Degenerate elliptic equations with singular nonlinearities

Degenerate elliptic equations with singular nonlinearities 281 For convex nonlinearities f (u) so that lim u→+∞ f (u) u =+∞, it is a long standing con...

0 downloads 31 Views 376KB Size
Calc. Var. (2009) 34:279–306 DOI 10.1007/s00526-008-0184-3

Calculus of Variations

Degenerate elliptic equations with singular nonlinearities Daniele Castorina · Pierpaolo Esposito · Berardino Sciunzi

Received: 25 April 2007 / Accepted: 5 May 2008 / Published online: 17 June 2008 © Springer-Verlag 2008

Abstract The behavior of the “minimal branch” is investigated for quasilinear eigenvalue problems involving the p-Laplace operator, considered in a smooth bounded domain of R N , and compactness holds below a critical dimension N # . The nonlinearity f (u) lies in a very general class and the results we present are new even for p = 2. Due to the degeneracy of p-Laplace operator, for p  = 2 it is crucial to define a suitable notion of semi-stability: the functional space we introduce in the paper seems to be the natural one and yields to a spectral theory for the linearized operator. For the case p = 2, compactness is also established along unstable branches satisfying suitable spectral information. The analysis is based on a blow-up argument and stronger assumptions on the nonlinearity f (u) are required. Mathematics subject classification

35B35 · 35B45 · 35J70 · 35J60

1 Introduction and statement of main results We deal with the analysis of solutions to the boundary value problem: ⎧   ⎨ − p u = −div |∇u| p−2 ∇u = λh(x) f (u) in  0
(1.1)

Authors are partially supported by MIUR, project “Variational methods and nonlinear differential equations”. D. Castorina · P. Esposito (B) Dipartimento di Matematica, Università degli Studi “Roma Tre”, Largo S. Leonardo Murialdo, 1-00146 Rome, Italy e-mail: [email protected] D. Castorina e-mail: [email protected] B. Sciunzi Dipartimento di Matematica, Università della Calabria, Via Pietro Bucci, 1-87036 Arcavacata di Rende (CS), Italy e-mail: [email protected]

123

280

D. Castorina et al.

where p > 1,  ⊂ R N is a smooth bounded domain, λ ≥ 0 is a parameter and ¯ → (0, +∞) is an Hölder continuous function. Throughout the paper, the nonh(x) :  linearity f (u) will be always assumed to be a non-decreasing, positive function defined on [0, 1) with a singularity at u = 1: lim f (u) = +∞.

(1.2)

u→1−

Nonlinear eigenvalue problems as (1.1) with p = 2 and f (u) a smooth nonlinearity unbounded at +∞: lim f (u) = +∞, have been largely studied in last thirty years. u→+∞

Since the pioneering work of Crandall and Rabinowitz [8] for f (u) = eu , there has been an intensive investigation to recover general smooth f (u). Let us set up the problem in order to explain the contributions already available in literature. Let f : [0, +∞) → (0, +∞) be a f (u) smooth non-decreasing function so that lim inf > 0. By Implicit Function Theorem, u→+∞ u there is a unique curve of positive solutions u λ of (1.1) branching off u = 0, for λ small. It is possible to define the extremal parameter in the following way: λ∗ = sup{λ > 0 : (1.1) has a positive classical solution},

(1.3)

and show that λ∗ < +∞. Since f (0) > 0, u = 0 is a subsolution of (1.1). By the method of sub/super solutions, the set of λ for which (1.1) is solvable coincides exactly with [0, λ∗ ), and the associated iterative scheme provides a minimal solution u λ (i.e. the smallest positive solution of (1.1) in a pointwise sense), for any λ ∈ [0, λ∗ ). Moreover, the family {u λ } is non-decreasing in λ, and u λ is a semi-stable solution of (1.1) in the sense: ⎧ ⎫  ⎨ ⎬ |∇φ|2 − λh(x) f˙(u λ )φ 2 : φ ∈ H01 (), φ 2 = 1 ≥ 0. (1.4) µ1 (u λ ) := inf ⎩ ⎭ 



The main issues in such a topic are the following: (1) compactness of the minimal branch u λ sup u λ ∞ < +∞

λ∈[0,λ∗ )

(1.5)

to guarantee that u ∗ = lim∗ u λ -the so-called extremal solution- is a classical solution of (1.1) λ↑λ

with λ = λ∗ ; (2) study of u ∗ when compactness (1.5) along the minimal branch fails. In general, u ∗ is a weak and still semi-stable solution: µ1 (u ∗ ) ≥ 0 (defined as in (1.4)). In the non compact situation, u ∗ can be also computed explicitly in some special cases (see [3,4]). When compactness holds, let us stress that µ1 (u ∗ ) = 0 to prevent the continuation of the branch u λ for λ > λ∗ . In such a case (see [8]), by Implicit Function Theorem there is a second curve Uλ , different from u λ , branching off u = u ∗ for λ in a small left neighborhood of λ∗ . The solutions Uλ turn out to be unstable, with Morse index one. The validity of (1.5) depends on the dimension N and the nonlinearity f (u): there is a critical “dimension” N # ∈ R so that compactness holds when N < N # and fails when N ≥ N # (for some h(x) and ). In [8,21], the critical dimension for the most typical examples f (u) = eu and f (u) = (1 + u)m are computed explicitly: N # = 10 when f (u) = eu and N # ≥ 11 when f (u) = (1 + u)m (the expression of N # in this case is rather involved). In [19], a thorough ODE analysis of solutions is achieved when  is a ball, h(x) = 1 and f (u) as above.

123

Degenerate elliptic equations with singular nonlinearities

281

f (u) = +∞, it is a long standing conjecture u that the critical dimension should 9 no matter f (u) is. The first contribution is due to Crandall and Rabinowitz in [8] who prove, under the additional assumption: For convex nonlinearities f (u) so that lim

u→+∞ satisfy N # >

f (u) f¨(u) f (u) f¨(u) ≤ lim sup = γ1 < ∞, 2 ˙ u→+∞ f (u) f˙2 (u) u→+∞ √ √ that (1.5) holds for any N < 4 + 2γ + 4 γ , provided γ1 < 2 + γ + γ . Recently, Ye and Zhou in [28] have improved Crandall-Rabinowitz statement: compactness holds for any √ N < 6 + 4 γ , where f (u) f¨(u) > 0. (1.6) γ = lim inf u→+∞ f˙2 (u) 0 < γ = lim inf

Let us remark that the critical dimension found by Ye and Zhou is 10 when f (u) = eu . While, for f (u) = (1 + u)m the dimension is not optimal but the optimal one can be easily recasted by a bootstrap argument. Without additional assumption, Nedev in [22] shows the validity of (1.5) for N = 2, 3 and Cabré in [5] has announced the result for N = 4. When restricting the problem to radial solutions on the ball (with h(x) = 1 for example), in [6] Cabré and Capella show compactness for any N < 10 and possibly non-convex f (u). Problem (1.1) for a singular nonlinearity f (u) = (1 − u)−m , m > 0, has been firstly considered by Joseph and Lundgreen in [19] in a radial setting. The analysis of the minimal branch u λ has been pursued in [17,21] and the associated critical dimension has been computed. In [12,14] compactness of any unstable branch of solutions to (1.1) with uniformly bounded Morse indices is shown. The study in [12,14,17] is motivated by the theory of so-called MEMS devices and is focussed on f (u) = (1 − u)−2 . A MEMS device (Micro-Electro Mechanical System) is composed by a thin dielectric elastic membrane held fixed on ∂ (at level 0) placed below an upper plate (at level 1). An external voltage is applied, whose strength is measured by λ. The membrane deflects toward the upper plate, measured by u(x) at any point x ∈ , and the deflection increases as λ increases. For an extremal λ∗ , the membrane could touch the upper plate: max u = 1, and 

the MEMS device would break down. The function h(x) -referred to as permittivity profileis directly related to the dielectricity of the membrane at the point x ∈ . Problem (1.1) with p = 2 and f (u) = (1 − u)−2 arises as a model equation to describe MEMS devices theory. We investigate here problem (1.1) for general p > 1 and nonlinearities f (u), singular at u = 1, with growth comparable to (1 − u)−m , m > 0. We refer the interested reader to [23] for a complete account on this theory. The difficulty is twofold. On one side, we allow more general functions f (u) and the right assumption (in the spirit of [8,28]) has to be understood. On the other side, the p-Laplace operator is a nonlinear degenerate operator. Problem (1.1) for general p > 1 has been considered in [15,16] for f (u) = eu and in [7] for f (u) of polinomial-type growth. We will borrow some ideas and techniques from [7] to deal with singular nonlinearities, and some of their arguments will be refined here. According to [11,20,26], Hölder continuity of first derivatives holds for any weak solution of (1.1) with f (u) ∞ < +∞. Hence, we will say that u is a classical solution of (1.1) if u has Hölder continuous first derivatives, 0 < u < 1 in , u = 0 on ∂ and u solves (1.1) in a weak sense:   1, p p−2 |∇u| ∇u∇φ = λ h(x) f (u)φ, ∀ φ ∈ W0 (). (1.7) 



123

282

D. Castorina et al.

Let u be a classical solution of (1.1). The linearization of (1.7) is not possible along every 1, p direction in W0 (), when 1 < p < 2, while it is possible not only along directions in 1, p W0 (), when p > 2. Then, the choice of a functional space Au , composed by admissible directions for which the linearization makes sense, is crucial. The stability of a minimal solution will be easier to establish as smaller the space Au is. However, the class Au should allow the choice of suitable test functions in order to prove a-priori energy estimates for semi-stable solutions. In [7], the authors find a good candidate for Au , which however does not allow a spectral theory for the “linearized operator”. We present here a different and more natural way to overcome the problem, by taking a space Au larger than in [7]. Letting ρ = |∇u| p−2 , we introduce a weighted L 2 -norm of the gradient: 1 

|φ| =  ρ|∇φ|2 2 . For 1 < p ≤ 2, Au is the following subspace of H01 (): Au = {φ ∈ H01 () : |φ| < +∞}.

