlinear equations is given in [2]. Hopf’s proof technique, as noted above, leads to other results of fundamental interest, particularly the celebrated Boundary Point Lemma and a Harnack principle for elliptic equations having two independent variables; for this last result, see the paper [34] of J. Serrin, reproduced in both [26] and [18]. A nonlinear version of the Harnack principle in two variables has also been given recently in [28]. 3. Some preliminary lemmas Here we turn to the study of the strong maximum principle and of the compact support principle for divergence structure quasilinear elliptic operators and for nonlinear terms f (u). In general, the results described cannot be obtained from the nonlinear theorems of the previous section, since the operators and equations in question for the most part

STRONG MAXIMUM PRINCIPLE

11

have specialized properties which are lost when they are written in the expanded form F(x, u, Du, D2 u) = 0 as required there. We shall assume from here on, and throughout the paper unless otherwise mentioned explicitly, that A and f satisfy (A1), (A2), (F 1), (F 2). Moreover, without loss of generality (since we deal with non–negative solutions) one may suppose that f (u) = 0

for u ≤ 0.

For convenience in what follows it is useful to extend the definition of the principal operator Φ to all values real values of ρ by setting Φ(ρ) = −Φ(−ρ) when ρ < 0, unless otherwise explicitly specified. Following and refining [27], we require several preliminary lemmas. Lemma 3.1. (i) For any constant σ ∈ [0, 1] there holds F (σu) ≤ σF (u),

u ∈ [0, δ].

(ii) Let w = w(t) be of class C 1 (0, T ), and write 0 = d/dt. If Φ ◦ w0 is of class C 1 (0, T ) then H ◦ w0 is of class C 1 (0, T ), and in this case (3.1)

[H(w0 (t))]0 = w0 (t)[Φ(w0 (t))]0

in (0, T ).

On the other hand, if H ◦ w0 is of class C 1 (0, T ) and w0 > 0, then Φ ◦ w0 is of class C 1 (0, T ) and (3.1) continues to be satisfied. To obtain (i), observe that σf (σu) ≤ σf (u) for u ∈ [0, δ], since f is non–decreasing. Integrating this relation from 0 to u yields the result. The first statement of (ii) is an immediate consequence of (1.4). The second part is also a consequence of (1.4) together with a small lemma: Let I be any interval of R and let Z a(t) t ∈ I, B(t) = b(s)ds,

where B ∈ C 1 (I); a, b ∈ C(I); and b > 0. Then a ∈ C 1 (I) and a0 = B 0 /(b ◦ a). This is easily demonstrated by using difference coefficients and the integral mean value theorem to get ∆B/∆t = b(a+θ∆a)∆a/∆t, 0 ≤ θ ≤ 1. The lemma then follows by dividing by b(a + θ∆a) and letting ∆t → 0. Lemma 3.2. Suppose f (u) > 0 for u > 0 and (in case H(∞) < ∞) that F (δ) < H(∞). If τ ≥ 1 and (1.6) holds, then also Z δ/τ ds = ∞. −1 (τ F (s)) H 0 Similarly, if 0 < σ ≤ 1 and (1.7) is satisfied, then Z δ ds < ∞. −1 (σF (s)) 0 H

Proof. For small ε > 0, we have by Lemma 3.1 (i), with σ = 1/τ , Z Z δ/τ Z δ/τ ds dt ds 1 δ ≥ = . −1 (τ F (s)) −1 (F (τ s)) −1 (F (t)) H H τ H ε/τ ε/τ ε Letting ε → 0 and applying (1.6) gives the first result. Again by Lemma 3.1 (i), Z δ Z Z δ ds ds 1 δσ dt ≤ = −1 (σF (s)) −1 (F (σs)) −1 (F (t)) H H σ H ε ε εσ and the second part now follows by letting ε → 0 and applying (1.7).

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P. PUCCI AND J. SERRIN

Lemma 3.3. Let T > 0 and assume (3.2)

q ∈ C(0, T ),

q>0

in (0, T ).

Then every classical distribution solution w = w(t) of the problem ( 0 = d/dt) ( [sign w(t)] · [q(t)Φ(w0 (t))]0 ≥ 0 in (0, T ), (3.3) w(0) = 0, w(T ) = m > 0 is such that (3.4)

w ≥ 0,

w0 ≥ 0

in (0, T ).

Even more there exists t0 ∈ [0, T ) with the property that

(3.5)

w≡0

in [0, t0 ];

w > 0,

w0 > 0

in (t0 , T ).

Proof. We first claim that w ≥ 0 in [0, T ]. If the conclusion fails, there would be t0 and t1 , with 0 ≤ t0 < t1 < T such that w(t0 ) = w(t1 ) = 0 and w < 0 in (t0 , t1 ). Then, multiplying (3.3) by w and integrating on [t0 , t1 ] yields by integration by parts (or simply by the distribution meaning of solutions with the test function w(t) on [t0 , t1 ]) Z t1 q(t)Φ(w0 (t))w0 (t)dt ≤ 0, t0

where the integrand is non–negative by (3.2) and the fact that ρΦ(ρ) > 0 for ρ = 6 0. That 0 is, necessarily w ≡ 0 on [t0 , t1 ]. Hence w ≡ 0 on [t0 , t1 ], since w(t0 ) = w(t1 ) = 0. This contradiction proves the claim. Define the set J = {t ∈ (0, T ) : w0 (t) > 0}. Then, obviously, J 6= ∅, since w(0) = 0 and w(T ) > 0, while also J is open in (0, T ) since w ∈ C 1 (0, T ). Let t0 = inf J, so t0 ∈ [0, T ) and w ≡ 0 in [0, t0 ], since we already know that w ≥ 0 in [0, T ]. Now, for any fixed t ∈ (t0 , T ) there obviously exists t1 ∈ (t0 , t) such that w0 (t1 ) > 0. By integration of (3.3) on [t1 , t], recalling that w ≥ 0 on (0, T ), we get q(t)Φ(w0 (t)) ≥ q(t1 )Φ(w0 (t1 )) > 0

by (3.2) and (A2), so that w0 > 0 on (t0 , T ]. In turn, by integration, w > 0 in (t0 , T ), proving (3.5). Remark. If in Lemma 3.3 the hypothesis (3.2) is strengthened to q ∈ C(0, T ),

in (0, T ),

q>0

q non–increasing,

then w0 is non–decreasing on [0, T ] and m . T Indeed from (3.3) and (3.4) it follows that q(t)Φ(w0 (t)) is non–decreasing, and then since q(t) is non–increasing also Φ(w0 (t)) is non–decreasing. But Φ is increasing, so w0 is non– decreasing. In turn, w is convex on [0, T ] and then (3.6) follows at once since w(T ) = m. 0 ≤ w0 (0) ≤

(3.6)

Lemma 3.4. Assume q ∈ C[0, T ],

(3.7)

q>0

in (0, T ).

Then along every classical distribution solution w of the problem ( [q(t)Φ(w0 (t))]0 − q(t)f (w(t)) ≤ 0 in (0, T ), (3.8) 0 w(0) = 0; 0 ≤ w ≤ δ, w ≥ 0 in (0, T ), there holds (3.9)

f (w(t)) Φ(w (t)) ≤ q(t) 0

Z

0

t

q(s) ds +

q(0) Φ(w0 (0+)), q(t)

STRONG MAXIMUM PRINCIPLE

13

where w0 (0+) is defined as lim supt→0+ w0 (t). In particular, if w0 (0) = 0 then (3.9) reduces to Z f (w(t)) t 0 (3.10) Φ(w (t)) ≤ q(s) ds. q(t) 0 Proof. Integrating (3.8) on [τ, t], with 0 < τ < t < T , yields Z t 0 0 (3.11) q(t)Φ(w (t)) − q(τ )Φ(w (τ )) ≤ q(s)f (w(s))ds, 0

and (3.9) follows at once by (F2), i.e., f (w(s)) ≤ f (w(t)) since 0 ≤ w(s) ≤ w(t) < δ, together with the lim sup as τ → 0. Lemma 3.5. Assume (3.7) and 0 + Z s q (s) 1 (3.12) q ∈ C (0, T ), − q(τ )dτ q(s)2 0

bounded on (0, t) for all t ∈ (0, T ).

Then along every classical distribution solution w ∈ C 1 (0, T ) of the problem (3.8) for which w0 (0) = 0 and the condition Φ(w0 ) is continuously differentiable

(3.13) is satisfied,

2

we have H(w0 (t)) ≤ B(t)F (w(t)),

(3.14) where

t ∈ (0, T ),

+ 0 Z q (s) s q(τ )dτ . B(t) = 1 + sup − q(s)2 0 s∈(0,t)

(3.15)

Note that if q 0 ≥ 0 , then (3.14) becomes H(w0 (t)) ≤ F (w(t)). Proof. Denote by E the energy function associated to w in (0, T ), namely E(t) = H(w0 (t)) − F (w(t)).

Since Φ(w0 ) ∈ C 1 (0, T ) by assumption, so also H(w0 ) ∈ C 1 (0, T ) by Lemma 3.1 (ii). Then by (3.1) and (3.8) one finds (since distribution derivatives of C 1 functions can be treated as ordinary derivatives) (3.16)

E 0 (t) = w0 {[Φ(w0 (t))]0 − f (w(t))} ≤ −

q 0 (t) Φ(w0 (t))w0 (t), q(t)

t ∈ (0, T ),

since by assumption w0 ≥ 0, q > 0 in (0, T ). Integrating (3.16) on (0, t), with 0 < t < T , yields Z t 0 q (s) 0 H(w (t)) ≤ F (w(t)) − Φ(w0 (s))w0 (s)ds (since w0 (0) = 0), q(s) 0 + Z t 0 Z q (s) s ≤ F (w(t)) + − q(τ ) dτ f (w(s))w0 (s) ds ≤ B(t)F (w(t)) 2 q(s) 0 0 by (3.10) and (3.15).

Proposition 3.6. Assume (3.7) and (3.12). Let w be a classical distribution solution of the problem ( [q(t)Φ(w0 (t))]0 − q(t)f (w(t)) ≤ 0 in (0, T ), (3.17) w(0) = 0, w(T ) = m > 0, w0 ≥ 0, 2For the main application of this lemma in Section 4 this condition holds without any difficulty; see Proposition 4.4.

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P. PUCCI AND J. SERRIN

for which (3.13) is satisfied. Suppose that f (u) > 0 for u > 0. If w0 (0) = 0 then Z δ ds (3.18) < ∞. −1 (F (s)) 0 H Proof. From the second line of (3.17) it is evident that there exists t0 ∈ [0, T ) such that w(t) = 0 for 0 ≤ t ≤ t0 while w > 0 in (t0 , T ). If t0 = 0, then w0 (0) = 0 by hypothesis, while if t0 > 0 then in turn w(t0 ) = w0 (t0 ) = 0 since w ∈ C 1 (0, T ). Let t2 ∈ (t0 , T ). Clearly there exists t1 ∈ (t0 , t2 ) such that m1 = w(t1 ) > 0 satisfies m1 < δ/B,

F (Bm1 ) < H(∞),

where B = B(t2 ) ≥ 1 is given in Lemma 3.5. From this lemma applied to the interval (t0 , t1 ), we thus obtain (see (3.14)) H(w0 (t)) ≤ B(t)F (w(t)) ≤ BF (w(t)) in (t0 , t1 ) since B(t) is obviously non–decreasing. In turn by Lemma 3.1 (i), with σ = 1/B, H(w0 (t)) ≤ F (Bw(t)) in (t0 , t1 ),

that is w > 0, w0 (t) ≤ H −1 (F (Bw(t))) on (t0 , t1 ). Using the fact that f (u) > 0 for u > 0 (and so also F (u) > 0 for u > 0), integration now yields Z m1 Z t1 Z Bm1 dw du w0 (t)dt = B = B ≤ B(t1 − t0 ) < ∞, −1 (F (Bw(t))) H −1 (F (u)) H −1 (F (Bw)) 0 t0 H 0

as required.

4. A singular two-point boundary value problem In this section we shall obtain existence and uniqueness theorems for the differential problems ( [q(t)Φ(w0 (t))]0 − q(t)f (w(t)) = 0 in (0, T ), (4.1) w(0) = 0, w(T ) = m > 0. and (4.2)

(

[q(t)Φ(w0 (t))]0 − a(t)q(t)f (w(t)) = h(t) w(0) = 0, w(T ) = m > 0.

in (0, T ),

The following two main existence theorems, Propositions 4.1 and 4.3, will be crucial in supplying radial comparison functions for the proofs in later sections. Importantly in these propositions, we are able to use a weakened version of condition (F2), namely (F3) f (0) = 0 and f is non–negative on some interval [0, δ), with δ possibly infinite. Accordingly it will be assumed in both Propositions 4.1 and 4.3 that m ∈ (0, δ). Finally, we shall suppose of the function q in (4.1) and (4.2) that it is of class C[0, T ] with q > 0 in [0, T ]. Put q0 = min q(t) > 0, [0,T ]

q1 = max q(t) > 0. [0,T ]

Of course, in addition to (F3), conditions (A1), (A2), (F1) will be maintained throughout the section. Proposition 4.1. (i). Let Φ(∞) = ∞. Then problem (4.1) admits a classical distribution solution with the properties (4.3)

w ∈ C 1 [0, T ],

Φ(w0 ) ∈ C 1 [0, T ];

w0 ≥ 0.

STRONG MAXIMUM PRINCIPLE

15

Moreover, for any such solution of (4.1) we have w0 (T ) > 0 and q1 ¯ kw0 k∞ ≤ Φ−1 [T f (m) + Φ(m/T )] , (4.4) q0 where f¯(m) = maxu∈[0,m] f (u). In particular, w0 ≤ 1 if m is sufficiently small. (ii) Suppose Φ(∞) = ω < ∞. Let m ∈ (0, δ) be such that q1 ¯ (4.5) [T f (m) + Φ(m/T )] < ω. q0 Then the conclusion of part (i) continues to hold. Proof. For the purpose of this proof only, we shall redefine the operator Φ for ρ < 0 by setting Φ(ρ) = ρ when ρ < 0; this can be done without loss of generality since the ultimate solution w satisfies w0 ≥ 0. Case (i). Let (4.6) µ1 = q1 [T f¯(m) + Φ(m/T )] and I = [0, µ1 ]. It is convenient also to redefine f so that f (u) = f (m) for all u ≥ m. This will not affect the conclusion of the proposition, since clearly any ultimate solution with w0 ≥ 0 satisfies 0 ≤ w ≤ m. We recall also the earlier agreement that f (u) = 0 for u ≤ 0. With these preliminaries settled, we can proceed to the main proof. We shall make use of the Leray–Schauder fixed point theorem, an idea suggested in this context by Montenegro. Denote by X the Banach space X = C[0, T ], endowed with the usual norm k · k∞ , and let T be the mapping from X to X defined by 3 (4.7)

T [w](t) = m −

Z

T

−1

Φ t

Z T 1 q(τ )f (w(τ ))dτ ds, µ− q(s) s

t ∈ [0, T ],

where µ = µ(w) ∈ I is chosen so that (4.8)

T [w](0) = 0.

