Weak solutions of semilinear elliptic equations with Leray

Weak solutions of semilinear elliptic equations with Leray-Hardy potentials and measure data Huyuan Chen Laurent V eron y Abstract We study existence ...

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Weak solutions of semilinear elliptic equations with Leray-Hardy potential and measure data Huyuan Chen, Laurent Veron

To cite this version: Huyuan Chen, Laurent Veron. Weak solutions of semilinear elliptic equations with Leray-Hardy potential and measure data. Mathematics in Engineering, AIMS, 2019, 1 (3), pp.391-418. �hal02013601v2�

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Weak solutions of semilinear elliptic equations with Leray-Hardy potentials and measure data Huyuan Chen∗ Laurent V´ eron



Abstract We study existence and stability of solutions of −∆u +

µ u + g(u) = ν in Ω, u = 0 on ∂Ω, |x|2 2

where Ω is a bounded, smooth domain of RN , N ≥ 2, containing the origin, µ ≥ − (N −2) 4 is a constant, g is a nondecreasing function satisfying some integral growth assumption and the weak ∆2 -condition, and ν is a Radon measure in Ω. We show that the situation differs depending on whether the measure is diffuse or concentrated at the origin. When g is a power we introduce a capacity framework to find necessary and sufficient conditions for solvability. Key Words: Leray-Hardy Potential; Radon Measure; Capacity; Weak solution. MSC2010: 35B44, 35J75.

Contents 1 Introduction

2

2 L1 data

8

3 The 3.1 3.2 3.3 3.4 3.5

subcritical case The linear equation . Dirac masses . . . . Measures in Ω∗ . . . Proof of Theorem B Proof of Theorem C

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12 12 14 16 18 22

∗ Department of Mathematics, Jiangxi Normal University, Nanchang 330022, China. E-mail: [email protected] † Laboratoire de Math´ematiques et Physique Th´eorique, Universit´e de Tours, 37200 Tours, France. E-mail: [email protected]

1

2

Leray-Hardy equations with absorption

4 The supercritical case 4.1 Reduced measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

22 22

Introduction

Schr¨odinger operators with singular potentials under the form u 7→ H(u) := −∆u + V (x)u

x ∈ R3

(1.1)

are at the core of the description of many aspects of nuclear physics. The associated energy, the sum of the momentum energy and the potential energy, endows the form Z  1 H(u) = |∇u|2 + V (x)u2 dx. (1.2) 2 R3 In classical physics V (x) = −κ|x|−1 (κ > 0) is the Coulombian potential and H is not bounded from below and there is no ground state. In quantum physics there are reasons arising from its mathematical formulation which leads, at least in the case of the hydrogen atom, to V (x) = −κ|x|−2 (κ > 0) and H is bounded from below provided κ ≥ − 14 . Furthermore, a form of the uncertainty principle is Hardy’s inequality Z Z 1 u2 |∇u|2 dx ≥ dx for all u ∈ C0∞ (R3 ). (1.3) 2 4 |x| 3 3 R R The meaning of this inequality is that if u is localized close to a point 0 (i.e., the right side term is large), then its momentum has to be large (i.e., the left side term is large), and the power |x|−2 is the consequence of a dimensional analysis (see [19], [20]). Such potential is often called a Leray-Hardy potential. The study of the mathematical properties of generalisations of the operator H in particular in N-dimensional domains generated hundred of publications in the last thirty years. In this article we define the Schr¨odinger operator L in RN by µ , |x|2

(1.4)

(N − 2)2 . 4

(1.5)

Lµ := −∆ + where µ is a real number satisfying µ ≥ µ0 := − 2

Note that (N −2) achieves the value 41 when N = 3. Let Ω ⊂ RN (N ≥ 2) be a bounded, smooth 4 domain containing the origin and g : R → R is a continuous nondecreasing function such that g(0) ≥ 0, we are interested in the nonlinear Poisson equation ( Lµ u + g(u) = ν in Ω, (1.6) u=0 on ∂Ω,