2− p Since  |∇φ|2 ≤ ∇u ∞ |φ|2 , (Au , | · |) is an Hilbert space. For p > 2, the weight ρ is in L ∞ () and satisfies ρ −1 ∈ L 1 (), as shown in [9]. According to [27], the space Hρ1 () = {φ ∈ L 2 () weakly differentiable : |φ| < +∞} is an Hilbert space and is the completion of C ∞ () with respect to the | · |-norm. For p > 2, the Hilbert space Au is the closure of C0∞ () in Hρ1 (). 1

For convenience, we replace the | · |-norm with the equivalent norm φ =< φ, φ > 2 , where  ∇u ∇u < φ, ψ >= |∇u| p−2 ∇φ∇ψ + ( p − 2)|∇u| p−2 · ∇φ · ∇ψ . |∇u| |∇u| 

For any p > 1, the Hilbert space Au is non-empty: u ∈ Au , and is compactly embedded in L 2 (), as we will derive in Appendix from the weighted Sobolev estimates of [10]. A first eigenfunction for the “linearized operator” then exists in Au : Theorem 1.1 Let u be a classical solution of (1.1). The infimum

φ 2 − λ  h(x) f˙(u)φ 2

µ1 (u) := inf 2 φ∈Au\{0} φ is attained at some positive function φ1 , and any other minimizer is proportional to φ1 . For our purposes, Theorem 1.1 is sufficient even if we guess a full spectral theory for the “linearized operator” to be in order. Once a right stability notion has been introduced, we show that the known results about the minimal branch are still available: Theorem 1.2 Let p > 1. Let f (u) be a non-decreasing, positive function on [0, 1) so that (1.2) holds. There exists λ∗ ∈ (0, +∞) so that, for any λ ∈ (0, λ∗ ), (1.1) has a unique minimal (classical) solution u λ and, for any λ > λ∗ no classical solution of (1.1) exists. The following upper bound holds: λ∗ ≤ f (0)−1 (inf h)−1 λ1 , 

(1.8)

where λ1 > 0 is the first eigenvalue of − p . Moreover, the family {u λ } is non-decreasing in λ and composed by semi-stable solutions: µ1 (u λ ) ≥ 0.

123

Degenerate elliptic equations with singular nonlinearities

283

We are now concerned with compactness issues and, in the spirit of (1.6), we assume on f (u): p−2 f (u) f¨(u) ln f (u) =γ > , lim inf lim inf = m > 0. (1.9) − p−1 u→1 u→1− ln 1 f˙2 (u) 1−u Set ⎧  mp ⎪ ⎨ p+m−1 γ + N# =  ⎪ ⎩ mp p+m−1 γ +

2 p−1 2 p−1

 p + γ ( p − 1) + γ − 1 −  2 √ + p−1 2 − p + γ ( p − 1) +

2 √ p−1 2 −

 

1 m −

if 1 < p ≤ 2 if p > 2, (1.10)

where u − =

|u|−u 2

is the negative part of u. The result we have is:

Theorem 1.3 Let p > 1 and f (u) be a non-decreasing, positive function on [0, 1) so that (1.9) holds. When 1 < p < 2 assume the convexity of f (u) near u = 1. Then sup u λ ∞ < 1,

λ∈[0,λ∗ )

(1.11)

provided N < N # . Remark 1.4 (1) Assumption γ > p−2 p−1 is necessary to obtain that (q− , q+ ) ∩ (− min{γ , 1}, +∞)  = ∅ in the basic integral estimate (3.30). In analogy with [22] it could be interesting to consider the case γ = p−2 p−1 when p ≥ 2 (as in [25] for regular nonlinearities f (u)). However, when p = 2 Nedev result [22] applies for N = 2, 3 and can not be seen as √ the limiting case of Ye-Zhou result [28] because lim (6 + 4 γ ) = 6. We are interested γ →0

here in obtaining the maximal regularity we can (depending on γ ) and we will consider only the case γ > p−2 p−1 . (2) Let us stress that the critical dimension N # given in (1.10) has a jump discontinuity at p = 2 for γ < 1 + m1 : the method we will use (inspired by [28]) leads to stronger estimates when 1 < p ≤ 2 and is based on the convexity of f (u) near u = 1. The case 1 < p < 2 and p−2 p−1 < γ ≤ 0 could also be considered (as in [24] for regular nonlinearities f (u)) but this improved approach could not be used. (3) Let us discuss assumption (1.9). Observe that, for singular polinomial nonlinearities f (u) = (1 − u)−m , there is a relation among m and γ : γ = 1 + m1 . In general, m and γ are not related, as the following convex nonlinearity shows: f (u) = (1 − u)−h(u) , h(u) =

   1 m 1 +m 2 m 2 −m 1 sin ln 1 + ln 1 + ln + , 2 1−u 2

where > 0 small and 0 < m 1 < m 2 < ∞. Observe that h(u) oscillates taking 1 2 −m 1 ˙ all the values in [m 1 , m 2 ]. Since |h(u) ln 1−u | ≤ 2 m1−u , for small there holds: 1 h(u) 1 ˙ f˙(u) = f (u)(h(u) ln 1−u + 1−u ) > 0 for any u ∈ [0, 1) (note that f (u) = eh(u) ln 1−u ). ˙ ¨ Since |h(u) ln 1 | + 2| h(u) | ≤ 3 m 2 −m21 , for small we get: 1−u

1−u

(1−u)

123

284

D. Castorina et al.

f¨(u) =

f˙2 (u) f (u)

¨ + f (u)(h(u) ln

1 1−u

˙

+ 2 h(u) 1−u +

h(u) ) (1−u)2

> 0 for any u ∈ [0, 1), and

˙ h(u) 1 ¨ h(u) ln 1−u + 2 h(u) f (u) f¨(u) 1−u + (1−u)2 γ = lim inf = 1 + lim inf  2 u→1− u→1− f˙2 (u) 1 ˙ + h(u) h(u) ln 1−u 1−u

= 1 + lim inf u→1−

Since lim inf u→1−

ln f (u) 1 1−u

ln

1 1 =1+ . h(u) m2

= lim inf h(u) = m 1 , the values of m and γ in (1.9) are indepenu→1−

dent. Moreover, this example features a general property (based on the validity of (1.2)):  lim inf

ln f (u)

u→1−

ln

1 1−u

−1

f (u) f¨(u) ≤ lim sup − 1, f˙2 (u) u→1− ≥ lim inf u→1−

 lim sup u→1−

ln f (u) ln

−1

1 1−u

f (u) f¨(u) − 1. f˙2 (u)

(1.12)

In strong analogy, let us remark that (1.6) on [0, +∞) implies:

ln f (u) lim inf u→+∞ ln u

−1

≤ 1 − lim inf

u→+∞

f (u) f¨(u) . f˙2 (u)

Unfortunately, to establish energy estimates we need an assumption on lim inf u→1−

ln f (u)

f (u) f¨(u) , f˙2 (u)

. It explains somehow why we need 1 ln 1−u to strengthen assumption (1.6) on [0, ∞) when considering nonlinearities on [0, 1). which does not imply any control on lim inf u→1−

When (1.11) holds, the extremal function u ∗ = lim∗ u λ is so that: max u ∗ < 1. Since 

λ↑λ

f (u ∗ ) ∞ < ∞, by regularity theory u ∗ is a classical solution of (1.1) with λ = λ∗ . Since u ∗ is the minimal solution, µ1 (u ∗ ) ≥ 0. When p = 2, the Implicit Function Theorem provides µ1 (u ∗ ) = 0 and, following the classical argument of [8], there is δ > 0 so that, for ¯ any λ ∈ (λ∗ − δ, λ∗ ), a second solution Uλ of (1.1) exists so that lim∗ Uλ = u ∗ in C 1 (). λ↑λ

For the analysis of the second branch Uλ , we will use a blowup approach developed in [12,14]. In order to identify a limiting equation on R N and to have some useful information on such a limit problem, for p = 2 we will require: lim

u→1−

f (u) f¨(u) =γ >1 f˙2 (u)

(1.13)

1

(1 − t) γ −1 f (t)

= c0 < +∞ 1 (1 − u) γ −1 f (u)   1  (1 − t) γ −1  f (t)   sup − 1 = 0 , ∀ M > 1. lim  1  u→1− 1−M(1−u)≤t≤u  (1 − u) γ −1 f (u) sup

sup

(1.14)

0≤u<1 0≤t≤u

123

(1.15)

Degenerate elliptic equations with singular nonlinearities

285

By (1.12), the inequality γ ≥ 1 in (1.13) always holds and m = lim

u→1−

Hence, N # in (1.10) reduces to:

ln f (u) ln

1 1−u

=

1 . γ −1

√ 2(γ + 2 + 2 γ ) N = γ #

(in case p = 2). The result we have is: Theorem 1.5 Let p = 2 and f (u) be a convex, non-decreasing, positive function on [0, 1) so that (1.13)–(1.15) hold. Let λn ∈ (0, λ∗ ) be a sequence and u n be associated solutions of (1.1). Assume that u n has Morse index at most 1: µ2,n ≥ 0 for any n ∈ N, where µ2,n = µ2 (− − λn h(x) f˙(u n )) is the 2nd eigenvalue of the linearized operator. Then sup u n ∞ < 1,

provided N <

√ 2(γ +2+2 γ ) . γ

n∈N

In [12], compactness of a solutions sequence u n with uniformly bounded Morse indices is shown to hold for f (u) = (1 − u)−2 , where h(x) is allowed to vanish at pi as |x − pi |αi , αi > 0, for i = 1, . . . , k. This is still true in such a more general context but, for the sake of shortness and simplicity, we will consider in Theorem 1.5 only the case of Morse index one ¯ Let us remark that assumptions (1.13)–(1.15) require that f (u) behaves at and h > 0 in . −

1



1

e main order like (1 − u) γ −1 as u → 1− : an example is given by f (u) = (1 − u) γ −1 ln 1−u , γ > 1. The paper is organized as follows. In Sect. 2 we illustrate how to adapt standard techniques to p-Laplace operator. In Sect. 3, we prove energy estimates and see how assumption (1.9) will allow us to prove Theorem 1.3. Let us stress that, by regularity theory, energy estimates on u λ can provide useless L ∞ -bounds on u λ (in our context u λ ∞ ≤ 1). In particular, the second assumption in (1.9) will be crucial in our argument. In Sect. 4, we describe the blow up approach and, by an instability property of the limiting equation, we will derive Theorem 1.5. Existence of first eigenfunctions as in Theorem 1.1 and some technical Lemmata of Sect. 4 will be proved in the Appendix.