We shall show that such a choice of µ is uniquely possible. Indeed for any fixed w ∈ X and for any µ ∈ I we have Z Z T µ1 f¯(m) T 1 − (4.9) q(t) dt ≤ µ− . q(τ )f (w(τ ))dτ ≤ q0 q(s) q0 s 0 It follows now that T [w] is well defined for each fixed µ in I. Moreover for µ = 0 we see that, for all w ∈ X, T [w](0) ≥ m. On the other hand, for µ = µ1 we find, for all w in X, Z T Z T q1 1 q(τ )f (w(τ ))dτ ds Φ(m/T ) + q1 T f¯(m) − T [w](0) = m − Φ−1 q(s) q(s) s 0 Z T ≤m− Φ−1 (Φ(m/T ))ds = 0, 0

3The simpler mapping

Z

T [w](t) =

0

t

Φ−1

1 κ+ q(s)

Z

s

q(τ )f (w(τ ))dτ

ds

0

with κ = κ(w) chosen so that T [w](T ) = m, is in fact less convenient in carrying out the proof.

16

P. PUCCI AND J. SERRIN

where we have used the condition (4.6), the definition of q1 , and the fact that 0 ≤ f (u) ≤ f¯(m). Since the integral on the right side of (4.7) is a strictly increasing function of µ for fixed w, it is now obvious that there exists a unique µ ∈ I such that (4.8) holds. Define the homotopy H : X × [0, 1] → X by Z T Z T 1 −1 µσ − σ q(τ )f (w(τ ))dτ ds, H[w, σ](t) = σm − Φ (4.10) q(s) t s where µσ = µ(w, σ) ∈ I is a number chosen such that H[w, σ](0) = 0.

Clearly, as above, such a value µσ exists and is unique, and the mapping H[w, σ] is accordingly well defined. By construction, any fixed point wσ = H[wσ , σ] is of class C 1 [0, T ], has the property that Φ(w0 ) ∈ C 1 [0, T ], and is a classical distribution solution of the problem ( [q(t)Φ(wσ0 (t))]0 − σq(t)f (wσ (t)) = 0 in [0, T ], (4.11) wσ (0) = 0, wσ (T ) = σm. Moreover, by Lemma 3.3, a fixed point w = H[w, 1] satisfies w, w0 ≥ 0, and so is a solution of problem (4.1) satisfying the conditioins (4.3), with w0 ≥ 0. It remains to show that such a fixed point w = w1 exists. We shall use Browder’s version of the Leray–Schauder theorem for this purpose (see Theorem 11.6 of [18]). To begin with, obviously µσ = 0 when σ = 0, and so H[w, 0](t) ≡ 0 for all w in X, that is H[w, 0] maps X into the single point w0 = 0 in X. (This is the first hypothesis required in the application of the Leray–Schauder theorem at the end of the proof.) We show next that H is compact and continuous from X × [0, 1] into X. Let (wk , σk )k be a bounded sequence in X × [0, 1]. Clearly µσk ∈ I; therefore again using the fact that 0 ≤ f (u) ≤ f¯(m) for all u ≥ 0, together with (4.9), it is clear that kH0 [wk , σk ]k∞ ≤ C 0 ,

where (recalling that Φ−1 (ρ) = ρ when ρ < 0) ¯ Z f (m) T (4.12) q(t)dt, Φ−1 (µ1 /q0 ) . C 0 = max q0 0 It is now an immediate consequence of the Ascoli–Arzel`a theorem that H maps bounded sequences into relatively compact sequences in X. We claim finally that H is continuous on X × [0, 1]. Indeed, let wj → w, σj → σ, (wj , σj ) ∈ X × [0, 1]. Then in (4.10) clearly σj f (wj ) → σf (w), since the modified function f is continuous4 on R. It must then be shown that µ(wj , σj ) → µ(w, σ). To this end, suppose for contradiction that this fails. Then, for some subsequence, still called (wj , σj ), we should have µ(wj , σj ) → µ ˜ 6= µ = µ(w, σ). In this case, from (4.8) one gets by subtraction Z T Z T 1 −1 Φ ˜−σ q(τ )f (w(τ ))dτ µ q(s) 0 s (4.13) Z T 1 −1 q(τ )f (w(τ ))dτ ds = 0 −Φ µ−σ q(s) s

But Φ−1 is a monotone increasing function of its argument, so clearly the integrand in (4.13) is either everywhere positive or everywhere negative, giving the required contradiction. 4It is here that the condition f (0) = 0 in (F3) is crucial. In fact the proposition fails otherwise, as

shown by the example f (u) ≡ 1, q ≡ 1, and A(ρ) ≡ 1. In this case every non–negative solution of (4.1) must have the form w(t) = at + 12 t2 , a ≥ 0, which gives the extraneous condition for solvability m = w(T ) = aT + 12 T 2 ≥ 21 T 2 .

STRONG MAXIMUM PRINCIPLE

17

To apply the Leray–Schauder theorem it is now enough to show that there is a constant M > 0 such that (4.14)

kwk∞ ≤ M

for all (w, σ) ∈ X × [0, 1], with H[w, σ] = w.

Let (w, σ) be a pair of type (4.14). But, as observed above, since w0 ≥ 0, clearly kwk∞ = w(T ) = σm ≤ m. Thus we can take M = m in (4.14). The Leray–Schauder theorem therefore can be applied and the mapping T [w] = H[w, 1] has a fixed point w ∈ X, which is the required solution of (4.1). That (4.3) holds for this solution was noted earlier in the proof. The last part of the theorem is a direct consequence of (4.7) evaluated at a fixed point w, together with the right hand inequality of (4.9) and the fact that µ ∈ I. Case (ii). The argument is exactly the same as before, with the single exception that in (4.9) the right hand side µ1 /q0 is now less than ω by virtue of (4.5). Thus, T is well–defined in X, and the rest of the proof is unchanged. In view of (4.3) we note that, for the given solution w, all derivatives with respect to t in (4.1) can equally well be understood as ordinary derivatives, no recourse to distribution solutions in fact being needed. The following lemma is important for the next proposition. Lemma 4.2. Let condition (F3) hold, and assume Φ(∞) = ∞. Suppose also that q ∈ C[0, 1] and that q is positive and non–increasing on [0, 1]. (i) Let w be any solution of (4.1) with m ∈ (0, δ) and T = 1. Then 0 −1 q(0) ¯ (4.15) [f (m) + Φ(m)] . w (1) ≤ Φ q(1) (ii) Let w be any solution of (4.1) with m ∈ (0, δ) and T = 1, but now with the initial condition w(0) = 0 replaced by w, w0 > 0 on [0, 1]. Then (4.15) continues to hold. Proof. Case (i) follows from the second part of Proposition 4.1 (i), and the identifications T = 1, q0 = q(1), q1 = q(0). The proof of case (ii) lies deeper, relying on an idea in [17]. Let v = v(t) be a solution of (4.1) with m ∈ (0, δ) and T = 1, given by Proposition 4.1 (i), which exists since Φ(∞) = ∞ in the present case. Also q0 = q(1), q1 = q(0) so that (4.4) implies 0 −1 q(0) ¯ (4.16) [f (m) + Φ(m)] , v (1) ≤ Φ q(1)

because T = 1. We shall show that

w0 (1) ≤ v 0 (1)

(4.17)

To see this, observe first by Lemma 3.3 that v ≡ 0 in [0, t0 ]; v, v 0 > 0 in (t0 , 1] for some t0 ∈ [0, 1). By assumption the given solution w is also such that w, w0 > 0 in [0, 1]. Hence we can introduce the C 1 functions t : [0, m] → [t0 , 1],

s : [w0 , m] → [0, 1],

respectively inverse to v and w on the sets where v and w are positive; here w0 = w(0) > 0. Clearly s(w0 ) = 0, t(w0 ) > t0 ; s(m) = t(m) = 1, and t0 (m) = 1/v 0 (1). s0 (m) = 1/w0 (1), If for contradiction (4.17) fails, then w0 (1) > v 0 (1) and s0 (m) < t0 (m). In this case, we claim that there would be an interval (u1 , m), with u1 ∈ (w0 , m), such that (4.18)

s(u) > t(u) > 0,

0 < s0 (u) < t0 (u) for u ∈ (u1 , m);

s0 (u1 ) = t0 (u1 ).

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P. PUCCI AND J. SERRIN

Indeed, since s(w0 ) < t(w0 ), the condition s0 (u) < t0 (u), which holds at u = m, cannot persist for all smaller values of u in the open interval (w0 , m). Thus there must be a first point u1 ∈ (w0 , m) where s0 (u1 ) = t0 (u1 ), and in turn the claim (4.18) follows at once. Now by integration of (4.1) along the solution v(t) from t(u1 ) to 1, we derive Z 1 Z m 0 q(t)f (v(t))dt = q(1)Φ(v 0 (1)) − q(t(u1 ))Φ(v 0 (t(u1 ))), q(t(u))f (u)t (u)du = u1

t(u1 )

with a similar relation for the solution w. By subtraction Z m [q(t(u))t0 (u) − q(s(u))s0 (u)]f (u)du = q(1)[Φ(v 0 (1)) − Φ(w0 (1))] u1

− [q(t(u1 )) − q(s(u1 ))]Φ(v 0 (t(u1 ))),

since w0 (s(u1 )) = v 0 (t(u1 )) by (4.18). The left hand side is non–negative by virtue of (F3), condition (4.18), and the fact that q is positive and non–increasing; while the right hand side is negative since v 0 (1) < w0 (1) by the contradiction assumption and again the fact that q is positive and non–increasing. This absurdity shows (4.17), and application of (4.16) then completes the proof. Proposition 4.3. Let q satisfy the conditions given in the paragraph before Proposition 4.1, and assume additionally that q is non–increasing. (i) Suppose Φ(∞) = ∞ and let T ≥ 1, m ∈ (0, δ). Then problem (4.1) admits a classical distribution solution with w ∈ C 1 [0, T ] and w ≥ 0. Moreover p1 ¯ [f (m) + Φ(m)] , (4.19) kw0 k∞ ≤ Φ−1 p0

where p0 = q(T ), p1 = q(T − 1). (ii) Suppose Φ(∞) = ω < ∞ and T ≥ 1. Let m ∈ (0, δ) be such that p1 ¯ (4.20) [f (m) + Φ(m)] < ω. p0 Then the conclusion of part (i) continues to hold. Proof. (i) Consider the auxiliary problem ( [q(t)Φ(v 0 (t))]0 − q(t)f (v(t)) = 0 (4.21) v(T − 1) = 0, v(T ) = m,

in (T − 1, T ),

where m ∈ (0, δ). We assert that (4.21) has a C 1 [T − 1, T ] solution with v 0 ≥ 0 and 0 −1 p1 ¯ kv k∞ ≤ Φ [f (m) + Φ(m)] . (4.22) p0

The existence in fact follows at once from Proposition 4.1 (i). To prove (4.22), it is enough to translate to the present case the estimate (4.4) in Proposition 4.1 (i). But for this we have obviously q0 = min q(t) = q(T ) = p0 , [T −1,T ]

q1 = max q(t) = q(T − 1) = p1 , [T −1,T ]

since q is non–increasing. Moreover, in (4.5) the length of the interval [T − 1, T ] is of course just 1. Hence (4.4) becomes exactly (4.22), as required. We now apply the comparison Lemma 4.2 to the solution w of Proposition 4.1 (i) and the solution v of (4.21) just determined. Their common interval of definition is just [T − 1, T ], an interval of precisely length 1. Clearly w(T ) = v(T ) = m. Moreover either w(T − 1) = 0 or w(t), w0 (t) > 0 for all t ∈ [T − 1, T ] – see Lemma 3.3. We thus infer that w0 (T ) ≤ v 0 (T ). But also w0 (t) ≤ w0 (T ) for all t ∈ [0, T ] in view of the comment after Lemma 3.3. Consequently w0 (t) ≤ v 0 (T )

STRONG MAXIMUM PRINCIPLE

19

and (4.19) now follows from (4.22). This proves case (i). ˆ defined by (ii) Let ω ˆ denote the left hand side of (4.20). We introduce a new operator Φ, for 0 ≤ ρ ≤ Φ−1 (ˆ ω) Φ(ρ) ˆ Φ(ρ) = (4.23) ˆ ω −1 for ω ). ρ ρ ≥ Φ−1 (ˆ Φ (ˆ ω) ˆ is continuous and increasing on [0, ∞), thus satisfying (A1) and (A2), and moreClearly Φ ˆ = ∞. over Φ(∞) ˆ Clearly a solution w exists, We apply part (i) to problem (4.1), but with Φ replaced by Φ. and by (4.19) it obeys −1 p1 ¯ 0 ˆ ˆ kw k∞ ≤ Φ [f (m) + Φ(m)] . (4.24) p0 Now from the given assumption (4.20) one finds p0 Φ(m) ≤ ω ˆ≤ω ˆ, p1

ˆ ω ), so Φ(m) since p0 ≤ p1 , because q is non–increasing. It follows that m ≤ Φ−1 (ˆ = Φ(m) by (4.23). Therefore (4.24) becomes −1 p1 ¯ 0 ˆ ˆ −1 (ˆ [f (m) + Φ(m)] = Φ ω ), kw k∞ ≤ Φ ω ) = Φ−1 (ˆ p0

again by (4.23). But this is just (4.19) for w, as required.

Proposition 4.4. Suppose that (F2) is satisfied. Let q ∈ C[0, T ] with q > 0 in [0, T ], and also assume condition (3.12) – or, slightly stronger, that q ∈ C 1 [0, T ). Suppose additionally that either f (u) = 0 when u ∈ (0, µ), µ > 0, or that (1.6) holds, that is Z δ ds (4.25) = ∞. −1 (F (s)) H 0

Then the solution of (4.1) given in either Proposition 4.1 or Proposition 4.3 has the properties (4.26)

w>0

in (0, T ],

w0 > 0

in [0, T ].