3

Leray-Hardy equations with absorption

where ν is a Radon measure in Ω. The reason for a measure framework is that the problem is essentially trivial if ν ∈ L2 (Ω), more complicated if ν ∈ L1 (Ω) and very rich if ν is a measure. When µ = 0, problem (1.6) reduces to ( −∆u + g(u) = ν in Ω, (1.7) u=0 on ∂Ω, which has been extensively studied by numerous authors in the last 30 years. A fundamental contribution is due to Brezis [4], Benilan and Brezis [2], where ν is bounded and the function g : R → R is nondecreasing, positive on (0, +∞) and satisfies the subcritical assumption in dimension N ≥ 3: Z +∞

N

(g(s) − g(−s))s−1− N −2 ds < +∞.

(1.8)

1

They obtained the existence, uniqueness and stability of weak solutions for the problem. When N = 2, V` azquez [26] introduced the exponential orders of growth of g defined by   Z ∞ −bt β+ (g) = inf b > 0 : g (t) e dt < ∞ , 1 (1.9)   Z −1 bt β− (g) = sup b < 0 : g (t) e dt > −∞ , −∞

and proved that if ν is any bounded measure in Ω with Lebesgue decomposition X ν = νr + αj δaj , j∈N

where νr is part of ν with no atom, aj ∈ Ω and the αj ∈ R satisfy 4π 4π ≤ αj ≤ , β− (g) β+ (g)

(1.10)

then (1.7) admits a (unique) weak solution. Later on, Baras and Pierre [1] studied (1.7) when g(u) = |u|p−1 u for p > 1 and they discovered that if p ≥ NN−2 the problem is well posed if and p only if ν is absolutely continuous with respect to the Bessel capacity c2,p0 with p0 = p−1 . It is a well established fact that, by the improved Hardy inequality [9] and Lax-Milgram theorem, the non-homogeneous problem Lµ u = f

in

Ω,

u=0

on ∂Ω,

(1.11)

with f ∈ L2 (Ω), has a unique solution in H01 (Ω) if µ > µ0 , or in a weaker space H(Ω) if µ = µ0 [18]. When f ∈ / L2 (Ω) a natural question is to find sharp conditions on f for the existence or nonexistence of solutions of (1.11) and the difficulty comes from the fact that the Hardy term |x|−2 u may not be locally integrable in Ω. An attempt done by Dupaigne in [18] is to consider problem (1.11) when µ ∈ [µ0 , 0) and N ≥ 3 in the sense of distributions Z Z uLµ ξ dx = f ξ dx, ∀ ξ ∈ Cc∞ (Ω). (1.12) Ω



4

Leray-Hardy equations with absorption

The corresponding semi-linear problem is studied in [3] with this approach. We adopt here a different point of view in using a different notion of weak solutions. It is known that the equation Lµ u = 0 in RN \ {0} has two distinct radial solutions:   |x|τ− (µ) if µ > µ0 ,   Φµ (x) = and Γµ (x) = |x|τ+ (µ) , N −2  |x|− 2 ln 1 if µ = µ , 0 |x| with N −2 τ− (µ) = − − 2

r

(N − 2)2 +µ 4

N −2 and τ+ (µ) = − + 2

r

(N − 2)2 + µ. 4

In the remaining of the paper and when there is no ambiguity, we put τ+ = τ+ (µ), τ+0 = τ+ (µ0 ), τ− = τ− (µ) and τ−0 = τ− (µ0 ). It is noticeable that identity (1.12) cannot be used to express that Φµ is a fundamental solution, i.e. f = δ0 since Φµ is not locally integrable if µ ≥ 2N . Recently, Chen, Quaas and Zhou found in [12] that the function Φµ is the fundamental solution in the sense that it solves Z Φµ L∗µ ξ dγµ (x) = cµ ξ(0) for all ξ ∈ C01,1 (RN ), (1.13) RN

where L∗µ ξ = −∆ξ − 2

dγµ (x) = Γµ (x)dx, and

( √ 2 µ − µ0 | S N −1 | cµ = N −1 S

τ+ hx, ∇ξi, |x|2

if µ > µ0 , if µ = µ0 .