2 Minimal branch In this section, we will establish Theorem 1.2 ( we refer to [7] for related results). Since the Implicit Function Theorem does not produce solutions of (1.1) for λ > 0 small and p  = 2, due to the degeneracy of p-Laplacian, we will use directly the sub/super solutions method. Since f (0) > 0, u = 0 is a sub-solution of (1.1). In order to produce a positive supersolution for λ small, let v be the solution of  − p v = h(x) f (0) in  (2.16) v=0 on ∂. ¯ Let us fix β > 0 small so that Problem (2.16) has a unique positive solution v ∈ C 1 (). u := βv satisfies u ∞ < 1. By the monotonicity of f (u), there holds: − p u = β p−1 h(x) f (0) ≥ λh(x) f (max u) ≥ λh(x) f (u) 

for any 0 < λ <

β p−1 f (0) f (max u) .

Namely, for λ small u is a positive super-solution of (1.1).

123

286

D. Castorina et al.

¯ Fix λ > 0 so that (1.1) has a super-solution u: 0 < u < 1 in . Let u 1 ∈ C 1 () be the unique, positive solution of: − p u 1 = λh(x) f (0) in , u 1 = 0 on ∂. Since − p u 1 ≤ λh(x) f (u) ≤ − p u, by the weak comparison principle 0 = u ≤ u 1 ≤ u. Introduce now the following iteration scheme: let u n , n ≥ 2, be the unique, positive solution of  − p u n = λh(x) f (u n−1 ) in  (2.17) un = 0 on ∂. We want to show that 0 = u ≤ u n ≤ u for any n ≥ 1. If such a property holds for some u n , by the weak comparison principle applied to: − p u n+1 = λh(x) f (u n ) ≤ λh(x) f (u) ≤ − p u, we get u n+1 ≤ u. Since u 1 ≤ u, by induction 0 = u ≤ u n ≤ u for any n ≥ 1. In the same way, u n is a non-decreasing sequence: u n+1 ≥ u n for any n ≥ 1. Set u λ (x) := limn→+∞ u n (x). Since u ∞ < 1, for any n ≥ 1 there holds: 

 |∇u n | p = λ 

 h(x) f (u n−1 )u n ≤ λ



h(x) f (u)u < +∞.  1, p

Up to a subsequence, we can assume that u n u λ weakly in W0 () and by Lebesgue Theorem f (u n ) → f (u λ ) in L 1 (), as n → +∞. Since ∇u n → ∇u λ in L q (), q < p, as n → +∞ (see [1]), Eq. (2.17) passes to the limit yielding to a classical solution 0 = u ≤ u λ ≤ u of (1.1) (classical in the sense specified in the Introduction). Since the scheme we defined is independent on u, we get that u λ ≤ u for any super-solution of (1.1). In particular, u λ defines the unique, positive minimal solution of (1.1). Resuming what we did, for λ > 0 small a minimal solution u λ exists. Then, λ∗ ∈ (0, +∞], given as in (1.3), is a well defined number. To establish an upper bound on λ∗ , let us compare (1.1) and  − p u = βu p−1 in  (2.18) u=0 on ∂. Since h(x) f (u) ≥ h(x) f (0) ≥ δu p−1 for any 0 ≤ u < 1, δ = f (0) inf h, a solution u of 

(1.1) is a super-solution of (2.18) for β = δλ. Let λ1 be the first eigenvalue of − p (λ1 is the least value β > 0 so that (2.18) has a non trivial solution) and let ϕ1 be an associated positive eigenfunction. For any β ≥ λ1 , ϕ1 is a sub-solution of (2.18). By Hopf Lemma and weak Harnack inequality, ∂ν u < 0 on ∂ and u > 0 in , where ν(x) is the unit outer normal of ∂ at x. Hence, for > 0 small, the function ϕ1 is still a first eigenfunction so that ϕ1 < u in . If λ∗ > λδ1 , the sub/super solutions method explained above works as well yielding to a positive eigenfunction ϕβ (ϕβ ≥ ϕ1 ) with associated eigenvalue β, for any δλ∗ > β > λ1 . Since it is well known that the only positive eigenfunction of − p is the first one, we reach a contradiction. Hence (1.8) holds. Since any classical solution u of (1.1), for some λ = λ¯ , is a super-solution of (1.1) for any 0 ≤ λ ≤ λ¯ , a super-solution exists for any λ ∈ [0, λ∗ ). The iterative scheme provides the existence of a (unique) classical, minimal solution u λ for any λ ∈ [0, λ∗ ) so that u λ ≤ u λ for any 0 ≤ λ ≤ λ < λ∗ . Next Lemma shows the semi-stability of u λ and complete the proof of Theorem 1.2:

123

Degenerate elliptic equations with singular nonlinearities

287

Lemma 2.1 Let λ ∈ [0, λ∗ ) and u λ be the minimal solution of (1.1). Then, u λ is semi-stable:  2   ∇u λ p−2 2 p−2 2 ˙ · ∇φ − λh(x) f (u λ )φ |∇φ| + ( p − 2)|∇u λ | |∇u λ | |∇u λ | 

≥ 0 ∀ φ ∈ Au λ .

(2.19)

u 1, p Proof Let M = {u ∈ W0 () : 0 ≤ u ≤ u λ a.e.} and F(u) = 0 f (s)ds. Since u λ ∞ < 1, F(u) is uniformly bounded for any u ∈ M: 0 ≤ F(u) ≤ F(u λ ) ≤ C < +∞ a.e. in . Introduce the energy functional:   1 |∇u| p − λ h(x)F(u), u ∈ M. E(u) = p 



The functional E is well defined, bounded from below and weakly lower semi-continuous in M. Then, it attains the minimum at some u ∈ M: E(u) = inf E(v). v∈M

The key idea is to prove: u λ = u. We need only to show that u is a classical positive solution of (1.1). Indeed, since u ≤ u λ and u λ is the minimal solution, necessarily u = u λ . Since u minimizes E(v) on the convex set M, the following inequality holds:   |∇u| p−2 ∇u∇(ψ − u) − λ h(x) f (u)(ψ − u) ≥ 0 ∀ ψ ∈ M. (2.20) 



Let us introduce the notation 





E (v)ϕ =

|∇v| p−2 ∇v∇ϕ − λ 

h(x) f (v)ϕ 

1, p

for any v ∈ M and ϕ ∈ W0 (). Let now ϕ ∈ C0∞ (). We use ψ = u + ϕ − (u + ϕ − u λ )+ + (u + ϕ)− ∈ M, > 0, as a test function in (2.20): 







0 ≤ E (u)(ψ − u) = E (u)ϕ − E (u)(u + ϕ − u λ )+ + E (u)(u + ϕ)− , and then



E (u)ϕ ≥

1  1  E (u)(u + ϕ − u λ )+ − E (u)(u + ϕ)−

(2.21)

holds. Since u λ solves (1.1) and f (u) is non-decreasing, by Lebesgue Theorem we have:  1  1   E (u)(u + ϕ − u λ )+ = E (u) − E (u λ ) (u + ϕ − u λ )+    |∇u| p−2 ∇u − |∇u λ | p−2 ∇u λ ∇ϕ ≥ {u λ ≤u+ ϕ}







 |∇u| p−2 ∇u − |∇u λ | p−2 ∇u λ ∇ϕ

(2.22)

{u λ =u}

as → 0+ , in view of u ≤ u λ and (|x| p−2 x − |y| p−2 y) · (x − y) > 0

∀ x, y ∈ R N , x  = y.

(2.23)

123

288

D. Castorina et al.

Recall now that, given u, u  ∈ W0 (), by Stampacchia’s Theorem it follows that ∇u = ∇u  a.e. in the set {u = u  }. Therefore ∇u = ∇u λ a.e. in {u = u λ }, and letting → 0+ in (2.22) we get:    1  lim inf E (u)(u + ϕ − u λ )+ ≥ |∇u| p−2 ∇u − |∇u λ | p−2 ∇u λ ∇ϕ = 0 →0+ 1, p

{u λ =u}





for any ϕ ∈ C0∞ (). Since 0 is a sub-solution of (1.1), comparing E (u) with E (0), similarly we have:  1  lim sup E (u)(u + ϕ)− ≤ − |∇u| p−2 ∇u∇ϕ = 0. →0+ {u=0}



Then, by (2.21) we get: E (u)ϕ ≥ 0, for any ϕ ∈ C0∞ (). By density we get that u is a nonnegative weak bounded solution of (1.1). By regularity theory and weak Harnack inequality, u is then a positive classical solution for 0 < λ < λ∗ . Once we have characterized u λ as the minimum point of E(u) in M, we are in a good position to show semi-stability of u λ . We should differentiate two times E(u) at u = u λ but a lot of care is needed because E(u) is not a C 2 -functional. Let 0 ≤ ϕ ∈ C0∞ () ∩ Au λ . Since u λ is a positive continuous function, u λ ≥ δ > 0 on Supp ϕ and u λ − tϕ ∈ M for t > 0 small. Compute the first derivative of F(t) = E(u λ − tϕ), for t > 0 small:    ˙ = −E  (u λ − tϕ)ϕ = − |∇u λ − t∇ϕ| p−2 (∇u λ − t∇ϕ)∇ϕ − λh(x) f (u λ − tϕ)ϕ . F(t)   ¨ ˙ ≥ 0, if Since F(t) ≥ F(0), for t > 0 small, and F(0) = −E (u λ )ϕ = 0, we have that F(0) ¨ F(0) exists. Let t = {x ∈  : 2t|∇ϕ|(x) < |∇u λ |(x)}. Observe that:

 I2 : = 



 |∇u λ − t∇ϕ| p−2 (∇u λ − t∇ϕ) − |∇u λ | p−2 ∇u λ ∇ϕ  t

\t



⎜ ≤ C ⎝ ⎛ ⎜ ≤C⎝



\t

|∇u λ | p−1 |∇ϕ| + t





⎟ t p−2 |∇ϕ| p ⎠

\t





⎟ |∇ϕ| p ⎠ ,

|∇u λ | p−2 |∇ϕ|2 +

\t



\t

because 21 |∇u λ | ≤ t|∇ϕ| and t p−2 |∇ϕ| p = (t|∇ϕ|) p−2 |∇ϕ|2 ≤ 22− p |∇u λ | p−2 |∇ϕ|2 + |∇ϕ| p in \t . Also {|∇u λ | = 0} is a zero measure set (see [9]). Since \t → {|∇u λ | = 0} in measure, as t → 0+ , then we have: \t → ∅, t → 

in measure, as t → 0+ .