Proof. Case 1. Let f (u) = 0 when u ∈ (0, µ). Then from (4.1) we have [q(t)Φ(w0 (t))]0 = 0 at least for t near 0. Hence in turn qΦ ◦ w0 = Constant > 0 for small t (if the constant is zero, then w0 = 0 for small t > 0, and then by continuation for all t > 0, which contradicts the boundary condition w = m at t = T ). Consequently w0 (0) = Φ−1 (Constant/q(0)) > 0, so from Lemma 3.3 and the fact that t0 = 0 in the present case, we get w0 (t) > 0 in [0, T ] and w > 0 in (0, T ] as required. Case 2. Let (4.25) hold. Note that (3.13) is satisfied in view of (4.3). Also we already know that w0 (0) ≥ 0 and 0 ≤ w ≤ m. In fact, the case w0 (0) = 0 cannot occur by Proposition 3.6 and assumption (4.25). Consequently w0 (0) > 0 and the required conclusion then follows as before. Remarks. If (A2) is strengthened by adding that q is in C 1 (0, T ) and Φ ∈ C 1 (R+ ) with Φ0 > 0 in R+ , then one finds easily that the solution w is in C 2 (0, T ). If also q 0 ≤ 0, as is frequently the case, then w00 ≥ 0. Proposition 4.1 can be improved by allowing a more general version of equation (4.1), namely [q(t)Φ(w0 )]0 − q(t)B(t, w, w0 ) = 0, provided that B is a continuous function of its variables such that −κΦ(ρ) ≤ B(t, u, ρ) ≤ κΦ(ρ) + f (u),

|ρ| ≤ 1,

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P. PUCCI AND J. SERRIN

for some constant κ > 0 and for f = f (u) satisfying the previous assumptions (F1) and (F3). The proof is essentially the same as before, with the exception that the space X = C[0, T ] must be replaced by X = C 1 [0, T ] while the required mapping H[w, σ] is now defined by Z T Z T 1 −1 0 µσ − σ Φ H[w, σ](t) = σm − B(τ, w(τ ), w (τ ))dτ ds. q(s) s t This of course makes it more delicate to prove that the mapping is compact, though the argument again follows from the Ascoli–Arzel`a theorem. Similarly, proving that any fixed point is uniformly bounded in X takes more effort, but no essentially new or difficult ideas, see [29]. An existence theorem for the problem (4.2) can be given, exactly following the ideas of Proposition 4.1. Proposition 4.5. Assume a, h, q ∈ C[0, T ] and h ≥ 0, a ≥ 0, q > 0 in [0, T ]. Then RT problem (4.2) with m ∈ (0, δ), and with m and 0 h(t) dt suitably small in case Φ(∞) < ∞, admits a classical distribution solution with the properties w ∈ C 1 [0, T ], w0 ≥ 0. The proof goes in almost the same way as before for Proposition 4.1, except one must take Z T µ1 = q1 [a1 T f¯(m) + Φ(m/T )] + h(t)dt, where a1 = max a(t). 0

t∈[0,T ]

The question of uniqueness of solutions of (4.1) and (4.2) is also of interest. For this result, we assume the main conditions (A1), (A2), (F1), (F2). Theorem 4.6. Assume a, h, q ∈ C(0, T ) and a ≥ 0, q > 0 in (0, T ). Then problems (4.1) and (4.2) admit at most one classical distribution solution with range in [0, δ). Proof. Let w and w ˜ be two solutions of (4.2) with ranges in [0, δ). Then by (4.2) together with (A2) and (F2), we obtain Z T ˜ 0 (t)]dt 0≤ q(t)[Φ(w0 (t)) − Φ(w ˜ 0 (t))] · [w0 (t) − w 0

=−

Z

T

0

˜ a(t)q(t)[f (w(t)) − f (w(t))] ˜ · [w(t) − w(t)]dt ≤ 0.

˜ in [0, T ] since Φ is strictly increasing. It now follows at once that w ≡ w

It is possible to prove uniqueness with condition (F2) replaced by the weaker hypothesis (F3), when m < δ¯ and q is non–increasing. We omit the discussion, the details being essentially the same as in Theorem 5.3 (ii) in the next section.

5. Radial solutions of an exterior Dirichlet problem In the next section we shall prove the necessity of Theorem 1.2 through the existence of classical solutions of the exterior Dirichlet problem for (1.1), with equality sign. Because of the separate and independent interest of this question, we devote the present section to its consideration. As in Section 4, we maintain conditions (A1), (A2), (F1). Moreover we consider in place of (F3) the slightly stronger condition (F3)0 f (0) = 0 and f is positive on some interval (0, δ), with δ possibly infinite. Clearly (F3)0 implies (F3), while as noted before (F2) also implies (F3). At the same time (F2) neither implies (F3)0 nor vice versa.

STRONG MAXIMUM PRINCIPLE

21

Theorem 5.1. (Exterior Dirichlet Problem). Assume condition (F3)0 is satisfied, and let ΩR = {x ∈ Rn : |x| > R}. Then for all R > 0 and m ∈ (0, δ), with m sufficiently small if Φ(∞) = ω < ∞, there is a classical radial solution u(x) = u(r) of the problem (5.1)

div{A(|Du|)Du} − f (u) = 0,

in ΩR , such that

u(R) = m,

(5.2)

u(x) → 0

u≥0

as |x| → ∞.

Moreover u0 < 0 whenever u > 0. The required smallness condition on m when ω < ∞ is given below by (5.3).

Proof. Let j = 1, 2, . . . , q(t) = (R + j − t)n−1 and denote by wj the solution of [q(t)Φ(wt (t))]t − q(t)f (w(t)) = 0, w(0) = 0, w(j) = m ∈ (0, δ), wt ≥ 0 in [0, j],

which exists by Proposition 4.3 and the fact that q(t) is decreasing. When ω < ∞ we must of course maintain condition (4.20), which in the present case take the form (since T = j ≥ 1, p0 = q(j) = Rn−1 , p1 = q(j − 1) = (R + 1)n−1 ), n−1 R ¯ ω. f (m) + Φ(m) < (5.3) R+1 It follows now that uj (r) = wj (t), t = R + j − r, is a solution of n−1 Φ(u0 (r))]0 − r n−1 f (u(r)) = 0 (0 = d/dr), [r u(R) = m, u(R + j) = 0, 0 u ≤ 0 in [R, R + j]

(here recall that Φ is defined for all real ρ, according to the agreement at the beginning of Section 3, namely Φ(ρ) = −Φ(−ρ) if ρ < 0). Now by (4.19) we have ! R + 1 n−1 ¯ 0 −1 [f (m) + Φ(m)] . (5.4) kuj k∞ ≤ Φ R Hence from the Arzel`a–Ascoli theorem (and a diagonal process) a subsequence of the functions uj converges uniformly to a non-negative, non–increasing Lipschitz continuous limit u on every compact subset of [R, ∞). We shall show that u is the required solution of (5.1), (5.2). Of course u : [R, ∞) → [0, m], with u(R) = m. In fact uj satisfies on [R, R + j] the following integral equation corresponding to (4.7), Z r Z s −1 1−n n−1 τ uj (r) = m − Φ s µj − f (uj (τ ))dτ ds. Moreover

uj0 (R)

=

R −1 1−n −Φ (R µj ),

R

so

µj = Rn−1 Φ(|u0j (R)|) > 0.

Then by (5.4) we get µj ≤ (R + 1)n−1 [f¯(m) + Φ(m)]. Hence, up to a subsequence, if necessary, the bounded sequence still called (µj )j must converge to some number µ ≥ 0. Letting j → ∞ the limit function u satisfies the integral equation Z r Z s −1 1−n n−1 u(r) = m − Φ s µ− τ (5.5) f (u(τ ))dτ ds. R

R

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P. PUCCI AND J. SERRIN

But then u is continuous on [R, ∞) by (5.5) and in turn then of class C 1 [R, ∞); thus u is also a classical distribution solution of ( [rn−1 Φ(u0 (r))]0 − rn−1 f (u(r)) = 0 in [R, ∞), (5.6) u(R) = m; u ≥ 0, u0 ≤ 0 in [R, ∞), by (5.5). Of course, the equation on the first line of (5.6) is equivalent to (5.1) for radial functions u = u(r). To complete the proof of the theorem it therefore remains to show that u0 < 0 when u > 0 and that u(r) → 0 as r → ∞. To obtain the first, note by virtue of (5.6) that should u0 = 0 at some point r0 where u > 0 then by (F3)0 we would have rn−1 Φ(u0 (r)) > 0 for all r > r0 sufficiently close to r0 , which is absurd. For the second part, it is first of all the case that u must decrease to some non–negative limit ` as r → ∞. Suppose for contradiction that ` > 0. By (F3)0 and the fact that u0 < 0 (since u > 0), by integrating (5.6) on [r, r + 1], with R ≤ r < ∞, we get n−1 Z r+1 r 1 0 0 Φ(u (r + 1)) − Φ(u (r)) = τ n−1 f (w(τ )) dτ r+1 (r + 1)n−1 r (5.7) n−1 Z r+1 r f (w(τ )) dτ. > r+1 r From (F3)0 and the fact that ` ≤ u ≤ δ along the solution, one sees that f (u(r)) > 0. Hence by (5.6) again, we find that rn−1 Φ(|u0 (r)|) is decreasing and in turn also |u0 | decreasing. That is, u0 is negative and increasing. Consequently one must have u0 (r) → 0 as r → ∞. Letting r → ∞ in (5.7) then yields 0 ≥ f (`) > 0, which is the required contradiction.

Theorem 5.2. Let the hypotheses of Theorem 5.1 be satisfied, and suppose also that condition (F2) is valid. Then the solution u given by Theorem 5.1 is everywhere positive provided that (1.6) holds. Conversely if (1.7) is satisfied, then u has compact support. The proof of the first part of this result will be given following Theorem 1.1 in the next section. Similarly, the proof of the second part of the result will be deferred until after the proof of Theorem 1.2. Remark. Condition (5.3) is not best possible, and can be improved to the form n−1 R m ¯ ω, T0 f (m) + Φ ≤ T0 R + T0 where T0 > 0 is a positive parameter which can be assigned arbitrarily; this follows easily by redoing Lemma 4.2 and Proposition 4.3 with the respective conditions T = 1 and T ≥ 1 replaced by T = T0 and T ≥ T0 . p As an example, when R << 1 and A(ρ) = 1/ 1 + ρ2 is the mean curvature operator, with f (u) = κu, κ > 0, and n = 2 (equation of a capillary surface under gravity), by taking T0 = aR with a >> 1 we get the solvability condition m < R; whereas from (5.3) one gets the weaker condition m < R/(1 + κ). An alternative approach to the radial exterior problem, containing a number of precise estimates in the case when ω < ∞ and Ω0 (0) > 0, has been given by Turkington [40]. We conclude the section by showing that the solution u = u(r) given in Theorem 5.1 is unique, under various natural conditions. The precise results are as follows. Theorem 5.3. Let m > 0 and R > 0 be fixed. (i) Assume (F3)0 is satisfied. Then there cannot be more than one radial solution of (5.1) in ΩR which has a bounded range in [0, δ) and satisfies u(R) = m. Moreover, any such solution is convex and obeys (5.2).

STRONG MAXIMUM PRINCIPLE

23

(ii) Assume (F3) is satisfied. Then there cannot be more than one radial solution of (5.1), (5.2) in ΩR which has range in [0, δ). (iii) Assume (F2). Then there cannot be more than one solution of (5.1), (5.2) in ΩR , whether radial or not, which has range in [0, δ). Proof. (i) Let u, v be two solutions of the type described. By the earlier arguments of this section it is evident that u is strictly convex whenever it is positive. Hence u0 ≤ 0 for otherwise u would become unbounded for large enough r, contrary to assumption. Then, as in the proof at the end of Theorem 5.1, we get u(x) → 0 as |x| → ∞, that is (5.2) holds. The same of course is true for the solution v. But then u ≡ v by virtue of Theorem 3.6.7 of [17], when we observe that equation (∗) in [17] is exactly (5.1) here, and condition (G1) there (with α replaced by δ) is just (F3) here.5 (ii) This is again just Theorem 3.6.7 of [17]. (iii) Uniqueness for this case is an immediate consequence of the following comparison result, which we state in a more general form than necessary, in anticipation of later pur poses. Theorem 5.4. (Weak comparison principle). Assume (F2) is satisfied. Let u and v be, respectively, classical solutions of (1.1) and (1.2) in a bounded domain Ω. Suppose also that u and v are continuous in Ω, with v < δ in Ω and u ≥ v on ∂Ω. Then u ≥ v in Ω. The conclusion also holds for exterior domains Ω, provided that additionally one has lim inf{u(x) − v(x)} ≥ 0

as |x| → ∞.

Before proving Theorem 5.4 it is convenient to give a simple preliminary lemma Lemma 5.5. Let ξ and η be vectors in Rn . Then whenever ξ 6= η.

{A(|ξ|)ξ − A(|η|)η} · (ξ − η) > 0

Proof. Since A(ρ) > 0 when ρ > 0 and ξ · η ≤ |ξ| · |η|, there follows by direct calculation {A(|ξ|)ξ − A(|η|)η} · (ξ − η) ≥ {Φ(|ξ|) − Φ(|η|)} (|ξ| − |η|)

and the conclusion now comes from the strict monotonicity of Φ.

Proof of Theorem 5.4. We follow the proof of Lemma 3 of [30], first supposing that Ω is bounded. Let w = u − v in Ω. If the conclusion fails, then there exists a point x1 ∈ Ω such that w(x1 ) < 0. Fix ε > 0 so small that w(x1 ) + ε < 0. Consequently, since w ≥ 0 on ∂Ω it follows that the function wε = min{w + ε, 0} is non–positive and has compact support in Ω. By the distribution meaning of solutions, taking the Lipschitzian function wε as test function, we get Z Z (5.8) {A(|Du|)Du − A(|Dv|)Dv}Dwε ≤ {f (v) − f (u)}wε . Ω

Ω

The left hand side of (5.8) is positive due to Lemma 5.5 and the fact that Dwε ≡ Dw = Du − Dv 6≡ 0 when w + ε < 0, while otherwise Dwε = 0 (a.e.). Moreover, when w + ε < 0 there holds 0 ≤ u < v − ε < δ; hence f (v) − f (u) ≥ 0 since f (s) is non–decreasing for s < δ by (F2). Thus the right hand side of (5.8) is non–positive, a contradiction. The case when Ω is an exterior domain is proved in almost exactly the same way. We leave the details to the reader. 5The proof of Theorem 3.6.7 in [17] relies on the preceding Theorems 3.6.1 – 3.6.5. All of these results

are straightforward, except possibly for Theorem 3.6.5. A simpler proof of the latter result can however be given, using the ideas of Theorem 3.6.4.