(1.14)

(1.15)

With the power-absorption nonlinearity in Ω∗ = Ω \ {0}, the precise behaviour near 0 of any positive solution of Lµ u + up = 0 in D0 (Ω∗ ) (1.16) is given in [22] when p > 1. In this paper it appears a critical exponent p∗µ = 1 −

2 τ−

(1.17)

with the following properties: if p ≥ p∗µ any solution of (1.16) can be extended by continuity as a solution in D0 (Ω). If 1 < p < p∗µ any positive solution of (1.16) either satisfies 2

lim |x| p−1 u(x) = `,

x→0

(1.18)

where ` = `N,p,µ > 0, or there exists k ≥ 0 such that lim

x→0

u(x) = k, Φµ (x)

(1.19)

5

Leray-Hardy equations with absorption

and in that case u ∈ Lploc (Ω; dγµ ). In view of [12], it implies that u satisfies Z  uL∗µ ξ + up ξ dγµ (x) = cµ kξ(0), ∀ ξ ∈ C01,1 (RN ).

(1.20)

RN

Note the threshold p∗µ and its role is put into light by the existence or non-existence of explicit 2 solutions of (1.16) under the form x 7→ a|x|b , where necessarily b = − p−1 and a = `. It is also proved in [22] that when µ > µ0 and g : R → R+ is a continuous nondecreasing function satisfying Z ∞



(g(s) − g(−s)) s−1−pµ ds < ∞,

(1.21)

1

then for any k > 0 there exists a radial solution of Lµ u + g(u) = 0

in D0 (B1∗ )

(1.22)

satisfying (1.19), where B1∗ := B1 (0) \ {0}. When µ = µ0 and N ≥ 3 it is proved in [22] that if there exists b > 0 such that Z 1   N −2 g −bs− N +2 ln s ds < ∞, (1.23) 0

then there a exists a radial solution of (1.22) satisfying (1.19) with γ = (N +2)b . In fact this 2 condition is independent of b > 0, by contrast to the case N = 2 and µ = 0 where the introduction of the exponential order of growth of g is a necessity. Moreover, it is easy to see that u satisfies Z  (1.24) uL∗µ ξ + g(u)ξ dγµ (x) = cµ γξ(0), ∀ξ ∈ C01,1 (RN ). RN

In view of these results and identity (1.13), we introduce a definition of weak solutions adapted to the operator Lµ in a measure framework. Since Γµ is singular at 0 if µ < 0, there is need of defining specific set of measures and we denote by M(Ω∗ ; Γµ ), the set of Radon measures ν in Ω∗ such that Z  Z ∗ Γµ d|ν| := sup ζd|ν| : ζ ∈ C0 (Ω ), 0 ≤ ζ ≤ Γµ < ∞. (1.25) Ω∗

Ω∗

If ν ∈ M+ (Ω∗ ), we define its natural extension, with the same notation since there is no ambiguity, as a measure in Ω by Z  Z ∗ ζdν = sup ηdν : η ∈ C0 (Ω ) , 0 ≤ η ≤ ζ for all ζ ∈ C0 (Ω) , ζ ≥ 0, (1.26) Ω

Ω∗

a definition which is easily extended if ν = ν+ − ν− ∈ M(Ω∗ ). Since the idea is to use the weight Γµ in the expression of the weak solution, the expression Γµ ν has to be defined properly if τ+ < 0. We denote by M(Ω; Γµ ) the set of measures ν on Ω which coincide with the above natural extension of νbΩ∗ ∈ M+ (Ω∗ ; Γµ ). If ν ∈ M+ (Ω; Γµ ) we define the measure Γµ ν in the following way Z  Z ∗ ζd(Γµ ν) = sup ηΓµ dν : η ∈ C0 (Ω ) , 0 ≤ η ≤ ζ for all ζ ∈ C0 (Ω) , ζ ≥ 0. (1.27) Ω