(2.24)

Since |∇u λ | p−2 |∇ϕ|2 +|∇ϕ| p ∈ L 1 () for any ϕ ∈ C0∞ ()∩ Au λ , then I2 → 0 as t → 0+ . Compute now:

123

Degenerate elliptic equations with singular nonlinearities



289

|∇u λ − t∇ϕ| p−2 (∇u λ − t∇ϕ) − |∇u λ | p−2 ∇u λ ∇ϕ t

I1 := t

1

 = −

dx

t

 ds |∇u λ − st∇ϕ| p−2 |∇ϕ|2

0

 + ( p − 2)|∇u λ − st∇ϕ| p−4 (∇u λ ∇ϕ − st|∇ϕ|2 )2 . Since C ≥ |∇u λ − st∇ϕ| ≥ |∇u λ | − t|∇ϕ| >

|∇u λ | 2

on t , observe that

|∇u λ − st∇ϕ| p−2 |∇ϕ|2 + ( p − 2)|∇u λ − st∇ϕ| p−4 (∇u λ ∇ϕ − st|∇ϕ|2 )2 ≤ 22− p |∇u λ | p−2 |∇ϕ|2 + ( p − 1)C p−2 |∇ϕ|2 ∈ L 1 () for any ϕ ∈ C0∞ () ∩ Au λ . By Lebesgue Theorem and (2.24) we get that 2    ∇u λ p−2 2 p−2 I1 → − · ∇ϕ |∇ϕ| + ( p − 2)|∇u λ | |∇u λ | |∇u λ | 

as t → 0+ .

˙ is clearly a C 1 We are now ready to conclude. Since the term λ  h(x) f (u λ −tϕ)ϕ in F(t) ¨ function at t = 0, the only difficulty to compute F(0) is given by the term

|∇u − t∇ϕ| p−2 (∇u λ − t∇ϕ)∇ϕ. The limit of I1 , I2 as t → 0+ provides the existence λ  ¨ of F(0) and:  2   ∇u λ p−2 2 p−2 2 ¨ · ∇ϕ − λh(x) f˙(u λ )ϕ ≥ 0, |∇ϕ| + ( p − 2)|∇u λ | F(0) = |∇u λ | |∇u λ | 

(2.25) for any 0 ≤ ϕ ∈ C0∞ () ∩ Au λ . For p ≥ 2, by definition of Au λ , C0∞ () is a dense subspace of Au λ in the · -norm. Then, inequality (2.25) holds for any 0 ≤ ϕ ∈ Au λ . This is still true for 1 < p ≤ 2 but more care is needed in the density argument. For 1 < p ≤ 2, let us observe that (2.25) still holds for any 0 ≤ ϕ ∈ L ∞ () ∩ Au λ with supp ϕ ⊂ . The 1, p argument to derive (2.25) works as well because Au λ ⊂ H01 () ⊂ W0 () for 1 < p ≤ 2. Finally, let us show that any function 0 ≤ ϕ ∈ Au λ can be approximated in · -norm by non-negative, essentially bounded functions ϕn with support in . Indeed, let ψn ∈ C0∞ () ¯ be so that ψn → ϕ in H01 (). By Hopf Lemma, there holds: |∇u λ | > 0 in \ 2δ for some δ > 0 small, where δ = {x ∈  : dist (x, ∂) > δ}. Let χ be a cut-off function so that χ = 1 in 2δ and χ = 0 in \δ . Define now ϕn = min{χϕ + (1 − χ)ψn , n} ∈ L ∞ (). We have that  |∇u λ | p−2 |∇(ϕn − ϕ)|2 

 =

 |∇u λ | p−2 |∇ϕ|2 +

{χ ϕ+(1−χ )ψn >n}





|∇u λ | p−2 |∇(1 − χ)(ψn − ϕ)|2

{χ ϕ+(1−χ )ψn ≤n}



|∇u λ | p−2 |∇ϕ|2 + C {χ ϕ+(1−χ )ψn >n}



(ψn − ϕ)2 + C \2δ

(∇ψn − ∇ϕ)2 → 0 \2δ

123

290

D. Castorina et al.

as n → +∞, because ψn → ϕ in H01 () and L 2 (), and  1 |{χϕ + (1 − χ)ψn > n}| ≤ 2 sup (χϕ + (1 − χ)ψn )2 → 0 n n∈N 

as n → +∞. Hence, 0 ≤ ϕn ∈ L ∞ () ∩ Au λ with supp ϕn ⊂ supp ψn ∩ δ ⊂ . ¨ Once (2.25) is established for any 0 ≤ ϕ ∈ Au λ , since F(0) is a quadratic form in ϕ and ϕ ± ∈ Au λ when ϕ ∈ Au λ , we get that(2.25) holds for any ϕ ∈ Au λ . The proof is done.   3 Compactness of minimal branch In this section, we will prove Theorem 1.3. Let us assume f (u) f¨(u) ≥ γ f˙2 (u) ∀ t ≤ u < 1, for some t = tγ ∈ (0, 1), where γ >

p−2 p−1

(3.26)

if p ≥ 2 and γ ≥ 0 if 1 < p < 2. Observe that in

p−2 p−1 .

particular γ > For suitable test functions, semi-stability of u λ and assumption (3.26) will provide integral bounds on the R.H.S. of (1.1). Let q > − min{γ , 1}. Introduce the following function:  0 if 0 ≤ u < t g(u) = u  q−1 2 q ˙ ¨ qf (s) f (s) + f (s) f (s)ds if t ≤ u < 1. t By (3.26), observe that q f q−1 (s) f˙2 (s) + f q (s) f¨(s) ≥ (q + γ ) f q−1 (s) f˙2 (s) ≥ 0 for any t ≤ s < 1. Then, g(u) is well defined and, for any t ≤ u < 1: √ g(u) ≥ q + γ

u f t

q−1 2

√ 2 q +γ  ˙ f (s) f (s)ds = q +1

q+1 2

(u) − f

q+1 2

 (t) .

(3.27)

  1, p Let us now test (1.1) against f q (u λ ) f˙(u λ ) − f q (t) f˙(t) χ{u λ ≥t} ∈ W0 ():  h(x) f q+1 (u λ ) f˙(u λ ) λ {u λ ≥t}





  |∇u λ | p q f q−1 (u λ ) f˙2 (u λ ) + f q (u λ ) f¨(u λ )

{u λ ≥t}

1 = p−1

 

|∇u λ |

p−2

|∇g(u λ )| + ( p − 2)|∇u λ | 2



p−2

2  ∇u λ · ∇g(u λ ) . (3.28) |∇u λ |

¯ and u λ ∞ < 1, g(u λ ) ∈ Au λ for any λ ∈ [0, λ∗ ). The semi-stability Since u λ ∈ C 1 () (2.19) of u λ , inserted into (3.28), and estimate (3.27) yield to:   1 h(x) f q+1 (u λ ) f˙(u λ ) ≥ h(x) f˙(u λ )g 2 (u λ ) (3.29) p−1 

{u λ ≥t}



123

4(q + γ ) ( p − 1)(q + 1)2

 {u λ ≥t}

 h(x) f˙(u λ ) f

q+1 2

(u λ ) − f

q+1 2

2 (t) .

Degenerate elliptic equations with singular nonlinearities

291

2 2 √ Setting q± = p−1 − 1 ± p−1 2 − p + γ ( p − 1), note that assumption γ > p−2 p−1 ensures q± ∈ R, (q− , q+ )  = ∅ and q+ > − min{γ , 1}. For any q ∈ (q− , q+ ), there holds: 4(q+γ ) > 1, and then: ( p−1)(q+1)2

4(q + γ )(1 − ) > 1, ( p − 1)(q + 1)2 for some > 0 small. Since lim f q+1 (u) = +∞, there exists t ∈ (t, 1) so that u→1−

f˙(u)( f

q+1 2

(u) − f

q+1 2

(t))2 ≥ (1 − ) f˙(u) f q+1 (u)

∀ t ≤ u < 1.

Combined with (3.29), finally we get that:  h(x) f q+1 (u λ ) f˙(u λ ) ≤ C ∀ q ∈ (q− , q+ ) ∩ (− min{γ , 1}, +∞).

(3.30)

{u λ ≥t}

By integration, (3.26) gives that: f˙(u) ≥ f˙(u) = f˙η (u) f˙1−η (u) for 0 ≤ η ≤ 1 and f˙1−η (u) ≥

f˙(t) f γ (t)

f˙1−η (t) f γ (1−η) (t)

f γ (u) for any t ≤ u < 1. Write

f γ (1−η) (u)

for any t ≤ u < 1, we get the following result: Theorem 3.1 Assume (3.26) for γ > p−2 p−1 if p ≥ 2 and γ ≥ 0 if 1 < p < 2. Given 0 ≤ η ≤ 1, then  h(x) f q (u λ ) f˙η (u λ ) < +∞ (3.31) sup λ∈[0,λ∗ )



for any 1 ≤ q < qη = γ (1 − η) +

2 p−1

+

2 √ p−1 2 −

p + γ ( p − 1).

For η = 0, Theorem 3.1 gives:  sup h(x) f q (u λ ) < +∞, ∀ 1 ≤ q < q0 = γ + λ∈[0,λ∗ )



2  2 + 2 − p + γ ( p − 1). p−1 p−1

(3.32) When 1 < p ≤ 2, estimate (3.32) can be improved with the following argument. Let us replace f (u) with f˜(u) = f (u) + u + Cu 2 , C > 0 large in order to have f˜ convex and strictly increasing on [0, 1) (we use here the property of convexity of f (u) near u = 1). Given s ≥ 1, by (1.1) let us compute (in a weak sense):    − p f˜s (u λ ) − f˜s (0) ≤ λs p−1 h(x) f˜( p−1)(s−1)+1 (u λ ) f˙˜ p−1 (u λ ) in  (3.33) f˜s (u λ ) − f˜s (0) = 0 on ∂. η

Since 0 ≤ u λ < 1, by (3.31) the R.H.S. in (3.33) is uniformly bounded in L p−1 (), for any q qη qη p 2− p 2 p − 1 ≤ η ≤ 1 and for any 1 ≤ s < ηη − 2− p−1 (note that η > p−1 implies η − p−1 > 1 1, p

for any 0 ≤ η ≤ 1). Since it is possible to find h λ ∈ W0 () so that − p h λ = λs p−1 h(x) f˜( p−1)(s−1)+1 (u λ ) f˙˜ p−1 (u λ ),

(3.34)

123

292

D. Castorina et al.

by weak comparison principle 0 ≤ f˜s (u λ ) − f˜s (0) ≤ h λ . Elliptic regularity theory for η p-Laplace operator (see [18]) applies to (3.34): L p−1 ()-bounds on the R.H.S. of (3.34) q p gives estimates on h λ and in turn, on f s (u λ ) for any 1 ≤ s < ηη − 2− p−1 . Let 1 < p ≤ 2. If N < provides:

p p−1 ,

then

η p−1

>

N p

for η = 1 and elliptic regularity theory

sup f (u λ ) ∞ < ∞.