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Theorem 5.4 is closely related to Theorem 10.1 of [18], and equally does not require differentiability conditions for the nonlinear terms; see also Theorem 10.5. 6. Proofs of Theorems 1.1 and 1.2 With the work of the preceding two sections available, we can now turn to the main results of the paper, proofs of the Strong Maximum Principle, Theorem 1.1, and the Compact Support Principle, Theorem 1.2. Proof of Theorem 1.1. We recall that Φ is defined for ρ < 0 by Φ(ρ) = −Φ(−ρ). The radial function v(x) = w(t), t = R − r, r = |x|, where w is given by Proposition 4.1 with m < δ, q(t) = (R − t)n−1 and T = R/2, satisfies the differential equation (1.2) in the annular set ER = {x ∈ Rn : R/2 ≤ |x| ≤ R}. Writing 0 = d/dt = −d/dr in accordance with Proposition 4.1, one has Dv(x) = −w0 (t)x/r for R/2 ≤ |x| ≤ R. Moreover w0 (t) > 0 for t ∈ [0, R/2] by Proposition 4.4 and the fact that Φ(w0 ) is continuously differentiable (see Proposition 4.1 (i)). Hence we find div A(|Dv|)Dv − f (v) = − div A(w0 )w0 x/r − f (w) (6.1)

=[Φ(w0 )]0 −

=

(n − 1) Φ(w0 ) − f (w) r

1 [q(t)Φ(w0 )]0 − f (w) = 0, q(t)

where at the second step we use D(A(w0 )w0 ) = −[Φ(w0 )]0 x/r. Of course one has Dv(x) = −w0 (R − r)x/r 6= 0 in [R/2, R]. This being shown, the proof of sufficiency is now exactly the same as in the standard demonstration of the strong maximum principle for linear equations (see e.g. the proof of Theorem 3.5 on page 35 in [18]); here one uses the fact that the comparison function v constructed above satisfies the following conditions, see the proof of Lemma 3.4 on page 34 in [18]: (i) v > 0 in [R/2, R) by Proposition 4.4, (ii) v = 0 when |x| = R by Proposition 4.1, (iii) ∂v/∂n = v 0 < 0 when |x| = R, where n is the outer normal to ∂ER , (iv) v = m when |x| = R/2 by Proposition 4.1, where m, R > 0 can be taken arbitrarily small and the origin of coordinates can be chosen arbitrarily in Ω. Note that the use of the weak maximum principle (Corollary 3.2 of [18]) is here replaced by application of Theorem 5.4. This completes the proof of the sufficiency part of Theorem 1.1. As remarked in the introduction, the necessity is due to Diaz [11]. Hence Theorem 1.1 is proved (see also comment 4 at the end of the section and the further remarks at the end of Section 7). Proof of first part of Theorem 5.2. Because of (1.6) the strong maximum principle is valid for (1.1). But since u(R) = m > 0 and because u is a non–negative (radial) solution of (1.1), it now follows that u > 0 on the entire domain of the solution. Proof of Theorem 1.2. To prove necessity, suppose (1.7) fails, that is (1.6) holds. By Theorem 5.1 and the first part of Theorem 5.2, therefore, there exists a positive classical solution u of (1.1) with equality sign (and thus also of (1.2) with equality), in the domain ΩR = {x ∈ Rn : |x| > R}, such that u(x) → 0 as |x| → ∞. This violates the compact support principle. Hence (1.7) is necessary. For the sufficiency we follow the proof of Theorem 2 of [30]. By (1.7) we can define Z δ ds (6.2) C= < ∞, −1 (F (s)) 0 H

STRONG MAXIMUM PRINCIPLE

25

where, if necessary, one can take δ > 0 smaller so that F (δ) < H(∞). Introduce w = w(r), 0 ≤ r ≤ C, by Z δ ds r= . (6.3) −1 (F (s)) w(r) H Differentiation gives −

w0 (r) =1 H −1 (F (w(r))

for 0 ≤ r ≤ C,

that is, w is of class C 1 [0, C], with w(0) = δ, w(C) = 0, 0 ≤ w ≤ δ, and w0 (r) < 0 for 0 ≤ r < C. Also H(|w0 |) = F (w), so H(|w0 |) is of class C 1 [0, C] with [H(|w0 |)]0 = f (w)w0 . Then from Lemma 3.1 (ii) with T = C, we see that Φ(|w0 |) is of class C 1 (0, C) and (6.4)

−[Φ(|w0 |)]0 = f (w)

for 0 < r < C.

Obviously w(r) → 0, w0 (r) → 0 and [Φ(|w0 |)]0 → 0 as r → C. Therefore, by defining w(r) ≡ 0 for r ≥ C, it is clear that w becomes a C 1 solution of (6.4) in (0, ∞). Now let u be the solution of (1.2) in an exterior domain Ω with u(x) → 0 as |x| → ∞. We must show that u has compact support in Ω. To begin with, clearly there exists R0 ≥ R such that u(x) < δ if |x| ≥ R0 . For any x ∈ Ω0 = {x ∈ Rn : |x| > R0 }, define v(x) = w(|x| − R0 ). Consequently, for x ∈ Ω0 , and r = |x|, we have (6.5)

div{A(|Dv|)Dv} − f (v) = −[Φ(|v 0 |)]0 +

(n − 1) Φ(v 0 ) − f (v) ≤ 0 r

in view of (6.4) (which now holds in (0, ∞)), and the fact that Φ(v 0 ) ≤ 0 when v 0 ≤ 0. Since 0 ≤ u(x) < δ = v(x) on ∂Ω0 , and since u(x), v(x) → 0 as |x| → ∞, we can apply the comparison Theorem 5.4 (with the roles of u and v interchanged) to obtain 0 ≤ u(x) ≤ v(x) in Ω0 . In particular u(x) = 0 when |x| ≥ R1 = R0 + C, as required. Proof of second part of Theorem 5.2. Recall that (F3) holds by hypothesis. Then because of (1.7) the compact support principle Theorem 1.2 is valid for equation (5.1). But since u is a non–negative (radial) solution of (5.1) with u(x) → 0 as |x| → ∞, it now follows that u has compact support in the domain |x| ≥ R. Remarks. 1. The sufficiency part of Theorem 1.2 is closely related to Theorem 4 of [31], by specializing the results there to the matrix aij = A(|ξ|)δij + [A0 (|ξ|)/|ξ|]ξi ξj which arises by expansion of the divergence term in (1.2). This specialization requires, however, two assumptions which are not needed here, first that the operator A be of class C 1 (0, ∞), and second, that the solutions in consideration should be of class C 2 at points of Ω where Du 6= 0. In the proof of Theorem 4 of [31] it is not evident that an appropriate comparison principle can be applied without the further assumption that the nonlinearity f be non– decreasing for small u > 0 – that is, for the validity of Theorem 4 of [31] this additional assumption, which is exactly (F2) above, seems to be required as well. For the special case of the degenerate Laplacian, see also [13]. The proof of sufficiency we have given is in fact not different in its underlying ideas from those in [4], [6], [13], [31], [41], the principal improvements here being the direct approach, the generality of the solution class, and the clarification of the method. We note also that Diaz, Saa and Thiel have stated a version of Theorem 1.1, see Theorem 6 of [14], but with insufficient proof. 2. The last sentence of the proof of Theorem 1.2 gives an a priori estimate for the support of the solution u. 3. Theorem 1.2 also applies when f satisfies the alternative conditions: (f1) f ∈ C(0, ∞), (f2) f is a maximal graph with f (0) = 0 and lim inf u→0 f (u) > 0 (or +∞);

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P. PUCCI AND J. SERRIN

rather than (F1), (F2). We can transform the vertical segment of f at u = 0 into a linear segment with finite slope, thus arriving at a function f¯ ≤ f satisfying (F1) and (F2). But then every solution of (1.2) remains a solution of (1.2) with f replaced by f¯, and the result of Theorem 1.2 continues to apply. A similar argument can be used also for maximal monotone graphs f , see [41]. 4. Another proof of the necessity of (1.6) for the Strong Maximum Principle. Suppose f (u) > 0 for u > 0 and that (1.6) fails, that is (1.7) holds. We can then introduce the n = function w = w(r), defined on [0, ∞), as in the proof of Theorem 1.2. For any x ∈ R+ n {x ∈ R : xn > 0}, let u(x) = w(xn ). By (6.4), u is obviously a solution of (1.1), with the n . Clearly u(0, . . . , 0, C) = w(C) = 0 and at the same equality sign, in the domain Ω = R+ time u 6≡ 0 in Ω. Hence the strong maximum principle fails. 5. The necessity of condition (1.6) for the Strong Maximum Principle can be obtained under a weaker hypothesis than (F2). In fact, it is enough to replace (F2) by (F2)0

f (0) = 0

and

F (s) > 0

for s ∈ (0, δ).

This is because the principal construction required for Diaz’ proof uses only condition (F2)0 ; see also the construction of the function w = w(r) noted just above. 6. The necessity also yields a direct and simple counterexample to the unique continuation question for the equation div{A(|Du|)Du} − f (u) = 0, when (1.7) holds. That is, the function u(x) = w(xn ) shows that a solution in a domain Ω may vanish in a subdomain without vanishing throughout Ω. Theorems 7.2 and 7.5 below give more sophisticated counterexamples. 7. Dead cores An elliptic equation or inequality is said to have a dead core solution u in some domain Ω ⊂ Rn provided that there exists an open subset Ω1 with compact closure in Ω such that u ≡ 0 in Ω1 ,

u>0

in Ω \ Ω1 .

The condition u > 0 could be replaced by u 6= 0, but for definiteness (and physical reality) we prefer the condition as stated. In what follows we maintain the original conditions (A1), (A2), (F1), (F2), unless otherwise stated. The additional condition (7.1)

f is positive in (0, δ)

will also be important. Lemma 7.1. (Dead core lemma). Suppose (7.1) and (1.7) are satisfied. For fixed σ in (0, 1), define Z δ ds Cσ = (7.2) (> 0). −1 (σF (s)) 0 H

Then for every C ∈ (0, Cσ ) there exists a number γ = γ(C) ∈ (0, δ) and a function w ∈ C 1 [0, C] such that γ → 0 as C → 0, (i) (ii)

w(0) = w0 (0) = 0,

w(C) = γ;

0 ≤ w0 ≤ H −1 (F (γ)),

(iii)

[Φ(w0 (t))]0 = σf (w(t))

for t ∈ (0, C),

(iv)

Φ(w0 (t)) ≤ σtf (w(t))

for t ∈ (0, C).

[Here we can assume without loss of generality that σF (δ) < H(∞).]

STRONG MAXIMUM PRINCIPLE

27

Proof. First note that the integral in (7.2) is convergent, in view of Lemma 3.2 and (1.7). For given C ∈ (0, Cσ ), we take γ ∈ (0, δ) so that Z γ ds ; 0

w0 (t) = 1, H −1 (σF (w(t)))

that is H(w0 ) = σF (w) and in turn [H(w0 )]0 = σf (w)w0 . Obviously part (ii) of the Lemma is satisfied; moreover, since w0 > 0 on (0, C], from Lemma 3.1(ii) we obtain part (iii). An integration using parts (ii), (iii) and (F2) shows that also Φ(w0 (t)) ≤ σtf (w(t)); see the proof of Lemma 3.4. This completes the proof. Theorem 7.2. Suppose (7.1) and (1.7) are satisfied. Let R > 0 be fixed. Then the equation (7.3)

div{A(|Du|)Du} − f (u) = 0

admits a non–negative dead core solution in BR . Proof. Fix σ = 1/n. Take 0 < C < min{Cσ , R} and put S = R − C. Define the radial function v(r) = w(r − S), r = |x| ∈ [S, R], where γ = γ(C) and w(t) are as given in Lemma 7.1. Then for r ∈ (S, R) n−1 div {A(|Dv(x)|)Dv(x)} − f (v(x)) = [Φ(v 0 (r))]0 + Φ(v 0 (r)) − f (v(r)) r r−S (7.4) ≤ σ 1 + (n − 1) − 1 f (v(r)) r ≤ (σn − 1)f (v(r)) = 0,

where we have used parts (iii) and (iv) of Lemma 7.1, and the fact that f (v(r)) > 0 since v((S, R]) ⊂ (0, δ). Of course also v(S) = v 0 (S) = 0,

v(R) = γ < δ.

Consider the radial solution u = u(r), r = |x|, of the problem ( div {A(|Du|)Du} − f (u) = 0, u(S) = 0, u(R) = m > 0, given by Proposition 4.1, with q(r) = rn−1 and with m ∈ (0, γ) suitably small (translate coordinates by r = t + S and take T = R − S = C). Also suppose (4.5) is obeyed if Φ(∞) < ∞. Now apply Theorem 5.4, with the roles of u and v interchanged. This gives 0 ≤ u(r) ≤ v(r), r ∈ [S, R]. Hence u0 (S) = 0 since v 0 (S) = 0. Therefore u can be extended as a solution of (7.3) to the entire set BR by putting u ≡ 0 in BS . This proves the existence of the required dead core solution of (7.3). Theorem 7.3. Suppose (7.1) and (1.7) are satisfied. Let R > 0 be fixed. Then any solution u of (1.2) in BR with range in [0, δ) and for which u(x) is suitably small on ∂BR , is a dead core solution. This is shown in the same way as Theorem 7.2.

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Corollary 7.4. Suppose condition (F2) is replaced by the assumption that f is non–decreasing in (−δ, δ). Assume also that uf (u) > 0 for u 6= 0 and that (1.7) holds for both ranges (0, δ) and (−δ, 0). Let u be a solution of [sign u(x)] · [div{A(|Du|)Du} − f (u(x))] ≥ 0

in BR with range in (−δ, δ). Then u vanishes in BS for some S ∈ (0, R), provided |u(x)| is suitably small on ∂BR . For p–regular equations (see Section 11), and therefore in particular without monotonicity conditions, this result was obtained by Diaz and Veron [15]. Lemma 7.1 gives a companion result to Proposition 4.4. Namely, let (7.1) and (1.7) be satisfied. Then if m is suitably small the solution of (4.1) given by Proposition 4.1 has the property w0 (0) = 0. The proof is obvious, after what has gone before. We conclude by noting the existence of compact support solutions of equation (1.2), with the equality sign. In fact, one can interpret a compact support solution as a dead core at infinity. Theorem 7.5. Suppose (7.1) and (1.7) are satisfied. Let R > 0 be fixed. Then (7.3) admits a (non–trivial) non–negative compact support solution in ΩR = {x ∈ Rn : |x| > R}.

This is just the second part of Theorem 5.2. A related result for the p–Laplace operator is well–known, see [13]. Of course, if (1.7) fails, the strong maximum principle shows that a non–negative compact support solution would in fact vanish identically. A dead core with bursts. It is known that when (7.1) and (1.7) hold and when f appropriately changes sign for u > δ, there are non–negative radially symmetric solutions v of (7.3) having compact support; see for example [17]. Let R1 be the support radius of such a solution. Next choose R and S in Theorem 7.2 so that S >> R1 , and let w denote the corresponding dead core solution. This being done, we can now replace the solution w on the set BR1 , where it vanishes, by the solution v, thus obtaining a new solution u which is then positive in BR1 and BR \ BS , and otherwise vanishes. This solution may be considered as a dead core with a symmetric burst centered at the origin. Of course, the same procedure may be repeated at other suitably chosen origins in BS , giving rise to multiple bursts. Naturally a given ball BS can accomodate only a certain number of bursts, but the larger are R and S the more bursts which can be allowed. Remarks. The existence of a dead core in Theorem 7.2 supplies still another counterexample to the strong maximum principle when (1.7) holds. It is worth pointing out here that this counterexample is in fact a solution of equation (7.3); that is one proves in this way a sharper version of the necessity of condition (1.6) for the strong maximum principle. The results of Theorems 7.2 and 7.3 can be extended to more general quasilinear cases, as anticipated in the Remark at the end of Section 4. See the forthcoming paper [29]. We wish to thank Professor L.A. Peletier for helpful discussions concerning the material of this section. 8. More general quasilinear inequalities Let D be a domain in Rn . Let [aij (x, u)], i, j = 1, · · · , n, be a continuously differentiable, symmetric coefficient matrix defined for x ∈ D, u ≥ 0, and which is positive definite in these variables, namely (8.1)

aij (x, u)ηi ηj > 0,

η ∈ Rn \ {0}.

We shall suppose furthermore that the principal operator A = A(ρ) satisfies the following strengthened versions of (A1), (A2), namely A ∈ C 1 (0, ∞), (A1)0

STRONG MAXIMUM PRINCIPLE

(A2)0

29

Φ0 (ρ) > 0 for ρ > 0, and Φ(ρ) → 0 as ρ → 0. 8.1. The strong maximum principle

Consider the differential inequality (8.2)

Di {aij (x, u)A(|Du|)Dj u} − B(x, u, Du) ≤ 0,

u ≥ 0,

in a domain Ω ⊂ D. We shall treat the following main conditions on the (continuous) function B(x, u, ξ): (B1) B(x, u, ξ) ≤ κΦ(|ξ|) + f (u), (B2) B(x, u, ξ) ≥ −κΦ(|ξ|) + g(u) for x ∈ Ω, u ≥ 0, and all ξ ∈ Rn with |ξ| ≤ 1, where κ > 0 and the nonlinearities f , g obey (F1) and (F2). It is interesting to observe that for the validity of the following results the function B(x, u, ξ) need not be non–decreasing in the variable u! This corresponds to the situation of Theorem 2.2 where the coefficient c(x) is not required to satisfy a sign condition for the validity of the conclusion. (For a statement of the strong maximum principle, see the second paragraph preceding Theorem 1.1.) Theorem 8.1. (Strong maximum principle). Assume (B1). For the strong maximum principle to hold for (8.2) it is sufficient that either f ≡ 0 in [0, µ), µ > 0, or that (1.6) is satisfied. Assume (B2). For the strong maximum principle to hold for (8.2) it is necessary that either g ≡ 0 for u ∈ [0, µ), µ > 0, or that Z δ ds (8.3) =∞ −1 (G(s)) 0 H Ru holds, where G(u) = 0 g(s)ds.