Ω∗

6

Leray-Hardy equations with absorption

If ν = ν+ − ν− , Γµ ν is defined acoordingly. Notice that the Dirac mass at 0 does not belong to M(Ω; Γµ ) although it is a limit of {νn } ⊂ M(Ω; Γµ ). We detote by M(Ω; Γµ ) the set of measures which can be written under the form ν = νbΩ∗ +kδ0 ,

(1.28)

where νbΩ∗ ∈ M(Ω; Γµ ) and k ∈ R. Before stating our main theorem we make precise the notion ∗ of weak solution used in this article. We denote Ω := Ω \ {0}, ρ(x) = dist(x, ∂Ω) and n o ∗ Xµ (Ω) = ξ ∈ C0 (Ω) ∩ C 1 (Ω ) : |x|L∗µ ξ ∈ L∞ (Ω) . (1.29) Clearly C01,1 (Ω) ⊂ Xµ (Ω). Definition 1.1 We say that u is a weak solution of (1.6) with ν ∈ M(Ω; Γµ ) such that ν = νbΩ∗ +kδ0 if u ∈ L1 (Ω, |x|−1 dγµ ), g(u) ∈ L1 (Ω, ρdγµ ) and Z Z  ∗  uLµ ξ + g(u)ξ dγµ (x) = ξd(Γµ ν) + kξ(0) for all ξ ∈ Xµ (Ω), (1.30) Ω



where L∗µ is given by (1.13) and cµ is defined in (1.15). A measure for which problem (1.6) admits a solution is a g-good measure. In the regular case we prove the following Theorem A Let µ ≥ 0 if N = 2, µ ≥ µ0 if N ≥ 3 and g : R → R be a H¨ older continuous nondecreasing function such that g(r)r ≥ 0 for any r ∈ R. Then for any ν ∈ L1 (Ω, dγµ ), problem (1.6) has a unique weak solution uν such that for some c1 > 0, kuν kL1 (Ω,|x|−1 dγµ ) ≤ c1 kνkL1 (Ω,dγµ ) . Furthermore, if uν 0 is the solution of (1.6) with right-hand side ν 0 ∈ L1 (Ω, dγµ ), there holds Z Z   ∗ |uν − uν 0 |Lµ ξ + |g(uν ) − g(uν 0 )|ξ dγµ (x) ≤ (ν − ν 0 )sgn(u − u0 )ξdγµ (x), (1.31) Ω

and Z



(uν −



uν 0 )+ L∗µ ξ



Z

+ (g(uν ) − g(uν 0 ))+ ξ dγµ (x) ≤



(ν − ν 0 )sgn+ (u − u0 )ξdγµ (x),

(1.32)



for all ξ ∈ Xµ (Ω), ξ ≥ 0. Definition 1.2 A continuous function g : R → R such that rg(r) ≥ 0 for all r ∈ R satisfies the weak ∆2 -condition if there exists a positive nondecreasing function t ∈ R 7→ K(t) such that |g(s + t)| ≤ K(t) (|g(s)| + |g(t)|)

for all (s, t) ∈ R × R s.t. st ≥ 0.

It satisfies the ∆2 -condition if the above function K is constant.

(1.33)

Leray-Hardy equations with absorption

7

The ∆2 -condition has been intruduced in the study of Birnbaum-Orlicz spaces [7], [23] and it is satisfied by power function r 7→ |r|p−1 r, p > 0, but not by exponential functions r 7→ ear . It plays a key role in the study of semilinear equation with a power type reaction term (see eg. [29], [30]). The new weak ∆2 -condition is more general and it is also satisfied by exponential functions. Theorem B Let µ > 0 if N = 2 or µ > µ0 if N ≥ 3 and g : R → R be a nondecreasing continuous function such that g(r)r ≥ 0 for any r ∈ R. If g satisfies the weak ∆2 -condition and Z ∞ ∗ ∗ (g(s) − g(−s))s−1−min{pµ , p0 } ds < ∞, (1.34) 1