λ∈[0,λ∗ )

In particular, compactness (1.11) holds for N < theory we get:

p p−1 . When

N=

p p−1 , by elliptic regularity

sup f (u λ ) L q () < ∞,

λ∈[0,λ∗ )

for any q ≥ 1. When N > η ∗∗ ( p−1 )

= gives that:

N η( p−1) N ( p−1)−ηp

p p−1 ,

for any p − 1 ≤ η ≤ 1 we have that

η p−1

<

N p

and

is well defined. Fix now p − 1 ≤ η ≤ 1. Elliptic regularity theory

sup f (u λ ) L q () < ∞, ∀ 1 ≤ q < q˜η =

λ∈[0,λ∗ )

qη 2− p − η p−1



η p−1

∗∗

.

(3.35)

We need now to maximize q˜η for p − 1 ≤ η ≤ 1 in order to achieve the better integrability. It is a tedious but straightforward computation to see that: p−1 2− p N ( p − 1) q˜η = N γ+ + p p−1 N ( p − 1) − ηp 2− p 2 2  p−1 γ+ +γ + + 2 − p + γ ( p − 1) . × −N p p−1 p−1 p−1 Then, the function q˜η is monotone in η. Define

√ p 2 + γ ( p − 1) + 2 2 − p + γ ( p − 1) Np = . p−1 2 − p + γ ( p − 1)

p p . If p−1 < N ≤ N p , the function q˜η is non-decreasing and achieves Observe that N p > p−1 the maximum at η = 1. If N > N p , q˜η decreases and achieves the maximum at η = p − 1. We can compute now:

q˜ :=

sup p−1≤η≤1

 =

q˜η 



√ N p + 2 2 − p + γ ( p − 1) N ( p−1)−  p 2 2 √ N N − p (2 − p)(γ − 1) + p−1 + p−1 2 −

 p + γ ( p − 1)

if

p p−1

< N ≤ Np

if N > N p .

When 1 < p ≤ 2, observe that q˜ ≥ q0 if and only if N ≤ N p . Let us define ⎧ p if N = p−1 , 1< p≤2 ⎪ ⎨ +∞   √ p N p + 2 2 − p + γ ( p − 1) if < N ≤ N p, 1 < p ≤ 2 q p = N ( p−1)− p p−1 ⎪ ⎩ γ + 2 + 2 √2 − p + γ ( p − 1) if either N > N p , 1 < p ≤ 2 or p > 2. p−1 p−1 Resuming (3.32), (3.35), the following result has been established:

123

Degenerate elliptic equations with singular nonlinearities

293

Theorem 3.2 Assume (3.26) for γ > p−2 p−1 if p ≥ 2 and γ ≥ 0 if 1 < p < 2. When p and 1 < p ≤ 2, there holds: 1 ≤ N < p−1 sup u λ ∞ < 1.

λ∈[0,λ∗ )

When either N ≥

p p−1 ,

1 < p ≤ 2 or p > 2, we have that: sup f (u λ ) L q () < +∞

(3.36)

λ∈[0,λ∗ )

for any 1 ≤ q < q p . p if 1 < p ≤ 2. We want to understand when compactness Let p > 1, and assume N ≥ p−1 (1.11) of u λ holds. We will use now the following assumption:

f (u) ≥

C0 (1 − u)m

∀ 0 ≤ u < 1,

(3.37)

for some m > 0 and C0 > 0. By (1.1) we have that (in a weak sense):  f (u λ ) 1 ) ≤ λh(x) (1−u in  − p (ln 1−u p−1 λ λ) 1 = 0 on ∂. ln 1−u λ

(3.38)

By (3.37) we get that 0 ≤ λh(x)

m+ p−1 f (u λ ) − p−1 m ≤ λC f (u λ ) m in , 0 (1 − u λ ) p−1

and then by (3.36) sup λh(x)

λ∈[0,λ∗ )

If

mq p m+ p−1

>

N p,

f (u λ ) L q () < +∞ (1 − u λ ) p−1

∀1≤q <

mq p . m+ p−1

arguing as before, by elliptic regularity theory [18] on (3.38) we get that sup ln

λ∈[0,λ∗ )

1 ∞ < +∞, 1 − uλ

or equivalently (1.11) on u λ holds. We need to discuss the validity of mpq p > N ( p − 1 + m).



(3.39)  p+2 2 − p + γ ( p − 1) √

p N Assume first 1 < p ≤ 2. If p−1 < N ≤ N p , then q p = N ( p−1)− p and (3.39) is satisfied when  p m N < N p1 = 1+ ( p + 2 2 − p + γ ( p − 1)) . p−1 m+ p−1

Compute: Np − If γ ≤ 1 +

N p1

√ p + 2 2 − p + γ ( p − 1) 1 mp γ −1− . =− m+ p−1 2 − p + γ ( p − 1) m

then N p1 ≤ N p and (3.39) holds when √ p + 2 2 − p + γ ( p − 1) 1 mp γ −1− −. N < Np − m+ p−1 2 − p + γ ( p − 1) m 1 m,

123

294

D. Castorina et al.

Observing that

mp 2 2  2 − p + γ ( p − 1) Np = γ+ + m+ p−1 p−1 p−1 √ mp p + 2 2 − p + γ ( p − 1) 1 − 1+ γ −1− , m+ p−1 2 − p + γ ( p − 1) m

we get that, for γ ≤ 1 + m1 , (3.39) holds when 2 mp 1 2  γ+ − , 2 − p + γ ( p − 1) + γ − 1 − N < N# = + p+m−1 p−1 p−1 m where N # is defined in (1.10). When γ > 1 + m1 , for N ≤ N p (3.39) is automatically 2 satisfied holds for any N ≤ N p . If γ > 1 + m1 and N > N p , we have that q p = γ + p−1 + 2 √ 2 − p + γ ( p − 1) and (3.39) holds when p−1 mp 2 2  # Np < N < N = 2 − p + γ ( p − 1) . γ+ + p+m−1 p−1 p−1 Hence (3.39) holds for any N < N # also when γ > 1 + m1 and the case 1 < p ≤ 2 has been completely discussed. 2 2 √ Assume now p > 2. Then, q p = γ + p−1 + p−1 2 − p + γ ( p − 1) and (3.39) holds when mp 2 2  2 − p + γ ( p − 1) . γ+ + N < N# = p+m−1 p−1 p−1 Finally, we can conclude. Let N # be defined by (1.10), where γ and m are given in (1.9), and let N < N # . For any > 0, assumption (1.9) implies that (3.26) and (3.37) are valid for γ − > p−2 p−1 and m − > 0, respectively. When 1 < p < 2 the convexity of f (u) near u = 1 ensures that we can also assume γ − ≥ 0. For > 0 small, N is less than the critical dimension N # associated through (1.10) to γ − , m − . Hence, (1.11) holds and Theorem 1.3 is established.  

4 Compactness of the unstable branch In this section we will give the proof of Theorem 1.5, namely the compactness of the first unstable branch (with Morse index one) for the problem (1.1) under the assumptions (1.13)–(1.15). The proof, adapted from the arguments in [12,14], will make use of two Lemmata which will be proved in the Appendix for the sake of simplicity. Let p = 2. Let λn ∈ (0, λ∗ ) be a sequence and let u n be associated solutions of (1.1) of Morse index at most one, i.e. µ2,n ≥ 0 for any n ∈ N, where µ2,n = µ2 (−−λn h(x) f˙(u n )) is the second eigenvalue of the linearized operator at u n . We want to prove that any such √ 2(γ +2+2 γ ) # . sequence is compact in the sense that sup u n ∞ < 1 for N < N = γ n∈N

Let us argue by contradiction and assume that this sequence is not compact, i.e. there exists xn ∈  such that u n (xn ) = max u n (x) −→ 1. Suppose xn → p ∈  and set 

εn = 1 − u n (xn ) −→ 0. n→∞ Notice that λn

123

n→∞

f (1 − εn ) −→ ∞. n→∞ εn

(4.40)

Degenerate elliptic equations with singular nonlinearities

295

Indeed, if it were bounded we would have λn → 0 since being f nondecreasing, we would have

f (1−εn ) −→ εn n→∞

∞ by (1.2). Then,

0 ≤ λn h(x) f (u n (x)) ≤ λn h ∞ f (1 − εn ) ≤ Cεn ¯ where u is From elliptic regularity, up to a subsequence, we would have u n → u in C 1 (), a weak harmonic function such that u = 0 on ∂ and max u = 1, which is a contradiction. Let us introduce the following rescaled function: 1  2 εn 1 − u n xn + λn f (1−εn ) y , Un (y) ≡ εn



y ∈ n = 

 − xn εn λn f (1−εn )

1 . 2

The following Lemma holds: 1 (R N ), Lemma 4.1 We have that n → R N and there exists a subsequence Un → U in Cloc where U is a solution of the problem  U = h( p) in R N 1 U γ −1

U (y) ≥ U (0) = 1

in R N .