The sufficiency was obtained in Theorem 10 of [30] under the additional technical assumption (2.5) of [30], and in Theorem 3 of [27] without the assumption (2.5) of [30]. In both papers, moreover, the matrix aij was assumed to be independent of the variable u. For other comments on earlier work, see the Introduction and also Section 4 of [30].

Proof. Sufficiency. We follow the proof of Theorem 3 of [27], using however a modified version of the auxiliary function constructed in Proposition 4.1. We first introduce the modified coefficient matrix a ˆij (x) ≡ aij (x, u(x)),

obviously still continuously differentiable in Ω. Let O be an arbitrary origin in Ω. Put ER = {x ∈ Rn : R/2 ≤ |x| ≤ R} where R is supposed sufficiently small that ER is in Ω. Define λ = min eigenvalue of [ˆ aij (x)] in ER = min eigenvalue of [aij (x, u(x))] in ER , Λ = max eigenvalue of [ˆ aij (x)] in ER = max eigenvalue of [aij (x, u(x))] in ER , and let α be a constant such that |ξj Di a ˆij (x)| ≤ α|ξ|

for all x ∈ ER and ξ ∈ Rn . Clearly such a constant α exists since u ∈ C 1 (Ω) and ER is a compact subset of Ω. It is easy to see that xj a ˆij (x) xi xj xj ˆij (x)) = (Di a + δij − 2 , Di a ˆij (x) r r r r so for x ∈ ER , xj n−1 ˆij (x) (8.4) Λ. ≤α+ Di a r r

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Define

(n − 1)Λ + (α + κ)R . λ We can now introduce the radial Hopf–type comparison function v(x) = w(t), t = R − r, r = |x|, where w is the unique solution (see Theorem 4.6) of (4.1), given by Proposition 4.1 when m < δ, q(t) = (R − t)k , T = R/2 and f is replaced by f /λ. Moreover, since Z δ ds =∞ −1 (λ−1 F (s)) 0 H k=

by Lemma 3.2 and (1.6), one sees that Proposition 4.4 applies to the solution w. Thus Dv(x) = −w0 (R − r)x/r 6= 0 in ER . Also, by restricting m to be even smaller if necessary – see Proposition 4.1 – one can maintain (8.5)

0 < |Dv| < 1.

Now we can carry out the following crucial calculation: Di {ˆ aij (x)A(|Dv|)Dj v} − κΦ(|Dv|) − f (v) n xi xj xj o Φ(w0 ) − κΦ(w0 ) − f (w) =a ˆij (x) 2 [Φ(w0 )]0 − Di a ˆij (x) r r xi xj f (w) k (8.6) 0 0 0 ≥a ˆij (x) 2 [Φ(w )] − Φ(w ) − r r λ xi xj f (w) 1 0 0 =a ˆij (x) 2 [q(t)Φ(w )] − =0 r q(t) λ by construction of w, that is aij (x)A(|Dv|)Dj v} − κΦ(|Dv|) − f (v) ≥ 0 Di {ˆ

(8.7) in ER , with

v ≥ 0, 0 < |Dv| < 1; v(R/2) = m, v(R) = 0. We next require a comparison result corresponding to Theorem 5.4, but applying to the more general inequality (8.2). Lemma 8.2. (Comparison lemma). Let u and v be respectively solutions of (8.2) and (8.7) in a bounded domain Ω, and let (B1) be satisfied. Suppose that u and v are continuous in Ω; and that Then u ≥ v in Ω.

0 ≤ v < δ,

0 < |Dv| < 1

in Ω;

u≥v

on ∂Ω.

Proof. By (8.7) we have Di {aij (x, u(x))A(|Dv|)Dj v} − κΦ(|Dv|) − f (v) ≥ 0,

0 ≤ v < δ,

in Ω, while from (8.2) and (B1),

Di {aij (x, u(x))A(|Du|)Dj u} − κΦ(|Du|) − f (u) ≤ 0,

|Dv| < 1,

u ≥ 0,

this being valid of course only when |Du| ≤ 1. In turn, since |Du| + |Dv| ≥ |Dv| > 0, we can apply Theorem 10.1 (together with the remark after Corollary 10.4). In particular, Lemma 8.2 follows from the identifications a = 0, b = 1, and ˆ z, ξ) = κΦ(|ξ|) + f (z), Aˆi (x, ξ) = A(|ξ|)aik (x, u(x))ξk ; B(x, |ξ| ≤ 1 provided we show that the matrix [Dξj Aˆi (x, ξ)] is positive definite for ξ 6= 0. But Dξj Aˆi (x, ξ) = aik (x, u(x))bkj (ξ),

where bkj (ξ) = A(|ξ|)δkj +

A0 (|ξ|) ξk ξj , |ξ|

ξ 6= 0.

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31

The matrix [bkj (ξ)] has eigenvalues A(|ξ|) (repeated n−1 times) and Φ0 (|ξ|). By assumption (A2)0 we have Φ0 (|ξ|) > 0 for ξ 6= 0, while also A(|ξ|) = Φ(|ξ|)/|ξ| > 0,

(A2)0 .

6 0, for ξ =

again by Hence [bij ] is positive definite for ξ 6= 0. Because [aij (x, u)] is assumed positive definite, it now follows that [Dξj Aˆi (x, ξ)] is positive definite for x ∈ Ω and ξ 6= 0, completing the proof. The point of Lemma 8.2 is that if |Dv| > 0 in Ω, then just as for Theorem 5.4 it is not necessary to have ellipticity at the value ξ = 0. We remark that it is exactly in the application of this lemma that the strengthened condition (A2)0 is needed. The rest of the proof of sufficiency in Theorem 8.1 is now essentially the same as the sufficiency part of Theorem 1.1. The main change is that at the last step we rely on Lemma 8.2 instead of Theorem 5.4. Necessity. This follows the corresponding arguments in Theorem 1.1. It is necessary to exhibit, for each x0 in D, a domain Ω in D with x0 in Ω, and a solution v of (8.2) in Ω such that v(x0 ) = 0 but v 6≡ 0 in Ω. The assumption to be made for this purpose is that (B2) holds, with g(u) > 0 for u > 0, together with the negation of (8.3), namely Z δ ds (8.8) < ∞. −1 (G(s)) 0 H Choose R < 1 so small that the closure of the domain Ω = BR (x0 ) is in D. As at the beginning of the proof, let λ = min eigenvalue of [aij (x, z)] in Ω,

Λ = max eigenvalue of [aij (x, z)] in Ω

for all values 0 ≤ z ≤ δ. Also let α be such that Rn ,

|ξj Di aij (x, u(x))| ≤ α|ξ|

0 ≤ u(x) ≤ δ, |Du| ≤ b = H −1 (G(δ)). As before, clearly such a value when x ∈ Ω, ξ ∈ a can be found. Finally, define σ = (nΛ + α + κ)−1 , where κ is given by (B2). Consider the dead core function v(r) = w(r − S), S ≤ r ≤ R, r = |x − x0 |, given in Theorem 7.2 (and using the notation there), but constructed with the function f replaced instead by g and with the new value of σ given above. Clearly v can be extended as a C 1 function to all of Ω by putting v ≡ 0 for 0 ≤ r < S. Then we find, see (7.4), Di aij (x, v(x))A(|Dv|)Dj v − B(x, v(x), Dv(x)) ≤ Di aij (x, v(x))A(|Dv|)Dj v + κΦ(|v 0 |) − g(v) by (B2) xi xj n−1 Φ(|v 0 |) − g(v) ≤ aij (x, v(x)) 2 [Φ(|v 0 |)]0 + α + κ + Λ (8.9) r r n−1 ≤ Λσg(v) + α + κ + Λ Cσg(v) − g(v) S ≤ [σ(nΛ + α + κ) − 1]g(v) = 0;

in obtaining (8.9), note first that when r = |x − x0 | < S there is nothing to show since v ≡ 0; on the other hand, for r ≥ S we apply the estimates of Lemma 7.1 in the same way as in previous proofs, together with the relations 0 < C < R ≤ 1 and 0 < C ≤ S; see the proof of Theorem 7.2. Since v has the dead core BS (x0 ), and is otherwise positive in Ω = BR , the proof is complete.

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Corollary 8.3. Assume that both (B1) and (B2) are satisfied, and that there exists c > 0 such that g(u) ≥ cf (u) for u ∈ [0, δ]. Then the strong maximum principle holds for (8.2) if and only if either f ≡ 0 in [0, µ], µ > 0, or (1.6) holds. We close the section with the following useful boundary point lemma, which will be required for the proof of Theorem 8.5 below. Corollary 8.4. (Boundary point lemma). Let x0 ∈ ∂Ω and suppose that Ω satisfies an interior sphere condition at x0 . Let u be a C 1 solution of (8.2) in Ω, with u > 0 in Ω and u = 0 at x0 . Assume that (B1) holds and that either f ≡ 0 in [0, µ), µ > 0, or that (1.6) is satisfied. Then ∂u/∂n < 0 at x0 , where n is the outer normal to ∂Ω at x0 . Proof. By the interior sphere condition there exist y ∈ Ω and R > 0 such that the open ball BR (y) ⊂ Ω and x0 ∈ ∂B. Let v be the solution of (8.7) given in Theorem 8.1 and put u ˜(x) = v(|x − y|). Then as from Lemma 8.2 it follows that u(x) ≥ u ˜(x) in BR (y) \ BR/2 (y) provided that m > 0 is sufficiently small. This completes the proof, since ∂ u ˜/∂n = v 0 (R) < 0. 8.2. The compact support principle There is a corresponding compact support principle for the reversed inequality (8.10)

Di {aij (x, u(x))A(|Du|)Dj u} − B(x, u, Du) ≥ 0,

u ≥ 0,

x ∈ Ω,

where Ω is unbounded, with ΩR = {x ∈ Rn : |x| > R} ⊂ Ω ⊂ D for some R > 0. (For the statement of the compact support principle, see the first paragraph before Theorem 1.2 in the Introduction.) The conditions on the matrix aij (x, u) now however must be somewhat strengthened since the compact support principle deals with neighborhoods of ∞. Specifically, we shall require that, for x ∈ Ω and 0 ≤ u < δ, (8.11)

λ|η|2 ≤ aij (x, u)ηi ηj ≤ Λ|η|2

for some positive constants λ, Λ. Moreover, for x ∈ Ω, and for functions u = u(x) such that 0 ≤ u(x) < δ and |Du(x)| ≤ b for some b, b ≥ 1 say, we assume that (8.12)

kDi aij (x, u(x))k ≤ α

for a constant α ≥ 0. Finally we shall suppose for the rest of the section that any solution u of (8.10) under consideration is such that |Du(x)| ≤ b in ΩR for some R > 0. (This condition can be dropped if the coefficient matrix [aij ] is independent of u. Of course, it is to be expected that solutions u(x) which approach 0 as |x| → ∞ will satisfy this condition for some domain ΩR and constant b, but this would certainly require further regularity assumptions on the equation.) Theorem 8.5. (Compact support principle). For the compact support principle to hold for (8.10) it is sufficient that (B2) is satisfied with g(u) > 0 for u > 0, and Z δ ds < ∞. (8.13) −1 (G(s)) 0 H On the other hand, if (B1) is satisfied with f (u) > 0 for u > 0, then for the compact support principle to hold for (8.10) it is necessary that (1.7) is satisfied.

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33

Proof. We first prove necessity. Here it will enough to show the existence of a radial solution v = v(r) of the problem in ΩR ( Di {aij (x, v(x))A(|Dv|)Dj v} − B(x, v, Dv) ≥ 0, in ΩR , (8.14) v(R) = m, v(r) → 0 as r → ∞; v > 0, v 0 < 0 in ΩR , where (B1) holds with f (u) > 0 for u > 0 and also, by negation, condition (1.6) is satisfied. To this end, as shown in (8.6) it is enough to consider the equation f (v) n−1 1 0 0 0 ≤ v < δ, −1 ≤ v 0 < 0, [Φ(v )] + α+κ+ Λ Φ(v 0 ) − = 0, λ r λ where λ and α are given by (8.11) and (8.12), respectively. That is, the problem becomes q (r)Φ(v 0 )]0 − q˜(r)f˜(v) = 0, in [R, ∞), [˜ (8.15) v(R) = m, v(r) → 0 as r → ∞, v > 0, −1 < v 0 < 0 in ΩR , where 0 = d/dr and q˜, f˜ are given by

−1 Λ

q˜(r) = r(n−1)λ

−1 r

e(a+κ)λ

,

f˜(v) = f (v)/λ.

Of course, f˜(v) continues to obey (1.7), by Lemma 3.2. The required solution can now be constructed (for suitably small m) exactly as in the proof of Theorem 5.1, with only the change that q(r) = rn−1 is replaced by the new function q˜(r), and f (v) by f˜(v). Note here, in particular, that (n−1)Λ/λ r q˜(r) −(α+κ)/λ =e , q˜(r + 1) r+1 which approaches the positive limit e−(a+κ)Λ/λ as r → ∞, cf. the corresponding calculation (5.7). This completes the proof of necessity. The proof of sufficiency is also somewhat delicate. Here the basic method is taken from Theorem 20 of [30], with some modifications to avoid applying the superfluous technical assumption (2.5) of [30]. We first construct an appropriate radial comparison function v = v(r). Fix σ ∈ (0, 1) by σ = (Λ + α + κ)−1 .

We take C < min{1, Cσ } and v(r) = w(R + C − r),

R ≤ r ≤ R + C,

where w is the function given in Lemma 7.1, corresponding to the given values of σ and C, and of course with f (u) replaces by g(u). Obviously v(R) = w(C) = γ (< δ) and v(R + C) = v 0 (R + C) = 0. We can thus suppose that v is extended to all r ≥ R by taking v(r) ≡ 0 for r > R + C. To check that v has the required property of an upper comparison function, we have with the help of Lemma 7.1 (and recalling that v 0 ≤ 0), Di aij (x, u(x))A(|Dv|)Dj v + κΦ(|v 0 |) − g(v) xi xj n−1 0 0 ≤ −aij (x, u(x)) 2 [Φ(|v |)] + α + κ − Λ Φ(|v 0 |) − g(v) r r xi xj (since C ≤ 1) ≤ aij (x, u(x)) 2 σg(v) + (α + κ)σg(v) − g(v) r ≤ [σ(Λ + α + κ) − 1]g(v) = 0; the steps in this calculation are essentially the same as those previously used to derive (8.9).