where p∗µ is given by (1.17), then for any ν ∈ M+ (Ω; Γµ ) problem (1.6) admits a unique weak solution uν . Note that min{p∗µ , p∗0 } = p∗µ for µ > 0 and min{p∗µ , p∗0 } = p∗0 if µ < 0. Furthermore, the mapping: ν 7→ uν is increasing. In the case N ≥ 3 and µ = µ0 we have a more precise result. Theorem C Assume that N ≥ 3 and g : R → R is a continuous nondecreasing function such that g(r)r ≥ 0 for any r ∈ R satisfying the weak ∆2 -condition and (1.8). Then for any ν = νbΩ∗ + cµ kδ0 ∈ M+ (Ω; Γµ ) problem (1.6) admits a unique weak solution uν . Furthermore, if νbΩ∗ = 0, condition (1.8) can be replaced by the following weaker one Z ∞ N +2 2N (g(t) − g(−t)) (ln t) N −2 t− N −2 dt < ∞. (1.35) 1

The optimality of these conditions depends whether the measure is concentrated at 0 or not. When the measure is of the form kδ0 the condition proved to be optimal in [22] and when it is of the type kδa with a 6= 0 optimality is shown in [28]. Normally, the estimates on the Green kernel plays an essential role for approximating the solution of elliptic problems with absorption and Radon measure data. However, we have avoided to use the estimates on the Green kernel for Hardy operators which are not easily tractable when 0 > µ ≥ µ0 , and our main idea is to separate the measure ν ∗ in M(Ω; Γµ ) and the Dirac mass at the origin, and then to glue the solutions with above measures respectively. This technique requires this new weak ∆2 -condition. In the previous result, it is noticeable that if k = 0 (resp. νbΩ∗ = 0) only condition (1.8) (resp. condition (1.35)) is needed. In the two cases the weak ∆2 -condition is unnecessary. In the power case where g(u) = |u|p−1 u := gp (u), ( Lµ u + gp (u) = ν in Ω, (1.36) u=0 on ∂Ω, the following result follows from Theorem B and C. Corollary D Let µ ≥ µ0 if N ≥ 3 and µ > 0 if N = 2. Any nonzero measure ν = νbΩ∗ +cµ kδ0 ∈ M+ (Ω; Γµ ) is gp -good if one of the following holds: (i) 1 < p < p∗µ in the case νbΩ∗ = 0;

8

Leray-Hardy equations with absorption

(ii) 1 < p < p∗0 in the case k = 0;  (iii) 1 < p < min p∗µ , p∗0 in the case k 6= 0 and νbΩ∗ 6= 0. We remark that p∗µ is the sharp exponent for existence of (1.35) when νbΩ∗ = 0, while the critical exponent becomes p∗0 when k = 0 and ν has atom in Ω \ {0}. The supercritical case of equation (1.36) corresponds to the fact that not all measures are gp -good and the case where k 6= 0 is already treated. Theorem E Assume that N ≥ 3. Then ν = νbΩ∗ ∈ M(Ω; Γµ ) is gp -good if and only if for any  > 0, ν = νχBc is absolutely continuous with respect to the c2,p0 -Bessel capacity. Finally we characterize the compact removable sets in Ω. Theorem F Assume that N ≥ 3, p > 1 and K is a compact set of Ω. Then any weak solution of Lµ u + gp (u) = 0 in Ω \ K (1.37) can be extended a weak solution of the same equation in whole Ω if and only if (i) c2,p0 (K) = 0 if 0 ∈ / K; (ii) p ≥ pµ∗ if K = {0}; (iii) c2,p0 (K) = 0 if µ ≥ 0, 0 ∈ K and K \ {0} = 6 {∅}; ∗ 6 {∅}. (iv) c2,p0 (K) = 0 and p ≥ pµ if µ < 0, 0 ∈ K and K \ {0} = The case (i) is already proved in [22, Theorem 1.2]. Notice also that if A 6= ∅ necessarily c2,p0 (A) = 0 holds only if p ≥ p0 . Therefore, if µ ≥ 0 there holds p ≥ p∗0 ≥ p∗µ , while if µ < 0, then p0 < p∗µ . The rest of this paper is organized as follows. In Section 2, we build the framework for weak solutions of (1.6) involving L1 data. Section 3 is devoted to solve existence and uniqueness of weak solution of (1.6), where the absorption is subcritical and ν is a related Radon measure. Finally, we deal with the super critical case in Section 4 by characterized by Bessel Capacity.