Moreover, there exists φn ∈ C0∞ () such that supp φn ⊂ B M > 0 and



 M

εn λn f (1−εn )

1 2

(xn ) for some

|∇φn |2 − λn h(x) f˙(u n )φn2 < 0

(4.41)



From this lemma (whose proof is in the Appendix) we get the existence of φn ∈ C0∞ () which is identically zero outside a small ball Brn (xn ), with rn → 0 as n → +∞, and the linearized operator is negative at φn . To conclude the proof we need the following estimate for Morse index one solutions, which says that the blow-up can essentially occur only along the maximum sequence xn : Lemma 4.2 Given 0 < δ < γ − 1, there exist C > 0 and n 0 ∈ N such that 1 − γ −δ

f (u n (x)) ≤ Cλn

|x − xn |

2 − γ −δ

(4.42)

for all x ∈  and n ≥ n 0 . From this lemma, thanks to estimate (4.42), we deduce that γ −δ−1

0 ≤ λn h(x) f (u n (x)) ≤ Cλn γ −δ h ∞ |x − xn |

2 − γ −δ

for any 0 < δ < γ − 1. Hence, λn h(x) f (u n (x)) is uniformly bounded in L s () for any 1 < s < γ N2 . From standard elliptic regularity theory we have that u n is uniformly bounded in W 2,s () for any 1 < s < γ N2 . By the Sobolev embedding Theorem u n is uniformly ¯ for any β ∈ (0, 2 − 2 ). Then, up to a subsequence, we have that bounded also in C 0,β () γ ¯ for any β ∈ (0, 2 − 2 ). u n u 0 weakly in H 1 () and u n → u 0 strongly in C 0,β () 0

γ

In case λn → 0, u 0 is a H01 ()-weak harmonic function and, by the Maximum Principle, has to vanish in . By uniform convergence, it holds that

123

296

D. Castorina et al.

u 0 ( p) = max u 0 = lim max u n = 1, 

n→∞ 

p = lim xn n→∞

(4.43)

and a contradiction arises. Hence, λn → λ > 0 and (4.43) implies p ∈ , since u 0 = 0 on ∂. By (4.42), it − 2 ¯ p}). Since follows that f (u 0 ) ≤ C|x − p| γ −δ in \{ p} and f (u n ) → f (u 0 ) in Cloc (\{ 2N

H01 (\{ p}) = H01 () and f (u 0 ) ∈ L N +2 (), then u 0 is an Hölderian H01 ()-weak solution of: ⎧ ⎨ −u 0 = λh(x) f (u 0 ) in  0 ≤ u0 ≤ 1 in  ⎩ u0 = 0 on ∂. Now, consider the first eigenvalue of the linearized operator at u 0 , namely ⎛ ⎞  ⎝ |∇φ|2 − λh(x) f˙(u 0 (x))φ 2 ⎠ . inf

µ1,λ (u 0 ) ≡ µ1 (− − λh(x) f˙(u 0 )) = φ∈C0∞ (): φ 2 =1



For convex nonlinearities f (u), uniqueness holds in the class of semi-stable H01 ()-weak solutions of (1.1) (see [14]). If µ1,λ (u 0 ) ≥ 0, we deduce from Theorem 1.2 that u 0 ≡ u λ , for some λ ∈ [0, λ∗ ]. But from Theorem 1.3 we know that max u λ < 1 for any λ ∈ [0, λ∗ ], and this contradicts max u 0 = 1.





So, we are left with the case µ1,λ (u 0 ) < 0, which means that there exists φ0 ∈ C0∞ () such that  |∇φ0 |2 − λh(x) f˙(u 0 )φ02 < 0 

But from (4.41) we already had the existence of φn ∈ C0∞ () such that supp φn ⊂ Brn (xn ) with rn −→ 0 and n→∞



|∇φn |2 − λn h(x) f˙(u n )φn2 < 0.



We want to replace φ0 with a truncated function φδ with δ > 0 small enough so that  |∇φδ |2 − λh(x) f˙(u 0 )φδ2 < 0 

and φδ ≡ 0 in Bδ 2 ( p) ∩ . So, for n large by Fatou’s Lemma it holds  |∇φδ |2 − λn h(x) f˙(u n )φδ2 < 0. 

Since φn and φδ have disjoint compact supports, from the variational characterization of the second eigenvalue we would have a contradiction: µ2,n < 0. This ends the proof. Take δ > 0 and set φδ = χδ φ0 with ⎧ if |x − p| ≤ δ 2 ⎪ ⎨0 log |x− p| if δ 2 ≤ |x − p| ≤ δ χδ (x) ≡ 2 − log δ ⎪ ⎩ 1 if |x − p| ≥ δ

123

Degenerate elliptic equations with singular nonlinearities

By Fatou’s Lemma we have



lim inf δ→0

λh(x) f˙(u 0 )φδ2 ≥



297



λh(x) f˙(u 0 )φ02 ,



whereas for the gradient term we have     2 2 2 2 2 |∇φδ | = φ0 |∇χδ | + χδ |∇φ0 | + 2 χδ φ0 ∇χδ ∇φ0 . 





We have the following estimates:  0 ≤ φ02 |∇χδ |2 ≤ φ0 2∞ 

and





δ 2 ≤|x− p|≤δ

C 1 ≤ 2 2 |x − p| log δ log 1δ

        2 χδ φ0 ∇χδ ∇φ0  ≤ 2 φ0 ∞ ∇φ0 ∞   log 1δ   

which give

B1 (0)



 |∇φδ |2 −→ δ→0



In conclusion,



lim sup δ→0

1 , |x|

|∇φδ |2 − λh(x) f˙(u 0 )φδ2 ≤



|∇φ0 |2 . 



|∇φ0 |2 − λh(x) f˙(u 0 )φ02 < 0.



For δ > 0 small enough, φδ is what we were searching for and this concludes the proof.   5 Appendix 5.1 Embedding of the space Au For p ≥ 2, we will show below that the space Au is compactly embedded in L 2 (), as it will follow by the weighted Sobolev estimates proved in [9]. The proof follows closely Theorem 9.16 in [2]. For the reader’s convenience we give the details: Lemma 5.1 Let p ≥ 2 and u be a solution of (1.1). For any 1 ≤ q < there exists C > 0 so that  φ 2L q ()  C |∇u| p−2 |∇φ|2 , ∀ φ ∈ Au .

2N ( p−1) (N −2)( p−1)+2( p−2)

(5.44)



Moreover, the embedding Au ⊂ Proof Since q¯ =

L q ()

2N ( p−1) (N −2)( p−1)+2( p−2)

is compact for any 1 ≤ q <

2N ( p−1) (N −2)( p−1)+2( p−2) .

> 2, for 2 < q < q¯ (5.44) follows by Theorem 2.2 in

[10]. Then, by Hölder inequality (5.44) follows also for 1 ≤ q ≤ 2.

123

298

D. Castorina et al.

Since q < q, ¯ fix δ > 0 so that q + δ < q, ¯ and set 1 (1 − α) =α+ , q q +δ

for 0 < α  1.

Now, let ω ⊂⊂  and consider h such that |h| < dist(ω, c ). By interpolation we have for φ ∈ L 2 () . φ(x + h) − φ(x) L q (ω) = τh (φ) − φ L q (ω)  τh (φ) − φ αL 1 (ω) τh (φ) − φ 1−α L q+δ (ω) Now we have that (see [9])

1 |∇u| p−2

∈ L 1 () and consequently Au ⊂ W 1,1 ,

with 

⎛ ⎞1 ⎛ 2   p−2 2⎠ ⎝ |∇φ|  |∇u| |∇φ| ·⎝







⎞1 2 1 ⎠ . |∇u| p−2

Recall that the | · |-norm and the · -norm, as defined in the Introduction, are equivalent. Therefore, for every φ ∈ Au with φ ≤ 1 we have τh (φ) − φ L 1 (ω)  |h| ∇φ L 1 ()  C0 |h| so that, exploiting (5.44) we get  1−α τh (φ) − φ L q (ω)  C0α |h|α 2 φ L q+δ ()  C|h|α . Since 1

φ L q (\ω)  φ L q+δ (\ω) |\ω| q

1 − q+δ

1

 C |\ω| q

1 − q+δ



for \ω sufficiently small, by Corollary 4.27 in [2] we deduce that the unit ball of Au is a compact set in L q (). Then, the embedding Au ⊂ L q () is compact for any 1 ≤ q < q. ¯   For 1 < p < 2, as already remarked in the Introduction, Au ⊂ H01 (). Since H01 () ⊂ L q () compactly for any 1 ≤ q < N2N −2 , by Lemma 5.1 we deduce Lemma 5.2 Let u be a solution of (1.1). The embedding Au ⊂ L 2 () is compact. 5.2 Proof of Theorem 1.1 • Step 1. Existence of a first eigenfunction Let us first note that f˙(u) ∈ L ∞ (), so that  2 φ − λ h(x) f˙(u)φ 2 

is bounded from below, for any φ ∈ Au with µ1 (u) =

123

inf

φ∈Au \{0}

Ru (φ) > −∞,





= 1. Therefore, µ1 (u) is well defined:

φ 2 − λ  h(x) f˙(u)φ 2

Ru (φ) = . 2 φ 2

Degenerate elliptic equations with singular nonlinearities

Consider now a minimizing sequence φn ∈ Au ,

299



2  φn

= 1, with

Ru (φn ) → µ1 (u) as n → +∞. Since f˙(u) ∈ L ∞ (), we have that sup φn < +∞.

n∈N

Therefore, up to a subsequence, we get that φn φ1

weakly in Au

and by Lemma 5.2 φn → φ1

strongly in L 2 ().

Now, the term λ  h(x) f˙(u)φ 2 is continuous in L 2 () and · is weakly lower semi

continuous in Au . Therefore, φ1 ∈ Au is so that  φ12 = 1 and Ru (φ1 ) ≤ µ1 (u). Hence, µ1 (u) is attained at φ1 . • Step 2. Every minimizer is positive (or negative) almost everywhere We show that φ1 > 0 (or φ1 < 0) in \ Z u , where Z u = {∇u = 0} is a zero measure set (see [9]). Assume

φ1 2 − λ  h(x) f˙(u)φ12

2 µ1 (u) =  φ1 so that for ψ ∈ Au it follows



< φ1 , ψ > −λ

h(x) f˙(u)φ1 ψ = µ1 (u)



Taking

φ1±

 φ1 ψ.