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In summary, we have Di aij (x, u(x))A(|Dv|)Dj v + κΦ(|v 0 |) − g(v) ≤ 0 (8.16)

in ΩR . Of course v ≡ 0 for |x| ≥ R1 = R + C, while v > 0 for R ≤ |x| < R1 , and v(R) = γ. It can also be observed that γ = δ if Cσ ≤ 1 but γ < δ otherwise. Now consider a solution u of the inequality (8.10) in an exterior domain Ω such that u(x) → 0 as |x| → ∞. Under the condition (B2) it is required to show that u has compact support in Ω. We can choose R0 > R so large that u(x) < γ in the set Ω0 = {|x| ≥ R0 }. Then, to simplify the notation one may consider the domain Ω0 to be the given domain Ω. It is now enough to show that u ≤ v as in the proof of Theorem 1.2, where v is the comparison function above, satisfying (8.16). For this purpose it is not possible to resort directly to Lemma 8.2, since Dv ≡ 0 for large |x|, while Du is unrestricted as to its null set. Accordingly we use an indirect argument. Define z = v − u in Ω. Clearly |z| ≤ γ. We claim that z ≥ 0. If this is not the case, then ε = − inf z < 0, Ω

0 < ε ≤ γ,

and we shall reach a contradiction. Note first that z = γ − u > 0 when |x| = R, and that z(x) → 0 as |x| → ∞; hence the infimum of z must be attained at some (interior) point x0 in Ω. Define ˆ = {R < |x| < R1 }, Ω Ω1 = {|x| > R1 }. ˆ ∪ ∂Ω1 ∪ Ω1 , so exactly the following three cases can occur: Then Ω = Ω

(1) The infimum of z is attained in Ω1 . (2) The infimum of z is not attained in Ω1 , but is reached at a point on ∂Ω1 . ˆ (3) The infimum of z is not attained in Ω1 , but is reached in Ω. In Case 1, let the infimum be attained at x0 in Ω1 . For x in Ω1 , define u(x) = −u(x) + ε. Then since v ≡ 0 in Ω1 , we see that u ≡ z + ε ≥ 0 has a zero minimum at x0 . Moreover, u(x) is such that 0 ≤ u ≡ −u + ε ≤ ε, while also by (8.10) Di {aij (x, −u + ε)A(|Du|)Dj u} + B(x, −u + ε, −Du) ≤ 0,

u ≥ 0,

in Ω0 . Subtracting the expression g(−u + ε) from both sides of the previous line, then gives (8.17)

˜ u, Du) ≤ −g(−u + ε) Di {aij (x, −u + ε)A(|Du|)Dj u} − B(x,

where ˜ u, ξ) ≡ −B(x, −u + ε, −ξ) + g(−u + ε) ≤ κΦ(|ξ|), B(x, ˜ u, ξ) satisfies (B1) with using the given condition (B2) at the second step. That is, B(x, f ≡ 0. Using the fact that g(u) ≥ 0 for 0 ≤ u ≤ γ < δ, we see that g(−u + ε) ≥ 0, so that finally from (8.17) there follows ˜ u, Du) ≤ 0 Di {aij (x, −u + ε)A(|Du|)Dj u} − B(x, in Ω0 (and hence in Ω1 ). Hence by the strong maximum principle (Theorem 8.1) applied to the domain Ω1 we obtain u ≡ 0. Thus u ≡ ε > 0 in Ω1 , which is impossible since u(x) → 0 as |x| → ∞. That is, Case 1 cannot occur. In Case 2, let the infimum of z be reached at x0 on ∂Ω1 . In this case, obviously u > 0 in Ω1 while u = 0 at x0 (we can of course consider u as a C 1 function on Ω1 ). Then, since Ω1 clearly satisfies an interior sphere condition at x0 , the boundary lemma (Corollary 8.4) gives ∂u/∂n < 0 at x0 . But this is also impossible, because Du ≡ Dz = 0 at x0 . In Case 3, necessarily v − u = z > −ε on the boundary of Ω1 , while as noted earlier ˆ while of v − u > 0 when |x| = R0 . Thus v − u ≥ −a, a ∈ [0, ε), on the boundary of Ω, ˆ This corresponds in essence to Lemma 8.2 for Ω = Ω, ˆ with course u < δ and Dv 6= 0 in Ω. the roles of u and v interchanged. We can thus apply Theorem 8.1, of course for the case

STRONG MAXIMUM PRINCIPLE

35

ˆ But this contradicts M = −a ≤ 0, the conclusion being that v − u ≥ M = −a > −ε in Ω. ˆ the condition of Case 3 that z = v − u attains its infimum −ε in Ω. We have thus shown that all three cases lead to a contradiction. Consequently z ≥ 0 in Ω, that is v ≥ u. In turn, u ≡ 0 for |x| > R1 , which completes the proof of the theorem. Corollary 8.6. Assume that both (B1) and (B2) are satisfied and that there exists c > 0 such that g(u) ≥ cf (u) > 0 for u > 0. Then the compact support principle holds for (8.10) if and only if (1.7) holds. We close the section with a counterexample showing the importance of the lower bound conditions (B1) and (B2). Consider the inequality (8.18)

∆p u + |Du|q1 − uq2 ≥ 0,

p > 1, q1 , q2 > 0.

Clearly, conditions (8.13), (B1) and (B2) are satisfied if and only if q1 ≥ p − 1 and q2 < p − 1. The compact support principle then holds for (8.18). On the other hand, for any q1 ∈ (0, p − 1) we can take q1 < q2 < p − 1. One easily checks that (8.18) then has positive solutions u(x) = const. |x|−l on ΩR = {x ∈ Rn : |x| > R} for l and R large. Hence the compact support principle fails even though condition (1.6), or equally (8.13), is fulfilled! 9. Riemannian weighted norms Let M be an n-dimensional Riemannian manifold of class C 1 , with contravariant metric tensor [g ij ] continuous in local coordinates x = (x1 , . . . , xn ). Let u be a real–valued C 1 function defined on some open connected submanifold Ω of M. The Riemannian norm of the gradient vector ∇u on Ω is then defined as the non–negative continuous function on Ω given in local coordinates by q ∂u |∇u|g = g ij Di uDj u, . Di u = ∂xi Consider the variational integral Z I[u] = {G(|∇u|g ) + F (u)}dM. Ω

The corresponding Euler–Lagrange equation is then (9.1)

divg {A(|∇u|g )∇u} − f (u) = 0,

where divg is the Riemannian divergence operator and A(ρ) = G 0 (ρ)/ρ, ρ > 0, as in the introduction, see (1.5). More explicitly, in local coordinates x = (x1 , . . . , xn ) in Ω, one √ has dM = gdx, where g = 1/det[g ij ]. Then a direct calculation of the Euler–Lagrange equation yields p 1 p (9.2) g(x)g ij (x)A(|∇u|g )Dj u − f (u) = 0, Di g(x)

that is, exactly (9.1). When A ≡ 1 the differential operator in (9.2) reduces just to the manifold Laplacian, see [43], page 232. A specific example is given by the variational integral Z 1 √ p |∇u|g + F (u) dM, where dM = gdx on Ω, p > 1, Ω p √ introduced by Mossino ([24], page 40), though without the volume factor g. Here of course A(ρ) = ρp−2 , p > 1. Other examples are given also in [25]. Obviously (9.2) is the special case of (1.13) when p p aij (x, u) = g(x)g ij (x), B(x, u, ξ) = g(x)f (u).

With this motivation in hand, we turn to the strong maximum principle for (1.13). As at the beginning of the section, we assume that (A1)0 and (A2)0 are valid, and additionally

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P. PUCCI AND J. SERRIN

that the tensors [g ij ] = [g ij (x, u)] and [aij ] = [aij (x, u)] are continuously differentiable, symmetric and positive definite in Ω × R+ 0 . In the context of (1.13) the domain Ω is now of course simply a connected open subset of Rn . The inequality (1.13) is more difficult to treat than (8.2), in that there are two different sets of hypotheses under which the strong maximum principle can be obtained. In the first, some mild conditions on the operator A = A(ρ) are required, satisfied in particular by both the p–Laplacian operator and the mean curvature operator. In the second case, a modification of condition (B1) is needed, together with stronger conditions on the metric tensor [g ij ]. It is convenient to consider the two cases separately. First, we introduce the additional structure hypotheses: (A3) (i) |A0 (ρ)|ρ2 ≤ c Φ(ρ) for some constant c ≥ 0 and for all ρ ∈ (0, 1], and (ii) for all σ0 ∈ (0, 1) there exists a value ν = ν(σ0 ) such that Φ0 (ρ) ≤ νΦ0 (σρ)

for all σ ∈ (σ0 , 1] and ρ ∈ (0, 1). Note that if Φ is concave, condition (A3)–(ii) is always satisfied with ν = 1; this is the case for example for the p–Laplacian operator when 1 < p ≤ 2, and for the mean curvature operator. On the other hand, for Φ(ρ) = ρp−1 , p > 2, we get ν(σ0 ) = σ02−p . It follows that (A3)–(ii) is satisfied for the p–Laplacian with ν = max{1, σ02−p }. Also (A3)–(i) is satisfied for the p–Laplacian with c = |p − 2| and for the mean curvature operator with c = 1, etc. Theorem 9.1. Let conditions (A1)0 , (A2)0 , (A3) and (B1) hold. Then the strong maximum principle is valid for inequality (1.13) provided that f (s) ≡ 0 for s ∈ [0, µ), µ > 0, or f (s) > 0 for s ∈ (0, δ) and (1.6) is satisfied.

Theorem 9.2. Let conditions (A1)0 and (A2)0 hold, let g ij = g ij (x) be independent of u and of class C 2 (Ω), and assume (B1) applies with Φ(|ξ|) replaced by Φ(|ξ|g ). Then the strong maximum principle is valid for inequality (1.13) provided that f (s) ≡ 0 for s ∈ [0, µ), µ > 0, or f (s) > 0 for s ∈ (0, δ) and (1.6) is satisfied.

Proof of Theorem 9.1. This closely follows the proof of Theorem 8.1, though with an additional term appearing in (8.7) due to the presence of the metric [g ij ], and with a slight (but not trivial) difference in the definition of the comparison function v = v(r). To begin with, we define the positive definite matrix gˆij (x) = g ij (x, u(x)), this of course being of class C 1 in the annular domain ER , see the proof of Theorem 8.1. Let θ2 and Θ2 be respectively the least and greatest eigenvalues of the positive definite matrix [ˆ g ij ] in ER , and write q ` = `(x) = |Dr|gˆ =

gˆij (x)xi xj /r2 .

Then in ER , (9.3)

θ ≤ ` ≤ Θ,

(Θ ≥ 1 without loss of generality).

Following the proof of Theorem 8.1, the estimate (8.5) continues to hold, and similarly, after a short calculation, (9.4)

|ξk Dk `| ≤ β|ξ|/`r

for some constant β ≥ 0, with β = 0 if g ij = δij . Finally it is convenient to define ν¯ = ν(θ/Θ), where ν is the function given in (A3)–(ii). Now let v(x) = w(t), t = (R − r)/Θ, r = |x|, where w is the unique solution of (4.1) given by Proposition 4.1 when q(t) = (R − Θt)k , T = R/2Θ, and f is replaced by (¯ ν Θ2 /λ)f . The constant k will be determined later.

STRONG MAXIMUM PRINCIPLE

37

Of course, Proposition 4.4 applies to the solution w in view of Lemma 3.2 and (1.6). Therefore Dv(x) = −w0 x/Θr 6= 0. Also, by restricting the boundary value w = m at T = R/2Θ to be sufficiently small, one can maintain kw0 k∞ ≤ 1 and so (9.5)

0 < |Dv| ≤ 1

in ER .

We can now turn to the important, but unfortunately somewhat complicated, calculation, applying for x ∈ ER , Di {ˆ aij (x)A(|Dv|gˆ)Dj v} − κΦ(|Dv|gˆ) − f (v) xj xi 1 − A0 (`w0 /Θ)|w0 |2 Di ` ˆij (x) Φ0 (`w0 /Θ)w00 · = 2a Θ r r n xj o 1 ˆij (x) A(`w0 /Θ)w0 − κΦ(`w0 /Θ) − f (v) − Di a Θ r xi xj cβ 1 αR + (n − 1)Λ ˆij (x) 2 Φ0 (`w0 /Θ)w00 − 3 Φ(`w0 /Θ) − ≥ 2a Φ(`w0 /Θ) Θ r ` r `r − κΦ(`w0 /Θ) − f (w) (by (A3)–(i), (8.4) and (9.3)) λ 0 0 00 1 cβ αR + (n − 1)Λ ≥ + Φ (w )w − + Rκ Φ(w0 ) − f (w) ν¯Θ2 r θ3 θ (by (9.3), (A3)–(ii) and Φ0 > 0) k¯ ν¯Θ2 λ 0 0 ¯ = [Φ(w )] − − f (w) (defining k) ν¯Θ2 r λ ν¯Θ2 1 λ 0 0 = [q(t)Φ(w )] − f (w) = 0, ν¯Θ2 q(t) λ 6 ¯ where we take k = k/Θ. The rest of the proof is essentially the same as for Theorem 8.1, with the single exception that now the matrix bkj (ξ) = bkj (x, ξ) in the proof of the analogue of Lemma 8.2 is given by A0 (`|ξ|) bkj (ξ) = A(`|ξ|)δij + ` ξk ξj . |ξ| The eigenvalues of [bkj ] are A(`|ξ|) and Φ0 (`|ξ|) so from (9.3) it is evident that [bkj ] is positive definite for ξ 6= 0 and all x ∈ ER .

Proof of Theorem 9.2. The idea of the proof is to replace the ball BR tangent to the support of u by a small geodesic ball {x ∈ Ω : s(x) ≤ S} centered at x0 and tangent to the singular set where u = 0, Du = 0; here s(x) denotes the geodesic distance (with respect to the metric induced by the matrix [g ij ]) from the given center x0 to nearby points x ∈ Ω. The existence of such a tangent ball can be shown exactly as in Hopf’s original proof, at least provided that |Ds| is equally bounded above and bounded away from zero. To show this fact, we observe by Gauss’ lemma (see [43], page 235) that (9.6)

|Ds(x)|g2 = g ij (x)Di s(x)Dj s(x) = 1,

6 x0 . x=

Thus, recalling that θ2 and Θ2 are the least and greatest eigenvalues of [g ij ], we get Θ−1 ≤ |Ds| ≤ θ−1 ,

as required. We can now proceed as in the proof of Theorem 9.1, with ER replaced by the geodesic annular set GS = {x ∈ Ω : S/2 ≤ s(x) ≤ S} and with v(x) = w(t),

0

t = S − s,

Dv = −w Ds,

T = S/2,

|Dv|g = w0

6If g ij = δ , then ` = 1, β = 0, θ = Θ = 1, ν ¯ = 1 and the calculation reduces exactly to (8.6), without ij the intervention of condition (A3).