2

L1 data

Throughout this section we assume N ≥ 2 and µ ≥ µ0 and in what follows, we denote by ci with i ∈ N a generic positive constant. We first recall some classical comparison results for Hardy operator Lµ . The next lemma is proved in [12, Lemma 2.1], and in [17, Lemma 2.1] if h(s) = sp . ¯ L : G × [0, +∞) 7→ [0, +∞) Lemma 2.1 Let G be a bounded domain in RN such that 0 6∈ G, be a continuous function satisfying for any x ∈ G, h(x, s1 ) ≥ h(x, s2 )

if

s1 ≥ s2 ,

and functions u, v ∈ C 1,1 (G) ∩ C(G) satisfy ( Lµ u + h(x, u) ≥ Lµ v + h(x, v) u≥v

on ∂G,

then u≥v

in G,

in

G.

9

Leray-Hardy equations with absorption

As an immediate consequence we have Lemma 2.2 Assume that Ω is a bounded C 2 domain containing 0. If L is a continuous function ∗ as in Lemma 2.1 verifying furthermore L(x, 0) = 0 for all x ∈ Ω, and u ∈ C 1,1 (Ω∗ ) ∩ C(Ω ) satisfies  Lµ u + L(x, u) = 0 in Ω∗ ,    u = 0 on ∂Ω, (2.1)    lim u(x)Φ−1 µ (x) = 0. x→0

Then u = 0. We recall that if u ∈ L1 (Ω, |x|−1 dγµ ) is a weak solution of ( Lµ u = f in Ω, u=0

on ∂Ω,

in the sense of Definition 1.1, it satisfies also Z Z ∗ uLµ (ξ) dγµ (x) = f ξ dγµ (x) Ω

for all ξ ∈ Xµ (Ω).

(2.2)

(2.3)



If u is a weak solution of (2.2) there holds in D0 (Ω∗ ),

Lµ u = f

(2.4)

and v = Γ−1 µ u verifies L∗µ v = Γ−1 µ f

in D0 (Ω∗ ),

(2.5)

a fact which is expressed by the commutating formula Γµ L∗µ v = Lµ (Γµ v).

(2.6)

The following form of Kato’s inequality, proved in [12, Proposition 2.1], plays an essential role in the obtention a priori estimates and uniqueness of weak solution of (1.6). Proposition 2.1 If f ∈ L1 (Ω, ρdγµ ), then there exists a unique weak solution u ∈ L1 (Ω, |x|−1 dγµ ) of (2.2). Furthermore, for any ξ ∈ Xµ (Ω), ξ ≥ 0, we have Z Z |u|L∗µ (ξ) dγµ (x) ≤ sign(u)f ξ dγµ (x) (2.7) Ω

and

Z Ω

u+ L∗µ (ξ) dγµ (x)



Z ≤

sign+ (u)f ξ dγµ (x).

(2.8)



The proof is done if ξ ∈ C01,1 (Ω), but it is valid if ξ ∈ Xµ (Ω). The next result is proved in [13, Lemma 2.3].

10

Leray-Hardy equations with absorption

Lemma 2.3 Assume that µ > µ0 and f ∈ C 1 (Ω∗ ) verifies 0 ≤ f (x) ≤ c2 |x|τ −2 , for some τ > τ− . Let uf be the solution of   Lµ u = f    u=0  u(x)   = 0. lim x→0 Φµ (x)

(2.9)

in Ω∗ , on ∂Ω,

(2.10)

Then there holds: (i) if τ− < τ < τ+ , 0 ≤ uf (x) ≤ c3 |x|τ

in

Ω∗ ;

(2.11)

(ii) if τ = τ+ , 0 ≤ uf (x) ≤ c4 |x|τ (1 + (− ln |x|)+ ) (iii) if τ > τ+ , 0 ≤ uf (x) ≤ c5 |x|τ+

in

in

Ω∗ ;

Ω∗ .