(5.45)



as a test function in (5.45), we get that   ± 2 ± 2 ˙ φ1 − λ h(x) f (u)(φ1 ) = µ1 (u) (φ1± )2 , 



φ1±

also minimizes Ru (φ). Then, there holds   < φ1± , ψ > −λ h(x) f˙(u)φ1± ψ = µ1 (u) φ1± ψ, ∀ ψ ∈ Au .

showing that



(5.46)



The differential operator in (5.46) is nondegenerate in \Z u . Moreover, by [9] \Z u is connected and |Z u | = 0, as already recalled. Therefore, the Strong Maximum Principle holds in \Z u , and φ1± is smooth in \Z u by standard regularity results. We have that either φ1± > 0 in \Z u or φ1± = 0 in \Z u . Indeed, C = {x ∈ \Z u : φ1± > 0} is clearly an open set. By the classic Strong Maximum Principle exploited in \Z u , we have ∂C ⊂ Z u ∪ ∂, and then, C = C¯ ∩ (\Z u ) is a closed set in the relative toplogy of \ Z u . Since \ Z u is connected, the set C either is empty or coincides with \ Z u . Since φ1  = 0 a.e. in , then either φ1 > 0 or φ1 < 0 in \ Z u .

123

300

D. Castorina et al.

• Step 3. The first eigenspace is one-dimensional Let φ1 be a first eigenfunction which can be assumed positive a.e. in : φ1 > 0 in \ Z u , by means of Step 2. Let now φ any other first eigenfunction. Since by Step 2 φ has constant sign, let us consider for example the case φ > 0 in \ Z u . Set β¯ = sup{β ≥ 0 : φ − βφ1 ≥ 0 a.e. in } < +∞. ¯ 1 is still a minimizer for Ru , by Step 2 we have that Since by linearity φ − βφ ¯ 1 = 0 a.e. in . ¯ 1 > 0 or φ − βφ either φ − βφ ¯ 1 > 0, we have that φ − (β¯ + )φ1 is still positive on a subset of positive If φ − βφ measure, for small > 0. As a minimizer, φ − (β¯ + )φ1 has constant sign and then, ¯ Therefore, φ = βφ ¯ 1 a.e. in . φ − (β¯ + )φ1 > 0 a.e. in , against the definition of β. 5.3 Proof of Lemma 4.1 The proof follows the arguments in [12,14], where similar results are proved for the nonlin1 ¯ earity f (u) = (1−u) 2 . Suppose that x n → p ∈  and consider the rescaled function around xn : Un (y) ≡ where βn =



εn λn f (1−εn )

1



2

1 − u n (xn + βn y) , εn

y ∈ n =

 − xn , βn

−→ 0 by means of (4.40). The function Un verifies

n→∞

n +βn y) Un = h(x f (1−εn ) f (1 − εn Un (y)) U (y) ≥ U (0) = 1

in n in n .

(5.47)

First of all, we need to prove that n −→ R N . It will be sufficient to prove that n→∞

βn dn−1 −→ 0, n→∞

where dn = dist (xn , ∂). Indeed, arguing by contradiction and up to a subsequence, assume that βn2 dn−2 → δ > 0. Obviously this implies that dn → 0 for n → ∞. Introduce the following rescaling Wn (y) ≡

1 − u n (xn + dn y) , εn

y ∈ An =

 − xn . dn

Since dn → 0 we have that An → T , where T is a hyperspace containing 0 so that d(0, ∂ T ) = 1. The function Wn satisfies the following equation λn dn2 h(xn + dn y) f (1 − εn Wn (y)) εn λn f (1 − εn )dn2 f (1 − εn Wn (y)) = h(xn + dn y) εn f (1 − εn ) ⎛ 1 1 ⎞ λn f (1 − εn )dn2 ⎝ f (1 − εn Wn (y))εnγ −1 Wnγ −1 ⎠ h(xn + dn y) = . 1 1 εn f (1 − εn )εnγ −1 Wnγ −1

Wn (y) =

123

Degenerate elliptic equations with singular nonlinearities

301

From the hypothesis and condition (1.14), for any sufficiently large n we get ⎛ 1 1 ⎞ λn f (1 − εn )dn2 ⎝ f (1 − εn Wn (y))εnγ −1 Wnγ −1 ⎠ 2 0≤ h(xn + dn y) ≤ c0 h ∞ < ∞. 1 εn δ f (1 − εn )εnγ −1 This means that the function Wn satisfies ⎧ ⎨Wn = ⎩

hn 1 γ −1

in An

Wn

Wn (y) ≥ C > 0

in An

with sup h n ∞ < ∞ and C = Wn (0) = 1. n∈N

Recall Lemma 4.1 in [14] (written there for the case γ = 23 ): Lemma 5.3 Let h n be a function on a smooth bounded domain An in R N . Let Wn be a solution of: ⎧ h n (x) ⎪ ⎪ in An , ⎪ Wn = 1 ⎨ γ −1 Wn ⎪ Wn (y) ≥ C > 0 in An , ⎪ ⎪ ⎩ Wn (0) = 1, for some C > 0. Assume that sup h n ∞ < +∞ and An → Tµ as n → +∞ for some n∈N

µ ∈ (0, +∞), where Tµ is an hyperspace so that 0 ∈ Tµ and dist (0, ∂ Tµ ) = µ. Then, either inf Wn ≤ C or inf ∂ν Wn ≤ 0, where ν is the unit outward normal of An .

∂ An ∩B2µ (0)

∂ An ∩B2µ (0)

Since Wn |∂ An ≡

1 −→ εn n→∞

∞ and Hopf Lemma provides ∂ν Wn > 0 on ∂ An , a contradiction

arises by means of Lemma 5.3. Hence, we have shown that βn dn−1 −→ 0, i.e. n → R N . n→∞ We now want to prove that there exists a subsequence {Un }n∈N such that Un → U in 1 (R N ), where U is a solution of the problem Cloc ⎧ ⎨U = h( p) in R N 1 (5.48) U γ −1 ⎩U (y) ≥ U (0) = 1 in R N . Fix R > 0 and, for n large, decompose Un = Un1 + Un2 , where Un2 satisfies  Un2 = Un in B R (0) in ∂ B R (0). Un2 = 0 By the Eq. (5.47) and the condition (1.14) we get that on B R (0):    f (1 − εn Un )   ≤ c0 h ∞ < ∞, 0 ≤ Un (y) ≤ h ∞  f (1 − εn )  and then, standard elliptic regularity theory gives that Un2 is uniformly bounded in C 1,β (B R (0)), β ∈ (0, 1). Up to a subsequence, we have that Un2 → U 2 in C 1 (B R (0)). Since Un1 = Un ≥ 1 on ∂ B R (0), by harmonicity Un1 ≥ 1 in B R (0). Through Harnack inequality, we have: 1 1 1 2 2 sup Un ≤ C R inf Un ≤ C R Un (0) = C R (1 − Un (0)) ≤ C R 1 + sup |Un (0)| < ∞. B R (0) 2

B R (0) 2

n

123

302

D. Castorina et al.

Hence, Un1 is uniformly bounded in C 1,β (B R (0)), β ∈ (0, 1). Up to a subsequence, we get 4

that Un1 → U 1 in C 1,β (B R (0)) for any R > 0. Up to a diagonal process and a further 4

1 (R N ). subsequence, we can assume that Un → U := U1 + U2 in Cloc Notice that, by the condition (1.15) we have: 1

f (1 − εn Un )) f (1 − εn Un )Unγ −1 h(xn + βn y) = h(xn + βn y) f (1 − εn ) f (1 − εn )

1 1 γ −1

Un

−→

n→∞

h( p) 1

U γ −1

0 (R N ). This means that U is a solution of (5.48). in Cloc The following unstability property, a special case of a more general result in [13], will be crucial:

Theorem 5.4 Let U be a solution of (5.48). Then, U is linearly unstable:     h( p) 2 1 2 ∞ N 2 µ1 (U ) = inf |∇φ| − φ = 1 < 0, γ φ : φ ∈ C 0 (R ), γ −1 U γ −1 provided 2 ≤ N < N # =

√ 2(γ +2+2 γ ) . γ

Theorem 5.4 provides the existence of φ ∈ C0∞ (R N ) such that  1 h( p) 2 |∇φ|2 − γ φ < 0. γ − 1 U γ −1 RN

Define − N 2−2

φn (x) ≡ βn

φ

x − xn βn



∈ C0∞ ().

Condition (1.13) rewrites as: lim

u→1−

f (u) f¨(u) (ln f˙) (u) = lim = γ, 2 − u→1 (ln f ) (u) ( f˙(u))

which implies by (1.2) and L’Höpital rule: lim

u→1−

ln f˙(u) = γ. ln f (u)

Hence, since γ > 1 we have that lim

u→1−

f˙(u) = lim f (u)γ −1+o(1) = +∞, f (u) u→1−

where o(1) → 0 as u → 1− . This means that by L’Höpital rule: lim

u→1−

(1 − u) f˙(u) 1−u = lim f (u) = lim f (u) u→1− u→1− f˙(u)

1 f (u) f¨(u) ( f˙(u))2

−1

1

1

=

1 . γ −1

(5.49)

Observe that by (1.15) and (5.49) it follows that 1 εn f˙(1 − εn Un ) 1 εn Un f˙(1 − εn Un ) f (1 − εn Un )εnγ −1 Unγ −1 1 → = γ γ 1 f (1 − εn ) f (1 − εn Un ) γ − 1 U γ −1 Unγ −1 εnγ −1 f (1 − εn ) (5.50) in Cloc (R N ) as n → +∞, in view of Un → U locally uniformly.

123

Degenerate elliptic equations with singular nonlinearities

303

Let us now evaluate the linearized operator at u n on φn :   εn f˙(1 − εn Un ) 2 φ |∇φn |2 − λn h(x) f˙(u n )φn2 = |∇φ|2 − h(xn + βn y) f (1 − εn )



n





|∇φ|2 − RN

1 γ −1

 RN

h( p) U

γ γ −1

φ 2 < 0 as n → +∞

by means of φ ∈ C0∞ (R N ) and (5.50). Hence, for any sufficiently large n we have:  |∇φn |2 − λn h(x) f˙(u n )φn2 < 0 

with supp φn ⊂ B Mβn (xn ), for some M > 0. This concludes the proof.