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by (9.6). The principal calculation, for x ∈ GS , is the following: Di {ˆ aij (x)A(|Dv|g )Dj v} − κΦ(|Dv|g ) − f (v)

= −Di {ˆ aij (x)Dj sA(w0 )w0 } − κΦ(w0 ) − f (w)

(9.7)

=a ˆij (x)Di sDj s[Φ(w0 )]0 − Di (ˆ aij (x)Di s)Φ(w0 ) − κΦ(w0 ) − f (w) α λ + ΛkD2 sk Φ(w0 ) − κΦ(w0 ) − f (w) ≥ 2 [Φ0 (w0 )]0 − Θ θ ¯ k λ ≥ 2 [Φ(w0 )]0 − Φ(w0 ) − f (w), Θ s

where k¯ is an appropriate constant. That such a constant exists depends on knowing that s ∈ C 2 (GS ), which is a consequence of the assumption that g ij is of class C 2 , see [43], Appendix II.1, and [33]. [Here it is essential to have g ij independent of u, for otherwise the constructed matrix [ˆ g ij ] would be only of class C 1 , however smooth the metric might be; thus in turn the corresponding geodesic distance sˆ(x) would be only of class C 1 away from x0 . Of course, due to the singularity at the center x0 the gradient Ds naturally is not continuous at x0 , while D2 s is unbounded of order 1/s as x approaches x0 (always assuming that g ij is of class C 2 ). These comments are reflected in the trivial Rn calculation that Dr = x/r is not continuous at the singularity x0 = 0, though it is bounded, and that the Hessian matrix D2 r = r−1 [δij − xi xj /r2 ]ij . The existence of the constant k¯ being shown, one can choose w = w(t) so that the right side of (9.7) vanishes, and the rest of the proof follows as before. The fact that Φ(|ξ|g ) replaces Φ(|ξ|) in condition (B1) causes no difficulty in the application of Theorems 8.1 and 10.1, since for |ξ| ≤ |η| there results Φ(|η|g ) − Φ(|ξ|g ) ≤ (Θ2 /θ)Φ0 (Θ|η|) · |η − ξ|,

that is Φ(|ξ|g ), as well as Φ(|ξ|), is Lipschitz continuous in ξ.

The strong maximum principle for the Riemannian equation (9.1), or for the corresponding inequality (9.8)

divg {A(|∇u|g )∇u} − f (u) ≤ 0

in Ω,

can be treated more simply than for the case of inequality (1.13), and under slightly lighter hypotheses. The result is as follows. Theorem 9.3. Let conditions (A1), (A2) and (F2) hold. Assume that the Riemannian manifold M is of class C 2 . Then the strong maximum principle is valid for inequality (9.8) provided that f (s) ≡ 0 for s ∈ [0, µ), µ > 0, or f (s) > 0 for s ∈ (0, δ) and (1.6) is satisfied. Proof. We begin as in the proof of Theorem 9.2, with the exception that (9.7) now becomes more simply, for x ∈ GS , p 1 p Di { g(x)g ij (x)A(|Dv|g )Dj v} − f (v) g(x) p 1 = −p Di { g(x)g ij (x)Dj s A(w0 )w0 } − f (w) (9.9) g(x) k¯ = [Φ(w0 )]0 − ∆s Φ(w0 ) − f (w) ≥ [Φ(w0 )]0 − Φ(w0 ) − f (w). s The remaining part of the proof involves the weak comparison theorem. In the present case this can be done with the help of Theorem 10.5 rather thanp the more difficult Theorem 10.1. ˆ ξ) = g(x)g ij (x)A(|ξ|g )ξ, that is, in To this end, we have to check (10.10) when A(x, Riemannian notation, p p g(x)hA(|η|g )η − A(|ξ|g )ξ, η − ξiM ≥ g(x) Φ(|η|g ) − Φ(|ξ|g ) · |η|g − |ξ|g

STRONG MAXIMUM PRINCIPLE

39

since hξ, ηiM ≤ |ξ|g |η|g , and (10.10) now follows because Φ is strictly increasing by (A2). In [25] a version of the strong maximum principle at infinity, the so–called Omori–Yau principle, has recently been given for singular elliptic inequalities including the p–Laplacian case as well as the mean curvature operator, and for smooth, connected, non–compact, complete Riemannian manifolds M. 10. Comparison and uniqueness theorems for singular divergence form operators 10.1. Comparison results Throughout the section we consider the pair of differential inequalities ˆ u, Du) ≤ 0, ˆ u, Du)} − B(x, div{A(x, (10.1) u ≥ 0, (10.2)

ˆ v, Dv) ≥ 0, ˆ v, Dv)} − B(x, div{A(x,

in a bounded domain Ω ⊂ Rn . Let the vector function ˆ z, ξ) : Ω × R × Rn → Rn A(x,

v ≥ 0,

be continuous in Ω × R × Rn and continuously differentiable with respect to z and ξ for all z and for ξ 6= 0. Also let ˆ z, ξ) : Ω × R × Rn → R B(x, be continuous in Ω × R × Rn and continuously differentiable with respect to ξ for |ξ| > 0 ˆ is elliptic in the sense that the in Rn . Suppose moreover throughout the section that A i ˆ matrix [Dξj A (x, z, ξ)] is positive definite for x ∈ Ω and ξ 6= 0 in Rn . Finally assume that ˆ z, ξ) is non–decreasing in the variable z for x ∈ Ω and |ξ| ≤ b. B(x, Then the following comparison principle holds. Theorem 10.1. (Comparison principle). Let u and v be respective solutions of (10.1) and (10.2) in Ω. Suppose that u and v are continuous in Ω, with |Du| + |Dv| > 0 in Ω, and either |Du| < b or |Dv| < b. Assume finally u ≥ v on ∂Ω. ˆ is independent of the variable z, then u ≥ v in Ω. If A More generally if the boundary condition is relaxed to u ≥ v − M on ∂Ω, where M is constant, then u ≥ v − M in Ω. ˆ and This is essentially Theorem 10.7 (i) of [18] with the exception that the functions A ˆ B are allowed to be singular at ξ = 0, this being compensated by the additional condition ˆ here, rather than A, B as in [18], in order to ˆ B |Du| + |Dv| > 0 in Ω. We have written A, avoid confusion with earlier notation in the paper. If Ω is unbounded, the boundary condition is understood to include the limit relation lim inf {u(x) − v(x)} ≥ −M

as |x| → ∞.

Before giving the proof it is convenient to state the following ˆ be a compact subset of Ω, and ξ, η vectors in Rn satisfying Lemma 10.2. Let Ω |ξ|, |η| ≤ b,

|tξ + (1 − t)η| ≥ d

for some positive constants b and d, with d ≤ b, and for all t ∈ (0, 1). Also suppose |z| ≤ `. ˆ such that Then there exist constants ν, ν ∗ depending only on b, d, ` and Ω ˆ z, η)} · (ξ − η) ≥ ν|ξ − η|2 ˆ z, ξ) − A(x, (10.3) {A(x, and

(10.4)

ˆ z, ξ) − B(x, ˆ z, η)| ≤ ν ∗ |ξ − η|. |B(x,

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Proof. By the integral mean value theorem, Z 1 ˆ z, ξ) − A(x, ˆ z, η) = ˆ z, tξ + (1 − t)η)(ξj − ηj )dt. A(x, Dξj A(x, 0

ˆ |z| ≤ ` and d ≤ |ζ| ≤ But the matrix [Dj Aˆi (x, z, ζ)] is uniformly positive definite for x in Ω, b, and the first conclusion then follows at once. Similarly Z 1 ˆ ˆ ˆ z, tξ + (1 − t)η)(ξj − ηj )dt. B(x, z, ξ) − B(x, z, η) = Dξj B(x, 0

ˆ |z| ≤ ` and d ≤ |ζ| ≤ b, and the second ˆ z, ζ) is uniformly bounded for x in Ω, Here Dξj B(x, inequality is proved. It may be remarked that in the special case of the p–Laplacian operator, that is, when ˆ A(ξ) = |ξ|p−2 ξ, we can take ν = dp−2 when p ≥ 2, and ν ∗ = (p − 1)bp−2 when p < 2.

Proof of Theorem 10.1. It is enough to treat M = 0, since the case for arbitrary values of M reduces to M = 0 by the substitution v = v − M . Now suppose for contradiction that the conclusion is false. Put w(x) = u(x) − v(x), whence ε = − inf w(x) > 0. x∈Ω

Then for ε ∈ (ε/2, ε) the function

wε = min{w + ε, 0}

is non–vanishing exactly in the set Σ = Σε = {x ∈ Ω : wε (x) < 0}.

Since w + ε > 0 on ∂Ω it is evident that Σ is pre–compact in Ω. We assert that if ε is suitably close to ε then (10.5)

|tDu + (1 − t)Dv| ≥ d,

|Du|, |Dv| ≤ b,

in Σ, where d > 0 is a constant (independent of ε) such that |Du| + |Dv| ≥ 4d in the pre–compact set Σa . To see this, observe first that Du − Dv = Dw = 0 on the closed subset E = {x ∈ Ω : w(x) = −ε} of Σ. Moreover, distance(E, ∂Σ) → 0 as ε → ε. Hence by continuity, |Du − Dv| < d in Σ provided ε ( > ε/2) is suitably near ε. In particular, for such values of ε we find (since surely max{|Du|, |Dv|} ≥ 2d in Σ) |tDu + (1 − t)Dv| ≥ max{|Du|, |Dv|} − |Du − Dv| ≥ d

in Σ,

which is the first part of (10.5). For the second part, consider (without loss of generality) the case where |Dv| < b in Ω. Define b = supx∈Σε/2 |Dv(x)|. Then b < b, and if we choose ε even nearer to ε, if necessary,

then also |Du − Dv| < b − b in Σ. But then |Du| ≤ |Dv| + |Du − Dv| ≤ b in Σ, as required. Continuing now as in the proof of Theorem 5.4, and using the non–positive test function wε , we have Z Z ˆ v, Dv) − B(x, ˆ u, Du)}wε ˆ ˆ {A(x, Du) − A(x, Dv)}Dwε ≤ {B(x, Σ Ω Z (10.6) ˆ u, Dv) − B(x, ˆ u, Du)}wε , ≤ {B(x, Σ

where in the last step of (10.6) we have used the facts that wε ≤ 0 and u ≤ v in Σ, and that ˆ is non–decreasing in the variable z. Then, with the help of Lemma 10.2, (10.6) implies B that Z Z 2 ∗ ν |Dwε | ≤ ν (10.7) |Dwε | · |wε |. Σ

Σ

STRONG MAXIMUM PRINCIPLE

41

Let Γ = Γε = {ε − ε < wε < 0}. Then Dwε = 0 on Σ \ Γ = E, so the integrals in (10.7) can equally be taken over the set Γ. Applying the Cauchy–Schwarz inequality to the right side of (10.7) yields Z Z ∗ 2 2 (ν /ν) |wε | ≥ (10.8) |Dwε |2 . Γ

Γ

From Poincar´e’s inequality (cf. (7.44) on page 164 of [18]) we obtain

ωn−1 |Γ|1/n ||Dwε ||Γ,2 = ωn−1 |Γ|1/n ||Dwε ||Σ,2 ≥ ||wε ||Σ,2 ≥ ||wε ||Γ,2 . Hence by (10.8) there results |Γ| ≥ ωn (ν/ν ∗ )n .

(10.9)

On the other hand, Γ → ∅ as ε → ε, a contradiction to (10.9). This completes the proof. In the following two theorems, the stated conditions on Du and Dv in Theorem 10.1 are removed. Essentially similar results were given earlier by Damascelli [9]; see also [10]. ˆ is independent of u, and that Theorem 10.3. (Comparison principle). Suppose that A i ˆ the matrix [∂ A /∂ξj ] is uniformly positive definite when 0 < |ξ| ≤ Const., u is bounded ˆ is uniformly Lipschitz and x is in any compact subset of Ω. Assume additionally that B continuous with respect to ξ on compact subsets of its variables and is non–decreasing in the variable u. If u ≥ v − M on ∂Ω, where M is constant, then u ≥ v − M in Ω. To prove Theorem 10.3 it is enough to observe that the conclusions of Lemma 10.2 hold without the restriction |tξ + (1 − t)η| ≥ d. In fact if ξ = η = 0 then (10.3) and (10.4) are trivially true, while otherwise certainly |tξ + (1 − t)η| > 0, in which case the conclusions follows from the hypothesis of uniformly positive definiteness and the Lipschitz continuity ˆ of B. This being shown, the proof of Theorem 10.1 then carries over unchanged, without the intervention of (10.5). The special case of the p–Laplacian operator is of particular importance. This is given in the following Corollary 10.4. Consider the inequalities ˆ u, Du) ≤ 0 ∆p u − B(x, ˆ v, Dv) ≥ 0 ∆p v − B(x,

in Ω, in Ω,

ˆ = B(x, ˆ z, ξ) is uniformly Lipschitz continuous in ξ on compact subsets where p ≤ 2, and B of its variables (and of course non–decreasing in the variable z). If u ≥ v − M on ∂Ω, where M is constant, then u ≥ v − M in Ω. Remark. If in Theorems 10.1 and 10.3 one adds the hypothesis that u ≥ 0, v < δ, then ˆ is needed only in the interval 0 ≤ z < δ − M ; see the proof of the monotonicity of B Theorem 5.4. Theorem 10.5. (Comparison principle). Let u and v be respective solutions of (10.1) ˆ is independent of z and and (10.2) in Ω. Suppose that u and v are continuous in Ω, that A ˆ is independent of ξ. Assume moreover that A ˆ is monotone in the variable ξ (but not B necessarily differentiable), i.e. (10.10)

ˆ ξ) − A(x, ˆ η)} · (ξ − η) > 0, {A(x,

If u ≥ v on ∂Ω, then u ≥ v in Ω.

when ξ 6= η.

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This follows at once from (10.10), exactly as in the proof of Theorem 5.4. Strong comparison theorems, under alternative hypotheses, have been obtained by Tolksdorf [38] and by Cuesta and Tak´ a˘c [8]. There is a final comparison theorem which avoids the conditions on Du and Dv in Theorem 10.1, but at the expense of a simpler boundary condition. Theorem 10.6. Let u be a solution of the inequality Di {aij (x, u)A(|Du|)Dj u} − B(x, u, Du) ≤ 0

in Ω.

Suppose that (8.1) is satisfied and that (10.11)

B(x, z, ξ) ≤ κΦ(|ξ|)

for x ∈ Ω, z < 0, and |ξ| ≤ 1. If u ≥ 0 on ∂Ω then u ≥ 0 in Ω. Proof. Assume for contradiction that u has a negative minimum M at some point x0 in Ω. Put w = u − M . Then w ≥ 0 in Ω, while w(x0 ) = 0. Using (10.11) one sees that w is a solution of the inequality Di {aij (x, w + M )A(|Dw|)Dj w} − κΦ(|Dw|) ≤ 0 in some neighborhood N of x0 (where 0 ≤ w < |M |). Hence by Theorem 8.1 we find w ≡ 0 in N , and then by chaining also w ≡ 0 in Ω, which is impossible by the boundary condition. Theorem 10.6 is false without condition (10.11), as follows from the example, equation (1.10), in the introduction. Indeed, essentially as noted there, this equation has the solution u(x) = C(|x|k − 1) on the unit ball, which vanishes on the boundary, and at the same time is negative in the interior. While we have not found a proof, we conjecture that the full result of Theorem 10.1 ˆ obeys (B1). should hold without the stated conditions on Du and Dv provided that B 10.2. Uniqueness of the Dirichlet problem The structure built up in the earlier parts of this section, and also in previous sections, allows one to present a number of uniqueness theorems for the Dirichlet problem (10.12)

ˆ u, Du) = 0 ˆ u, Du) − B(x, divA(x, u(x) = ϕ(x)

in Ω, on ∂Ω,

where ϕ ∈ C(∂Ω).