 

5.4 Proof of Lemma 4.2 Fix 0 < δ < γ − 1. Let us prove estimate (4.42): there exist C > 0 and n 0 ∈ N such that 1 − γ −δ

f (u n (x)) ≤ Cλn

|x − xn |

2 − γ −δ

for any x ∈  and for any n ≥ n 0 . Let us argue by contradiction and assume that (4.42) does not hold. Up to a subsequence, we get the existence of a minimizing sequence yn ∈  such that:  − 1 − 1 − 2 − 2 λn γ −δ f (u n (yn ))−1 |xn − yn | γ −δ = λn γ −δ min f (u n (x))−1 |x − xn | γ −δ −→ 0. n→∞

y∈

(5.51) This means that f (u n (yn )) → +∞ as n → +∞, so that we have blow-up along the sequence yn , i.e. µn ≡ 1 − u n (yn ) −→ 0. n→∞

By (5.49) and L’Höpital rule we get that lim

u→1−

ln f (u) ln

1 1−u

=

1 , γ −1

and then, we have: C 1−u for some C = Cδ > 0. Hence, this yields to: f γ −δ−1 (u) ≤

in [0, 1)

µn f (1 − µn )γ −1−δ ≤ C.

(5.52)

From (5.51)–(5.52), we have that βˆn := λn

1 f (1 − µn ) 1 = (λnγ −δ f (1 − µn ))γ −δ → ∞ as n → +∞. µn µn f (1 − µn )γ −1−δ

¯ as n → +∞. Consider the following rescaled function Assume that yn → q ∈  1 − u n (yn + βˆn y)  − yn ˆn = Uˆ n (y) ≡ , y∈ µn βˆn

123

304

D. Castorina et al.

We want to prove the following crucial convergence: βˆn lim = lim n→∞ |x n − yn | n→∞

1

µn2 1

1

λn2 f (1 − µn ) 2 |xn − yn |

= 0.

(5.53)

From (5.51) and (5.52), we have that   βˆn2 µn 1 = = µn f (1−µn )γ −1−δ |xn − yn |2 λn f (1−µn )|xn − yn |2 λn f (1−µn )γ −δ |xn − yn |2 C ≤ −→ 0. λn f (1 − µn )γ −δ |xn − yn |2 n→∞ Namely, (5.53) holds. We enumerate now several properties of the crucial choice 1 −1 Rn := βˆn 2 |xn − yn | 2 : (a) Rn −→ ∞ from (5.53); n→∞

1

1

1

1

(b) Rn βn = βn2 |xn − yn | 2 ≤ βn2 (diam ) 2 −→ 0; n→∞ − 1  1 2 |xn −yn | R n βn − 21 2 (c) |xn −yn | = βn |xn − yn | = −→ 0. βn n→∞

ˆ n ∩ B Rn (0) we have that: Let us now focus our attention on the function Uˆ n . If y ∈  ˆ 1−u (y + β y) n n n ≥ 1; or u n (yn ) < u n (yn + βˆn y) either u n (yn + βˆn y) ≤ u n (yn ), which implies 1−u n (yn ) and assumption (1.15) implies 1

(1 − u n (yn + βˆn y)) γ −1 f (u n (yn + βˆn y)) (1 − u n (yn ))

1 γ −1

f (u n (yn ))



1 , c0

or equivalently 1 − u n (yn + βˆn y) 1 ≥ γ −1 1 − u n (yn ) c0



f (u n (yn ) f (u n (yn + βˆn y))

γ −1

.

In the latter situation, from the definition of yn we have: 2

2

f (u n (yn ))|xn − yn | γ −δ ≥ f (u n (yn + βˆn y))|yn + βˆn y − xn | γ −δ and, since (b) and (c) imply that for any n ≥ n 0 : βˆn |y| βˆn Rn |yn + βˆn y − xn | 1 ≥1− ≥1− ≥ , |y| ≤ Rn , |yn − xn | |xn − yn | |xn − yn | 2 we get that  2 γ −1 γ −1 |yn + βˆn y − xn | γ −δ f (u n (yn )) 1 1 − u n (yn + βˆn y) 1 ≥ γ −1 ≥ γ −1 1 − u n (yn ) |yn − xn | f (u n (yn + βˆn y)) c0 c0 γ −1 1 1 2 ≥ γ −1 ( ) γ −δ . 2 c 0

ˆ n ∩ B Rn (0): We finally get that for any n ≥ n 0 and any y ∈   ! 1 − u n (yn + βˆn y) 1 1 2 γγ −1 −δ ˆ ≥ D0 = min 1, γ −1 ( ) Un (y) = 1 − u n (yn ) 2 c0

123

Degenerate elliptic equations with singular nonlinearities

305

Setting dˆn = dist (yn , ∂), consider the rescaled function     1 − u n yn + dˆn y ˆ n ∩ B Rn (0) . Wˆ n (y) ≡ , y ∈ Aˆ n = βˆn dˆn−1  µn Since Wˆ n (y) ≥ D0 in Aˆ n , we can apply again Lemma 5.3 to get: βˆn dˆn−1 −→ 0, i.e. n→∞

ˆ n ∩ B Rn (0) → R N as n → ∞. Now, proceeding as in the proof of Lemma 4.1, we get that  1 (R N ), where U ˆ solves Uˆ n → Uˆ in Cloc  in R N Uˆ = h(q) 1 Uˆ γ −1

Uˆ (y) ≥ D0

in R N .

Moreover, there exists ψn ∈ C0∞ (R N ) such that  |∇ψn |2 − λn h(x) f (u n )ψn2 < 0 

with suppψn ⊂ B M βˆn (yn ) for some M > 0. But from Lemma 4.1 we already had φn ∈ C0∞ (R N ) with the same property and such that supp φn ⊂ B M  βn (xn ) for some M  > 0. Since the nonlinearity f (u) in non-decreasing and 1 − εn = u n (xn ) ≥ u n (yn ) = 1 − µn , by (5.53) we get that:

1

εn2 1

1

λn2 f (1 − εn ) 2 |xn − yn |

=

εn µn

1 2

f (1 − µn ) f (1 − εn )

1

1

µn2

2

1

1

λn2 f (1 − µn ) 2 |xn − yn |

1



µn2 1

1

λn2 f (1 − µn ) 2 |xn − yn |

−→ 0.

n→∞

This means that φn and ψn have disjoint compact support for n large, which contradicts the Morse index-one property of the solutions u n and concludes ths proof.  

References 1. Boccardo, L., Murat, F.: Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal. 19(6), 581–597 (1992) 2. Brezis, H.: Analyse fonctionnelle. Collection Mathématiques Appliquées pour la Maîtrise. Masson, Paris (1983) 3. Brezis, H., Cazenave, T., Martel, Y., Ramiandrisoa, A.: Blow up for u t − u = g(u) revisited. Adv. Differ. Equ. 1(1), 73–90 (1996) 4. Brezis, H., Vazquez, J.L.: Blow-up solutions of some nonlinear elliptic problems. Rev. Mat. Univ. Compl. Madrid 10(2), 443–469 (1997) 5. Cabré, X.: Conference Talk, Rome (2006) 6. Cabré, X., Capella, A.: Regularity of radial minimizers and extremal solutions of semilinear elliptic equations. J. Funct. Anal. 238(2), 709–733 (2006) 7. Cabré, X., Sanchón, M.: Semi-stable and extremal solutions of reactions equations involving the p-Laplacian. Commun. Pure Appl. Anal. 6(1), 43–67 (2007) 8. Crandall, M.G., Rabinowitz, P.H.: Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems. Arch. Ration. Mech. Anal. 58(3), 207–218 (1975) 9. Damascelli, L., Sciunzi, B.: Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations. J. Differ. Equ. 206(2), 483–515 (2004)

123

306

D. Castorina et al.

10. Damascelli, L., Sciunzi, B.: Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-Laplace equations. Calc. Var. Partial Differ. Equ. 25(2), 139–159 (2006) 11. Di Benedetto, E.: C 1+α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7(8), 827–850 (1983) 12. Esposito, P.: Compactness of a nonlinear eigenvalue problem with a singular nonlinearity. Commun. Contemp. Math. 10(1), 17–45 (2008) 13. Esposito, P.: Linear instability of entire solutions for a class of non-autonomous elliptic equations. In: Proceedings of Royal Society Edinburgh Sect. A (to appear) 14. Esposito, P., Ghoussoub, N., Guo, Y.: Compactness along the branch of semi-stable and unstable solutions for an elliptic problem with a singular nonlinearity. Comm. Pure Appl. Math. 60(12), 1731–1768 (2007) 15. García-Azorero, J., Peral, I.: On an Emden-Fowler type equation. Nonlinear Anal. 18(11), 1085–1097 (1992) 16. García-Azorero, J., Peral, I., Puel, J.P.: Quasilinear problems with exponential growth in the reaction term. Nonlinear Anal. 22(4), 481–498 (1994) 17. Ghoussoub, N., Guo, Y.: On the partial differential equations of electrostatic MEMS devices: stationary case. SIAM J. Math. Anal. 38(5), 1423–1449 (2006/2007) 18. Grenon, N.: L r estimates for degenerate elliptic problems. Potential Anal. 16(4), 387–392 (2002) 19. Joseph, D.D., Lundgren, T.S.: Quasilinear Dirichlet problems driven by positive sources. Arch. Ration. Mech. Anal. 49, 241–2689 (1972/1973) 20. Lieberman, G.M.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12(11), 1203–1219 (1988) 21. Mignot, F., Puel, J.P.: Sur une classe de problèmes non linéaires avec non linéarité positive, croissante, convexe. Comm. Partial Differ. Equ. 5(8), 791–836 (1980) 22. Nedev, G.: Regularity of the extremal solution of semilinear elliptic equations. C. R. Acad. Sci. Paris Sér I Math. 330(11), 997–1002 (2000) 23. Pelesko, J.A., Bernstein, D.H.: Modeling MEMS and NEMS. Chapman Hall and CRC Press, London (2002) 24. Sanchón, M.: Boundedness of the extremal solution of some p-Laplacian problems. Nonlinear Anal. 67(1), 281–294 (2007) 25. Sanchón, M.: Regularity of the extremal solution of some nonlinear elliptic problems involving the p-Laplacian. Potential Anal. 27(3), 217–224 (2007) 26. Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ. 51(1), 126–150 (1984) 27. Trudinger, N.S.: Linear elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa (3) 27, 265–308 (1973) 28. Ye, D., Zhou, F.: Boundedness of the extremal solution for semilinear elliptic problems. Commun. Contemp. Math. 4(3), 547–558 (2002)

123