ˆ of Du, and that (10.10) holds. Then ˆ is independent of u and B Theorem 10.7. Suppose A problem (10.12) can have at most one solution. ˆ u, ξ) = This is an immediate consequence of Theorem 10.5. The special case when A(x, A(|ξ|)ξ (and A satisfies conditions (A1) and (A2) in the introduction, for example the case of the p–Laplacian) also follows directly from Theorem 5.4. ˆ is independent of u, and that the matrix [∂ Aˆi /∂ξj ] is Theorem 10.8. Suppose that A uniformly positive definite when 0 < |ξ| ≤ Const. and x is in any compact subset of Ω. ˆ is uniformly Lipschitz continuous with respect to ξ on compact Assume additionally that B subsets of its variables (and of course non–decreasing in the variable u). Then the problem (10.12) can have at most one solution. The special case of the p–Laplacian operator is of particular importance. This is given in the following

STRONG MAXIMUM PRINCIPLE

Corollary 10.9. The Dirichlet problem ˆ u, Du) = 0 ∆p u − B(x, u(x) = ϕ(x)

43

in Ω, on ∂Ω,

ˆ z, ξ) is uniformly Lipschitz continuous in ξ on compact subsets ˆ = B(x, where p ≤ 2 and B of its variables, can have at most one solution. When the boundary data takes the canonical form u = 0 on ∂Ω, then the condition of uniform positive definiteness in the previous theorem can be dropped. The result is as follows. Theorem 10.10. Consider the equation Di {aij (x, u)A(|Du|)Dj u} − B(x, u, Du) = 0

in Ω,

with (8.1) satisfied. Assume also (10.13)

[sign z] · B(x, z, ξ) ≥ −κΦ(|ξ|)

for x ∈ Ω, z ∈ R and |ξ| ≤ 1. Then the Dirichlet problem u = 0 on ∂Ω has the unique solution u ≡ 0. This follows immediately from Theorem 10.6, once it is shown that u ≡ 0 is a solution. But this is a consequence of the fact that B(x, 0, 0) = 0. Indeed by (10.13) one has [sign z] · B(x, z, 0) ≥ 0

so that B(x, z, 0) changes sign as z passes through zero, which by continuity gives B(x, 0, 0) = 0. 11. p–regular equations For a large set of equations displaying p–homogeneity, p > 1, including in particular equations involving the p–Laplacian ∆p , there is an elegant Strong Maximum Principle which corresponds closely to the case of regular equations discussed in the introduction. In particular, we consider the singular differential inequality ˆ u, Du) ≤ 0 in Ω, ˆ u, Du) − B(x, divA(x, (11.1) u ≥ 0,

ˆ have the following homogeneity and ellipticity ˆ and B where the (measurable) functions A n and ξ ∈ properties for all x ∈ D, u ∈ R+ R 0 ˆ u, ξ) · ξ ≥ a1 |ξ|p − a2 up A(x,

(11.2)

ˆ u, ξ)| ≤ a3 |ξ|p−1 + a4 up−1 |A(x, ˆ u, ξ) ≤ b1 |ξ|p−1 + b2 up−1 B(x,

with a1 , a3 > 0; a2 , a4 , b1 , b2 ≥ 0 (see [35], where these conditions apparently appear first). Trudinger [39], closely using the ideas of [35], has proved under these conditions the following beautiful Harnack inequality for continuous (non–negative) solutions u of (11.1) which are in the Sobolev space W 1,p (Ω): For any ball BR , such that 0 < R ≤ 1 and B3R ⊂ Ω, there holds

(11.3)

||u||B2R ,γ ≤ C|R|n/γ min u(x), BR

where C depends only (p, n, γ, a1 , a2 , a3 , a4 , b1 , b2 ) and γ ∈ (0, (p − 1)n/(n − p)) (or (0, ∞) if p ≥ n).

This immediately implies the following Strong Maximum Principle.7 Theorem 11.1. (Strong maximum principle). Let u be a (non–negative) solution of (11.1) in Ω, as defined above. Then either u ≡ 0 in Ω or u > 0 in Ω. 7The special case a = a = 0 and B ˆ = 0 was noted by Granlund [19]. 2 4

44

P. PUCCI AND J. SERRIN

Proof. Indeed suppose that u = 0 at some point x0 in Ω. Let B3R be a ball centered at x0 , with R so small that B3R is in Ω. Then minBR u(x) = 0, so in turn ||u||B2R ,γ = 0 by (11.3). That is, u = 0 in B2R . Chaining then gives the conclusion u ≡ 0 in Ω, proving the theorem. Remark. If we consider classical distribution solutions of (11.1), rather than the weaker class above, then conditions (11.2) need only apply for small u ≥ 0, say u < δ, and for |ξ| ≤ 1, say. ˆ for values u ≥ δ and ˆ and B To prove Theorem 11.1 for this case, we first modify A |ξ| > 1, so that the modified functions remain measurable but now also satisfy (11.2) for the complete set of variables. Then, corresponding to any classical (non–negative) solution of (11.1) for which u(x0 ) = 0, there is some neighborhood N of x0 where u < δ and |ξ| ≤ 1. Therefore u satisfies the modified equation in N , for which the full conditions (11.2) hold. Thus u ≡ 0 in N by Theorem 11.1, and then u ≡ 0 in Ω, by chaining. Theorem 11.1 is obviously broad and powerful. On the other hand, it has some drawbacks in comparison with Theorems 1.1 and 1.2 (or Theorem 8.1 and 8.5). Specifically it applies only to operators A(ρ) which obey Const. ρp−1 ≤ Φ(ρ) ≤ Const. ρp−1

(11.4)

for some positive constants, and similarly it requires that the function f (u) in (1.1), or in (B1), must satisfy f (u) ≤ up−1 for small u > 0. Finally, of course, it does not lend itself to the precise necessary and sufficient condition (1.6), even in case A obeys (11.4) There is a corresponding comparison theorem of interest, valid under the stronger conditions following: ˆ u, ξ) ≤ b1 |ξ|p−1 , ˆ u, ξ) · ξ ≥ a1 |ξ|p , A(x, B(x, (11.5) where a1 > 0 and b1 ≥ 0. Theorem 11.2. (Comparison principle). Let u be a solution of the inequality (11.1), ˆ satisfy (11.5) for x ∈ Ω and u < M . ˆ and B where A If u ≥ M on ∂Ω, then u ≥ M in Ω. Gilbarg and Trudinger give a related result ([18], Theorem 10.9), but with a more difficult proof.

Proof. Suppose for contradiction that the result fails. We then follow the proof of Theorem 10.1, with w(x) = u(x)−M , however without the intervention of (10.5). Corresponding to (10.6), one finds, using the non–positive test function wε , Z Z ˆ u, Du)wε . ˆ (11.6) A(x, u, Du) · Dwε ≤ − B(x, Σ

Ω

Then, with the help of (11.5) and the fact that Du = Dw on Σ, the inequality (11.6) implies that Z Z p a1 (11.7) |Dwε | ≤ b1 |Dwε |p−1 · |wε |. Σ

Σ

Let Γ = Γε = {ε − ε < wε < 0}. Then Dwε = 0 on Σ \ Γ = E, so the integrals in (10.7) can equally be taken over the set Γ. Applying H˝older’s inequality to the right side of (11.7) yields, cf. (10.8), b1 ||wε ||Γ,p ≥ a1 ||Dwε ||Γ,p .

(11.8)

From Poincar´e’s inequality (7.44) of [18], we obtain

ωn−1 |Γ|1/n ||Dwε ||Γ,p = ωn−1 |Γ|1/n ||Dwε ||Σ,p ≥ ||wε ||Σ,2p ≥ ||wε ||Γ,2p .

Hence by (11.8) there results (11.9)

|Γ| ≥ ωn (a1 /b1 )n .

STRONG MAXIMUM PRINCIPLE

45

On the other hand, Γ → ∅ as ε → ε, a contradiction to (11.9). This completes the proof.

12. Special cases 12.1. The linear case Consider the linear inequality (12.1)

Di {aij (x)Dj u} + bi (x)Di u + c(x)u ≤ 0,

u ≥ 0,

for x ∈ Ω, where the matrix [aij ] is continuously differentiable and satisfies (8.1), bi , c ∈ C(Ω) for all i = 1, . . . , n. This is the special case of (8.2) where A(ρ) ≡ 1, B(x, u, ξ) = −bi (x)ξi − c(x)u. Here we can apply the result of Theorem 8.1, assuming also that bi (x) and c(x) are locally bounded. By slightly shrinking the domain Ω we can then suppose that κ = max sup |bi (x)| < ∞, i

c = − inf {c(x), 0} < ∞, Ω

Ω

√ and moreover define f (u) = cu. Then Φ(ρ) = ρ, H −1 (ρ) = 2ρ and F (u) = cu2 /2, so that (B1) and (1.6) hold as required. This gives the strong maximum principle for (12.1), closely related to the classical Theorem 2.2 of E. Hopf. Indeed, assuming as above that aij is continuously differentiable, then the strong maximum principle for C 2 solutions of (12.1) is an immediate consequence of Theorem 2.2, while conversely the strong maximum principle for C 1 distribution solutions of (2.1) follows at once from Theorem 8.1. These comments moreover lead us to expect that the proof of Theorem 8.1 can be simplified for the special linear case. In fact, the principal inequality (8.6) in the proof of Theorem 8.1 suggests that the required comparison function v for the Hopf proof can be obtained for the linear case by exhibiting an explicit solution of the inequality cv k {|v 0 |}0 + |v 0 | + ≤ 0. r λ (since Φ(ρ) = ρ in the present linear case). A natural choice for v is # " R ϑ R v(r) = α −1 , ≤ r ≤ R, (12.2) r 2 where ϑ and R are to be determined. Then calculation

v 0 (r)

αϑ = R

ϑ+1 R and so after a short r

ϑ R cv k − (ϑ + 1) + r r2 λ ϑ c R k − (ϑ + 1) ≤ αϑ + . r r2 λϑ

cv k |v | + |v 0 | + = αϑ r λ 0 0

This will be ≤ 0 provided that ϑ = 2k − 1,

R2 ≤

λk(2k − 1) . c

Thus the rational comparison function (12.2) can be used for the linear inequality (12.1), alternative to the standard exponential function 2 2 v(r) = ε e−αr − e−αR , see page 148 of [20], or page 34 of [18].

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12.2. The degenerate Laplacian case A similar simplification can be used for the canonical inequality (12.3)

∆p u − f (u) ≤ 0,

u ≥ 0,

for the p–Laplace operator, p > 1. For our present purpose, we assume that (12.4)

f (u) ≤ cup−1 ,

the borderline case for (1.6). The comparison function v = v(r), r = |x|, for (6.1) again can be taken in the form (12.2). Then we have p−1 (p−1)(ϑ+1) R αϑ 0 0 p−1 Φ(|v |) = |v | = ; R r Thus as before, we find after a short calculation that (p−1)ϑ f (v) c n − 1 − (p − 1)(ϑ + 1) n−1 0 p−1 R 0 0 Φ(|v |) + ≤ (αϑ) [Φ(|v |)] + + . r λ r rp λϑp−1 This again will be ≤ 0 provided that

2(n − 1) − 1, ϑ= p−1

R≤

(n − 1)λ c

1/p

0

ϑ1/p .

That is, ∆p v − f (v) ≥ 0 for R/2 ≤ |x| ≤ R, and the proof of the strong maximum principle, Theorem 1.1, now applies unchanged, but without using Proposition 4.1. In summary, for the borderline case (12.4) of inequality (12.3), we get an elementary proof of V´azquez’ strong maximum principle, avoiding the delicate arguments of Sections 3 and 4, or of [41]. Note that the simple comparison function (12.2) does not suffice for general operators or for more complicated nonlinearities. This observation indicates the need for the new construction of v = v(r) used in the proof of Theorem 1.1. Of course, for more complicated linearities it is also necessary to use the comparison Theorem 10.1 rather than the simpler Theorem 5.4. Acknowledgement. The first author was supported by the Italian MIUR project titled “Metodi Variazionali ed Equazioni Differenziali non Lineari”. The paper is based on a mini–course given by P. Pucci in September 2002, at Grado, Italy, in the workshop “Stationary and evolution problems”, supported by the GNAMPA of the Istituto Nazionale di Alta Matematica “F. Severi”; and on the lecture of J. Serrin given in June 2003 in the series “Lezioni Leonardesche”, organized by the Mathematics Departments of the two Universities in Milan and of the Politechnic of Milan. References [1] Bandle, C. and I. Stakgold, The formation of the dead core in parabolic reaction–diffusion problems, Trans. Amer. Math. Soc., 286 (1984), 275–293. [2] Bardi, M. and F. Da Lio, On the strong maximum principle for fully nonlinear degenerate elliptic equations, Arch. Math., 73 (2000), 276–285. [3] Barles, G., G. Diaz and J.I. Diaz, Uniqueness and continuum of foliated solutions for a quasilinear elliptic equation with a nonlipschitz nonlinearity, Comm. Partial Diff. Equations, 17 (1992), 1037–1050. [4] Benilan, P., H. Brezis and M. Crandall, A semilinear equation in L1 (Rn ), Ann. Scuola Norm. Sup. Pisa, 4 (1975), 523–555. [5] Cellina, A., On the strong maximum principle, Proc. Amer. Math. Soc., 130 (2001), 413–418. ´ zar, C., M. Elgueta and P. Felmer, On a semilinear elliptic problem in Rn with a non[6] Corta Lipschitzian nonlinearity, Adv. in Diff. Equations, 1 (1996), 199–218. [7] Courant, R. and D. Hilbert, Methoden der Mathematischen Physik, I, II, Springer–Verlag, 1924, 1937, and Methods of Mathematical Physik, Vols. 1, 2, Wiley–Interscience, N.Y., 1962.

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[37] Serrin, J., Commentary on the Hopf strong maximum principle, in Selected Works of Eberhard Hopf with commentaries, ed. by C.S. Morawetz, J.B. Serrin and Y.G. Sinai, Amer. Math. Soc., Providence, 2002. [38] Tolksdorf, P., On the Dirichlet problem for quasilinear elliptic equations in domains with conical boundary points, Comm. Partial Differ. Equations, 8 (1983), 773–817. [39] Trudinger, N., On Harnack type inequalities and their applications to quasilinear elliptic equations, Comm. Pure Appl. Math., 20, 721–747. [40] Turkington, B., Height estimates for exterior problems of capillary type, Pac. J. Math., 88 (1980), 517–540. ´ zquez, J-L., A strong maximum principle for some quasilinear elliptic equations, Applied Mathe[41] Va matics and Optimization, 12 (1984), 191–202. [42] Walter, W., Differential and Integral Inequalities, Springer Verlag, Berlin, 1964 (German), 1970 (English). [43] Willmore, T.J., An Introduction to Differential Geometry, Oxford University Press, Oxford,, 1993. ` degli Studi di Perugia, Via VanDipartimento di Matematica e Informatica, Universita vitelli 1, Perugia, Italy E-mail address: [email protected] University of Minnesota, Department of Mathematics, Minneapolis, USA E-mail address: [email protected]