(2.13)

Proof of Theorem A. Let H1µ,0 (Ω) be the closure of C0∞ (Ω) under the norm of sZ Z u2 kukH1µ,0 (Ω) = |∇u|2 dx + µ dx. 2 Ω Ω |x| Then H1µ,0 (Ω) is a Hilbert space with inner product Z Z hu, viH1µ,0 (Ω) = h∇u, ∇vidx + µ





(2.12)

uv dx |x|2

(2.14)

(2.15)

and the embedding H1µ,0 (Ω) ,→ Lp (Ω) is continuous and compact for p ∈ [2, 2∗ ) with 2∗ = N2N −2 when N ≥ 3 and any p ∈ [2], ∞ if N = 2. Furthermore, if η ∈ C01 (Ω) has the value 1 in a neighborhood of 0, then ηΓµ ∈ H1µ,0 (Ω). We put Z v G(v) = g(s)ds, 0

then G is a convex nonnegative function. If ρν ∈ L2 (Ω) we define the functional Jν in the space H1µ,0 (Ω) by  Z Z  1 2 kvkH1 (Ω) + G(v)dx − νvdx if G(v) ∈ L1 (Ω, dγµ ), µ,0 Jν (v) = (2.16) 2 Ω Ω  ∞ if G(v) ∈ / L1 (Ω, dγµ ). The functional J is strictly convex, lower semicontinuous and coercive in H1µ,0 (Ω), hence it admits a unique minimum u which satisfies Z Z hu, viH1µ,0 (Ω) + g(u)vdx = νvdx for all v ∈ H1µ,0 (Ω). Ω



11

Leray-Hardy equations with absorption

If ξ ∈ C01,1 (Ω) then v = ξΓµ ∈ H1µ,0 (Ω), then  Z  Z µΓµ h∇u, ∇Γµ i + hu, ξΓµ iH1µ,0 (Ω) = h∇u, ∇ξidγµ (x) + ξdx, |x|2 Ω Ω and

Z

Z Z h∇u, ∇Γµ iξdx = − h∇ξ, ∇Γµ iudx − uξ∆Γµ dx,



C0∞ (Ω)

(2.17)

since is dense in 1 L (Ω, dγµ ), hence uL∗µ ξ ∈

Ω 1 Hµ,0 (Ω). Furthermore, L1 (Ω, dγµ ). Therefore

Z

uL∗µ ξ

since u ∈

Ω Lp (Ω)

for any p < 2∗ , |x|−1 u ∈

Z



νξdγµ .

+ g(u)ξ dγµ =



(2.18)



Next, if ν ∈ L1 (Ω, ρdγµ ) we consider a sequence {νn } ⊂ C0∞ (Ω) converging to ν in L1 (Ω, ρdγµ ) and denote by {un } the sequence of the corresponding minimizing problem in H1µ,0 (Ω). By Proposition 2.1 we have that, for any ξ ∈ Xµ (Ω), Z Z  ∗ |un − um |Lµ ξ + (g(un ) − g(um ))sgn(un − um )ξ dγµ ≤ (νn − νm )sgn(un − um )ξdγµ . Ω



(2.19) We denote by η0 the solution of L∗µ η = 1

in Ω,

η=0

on ∂Ω.

(2.20)

Its existence is proved in [12, Lemma 2.2], as well as the estimate 0 ≤ η0 ≤ c6 ρ for some c6 > 0. Since g is monotone, we obtain from (2.19) Z Z (|un − um | + |g(un ) − g(um )|η0 ) dγµ ≤ |νn − νm |η0 dγµ . (2.21) Ω



Hence {un } is a Cauchy sequence in L1 (Ω, dγµ ). Next we construct a solution η1 to L∗µ η = |x|−1

in Ω∗ ,

η=0

on ∂Ω.

For this aim, we consider for 0 < θ < 1 the function yθ (x) = L∗µ yθ = |x|−θ

in B1 ,

yθ = 0

1−|x|2−θ (N −θ+2τ+ (µ))

(2.22) which verifies verifies

on ∂B1

(we can always assume that Ω ⊂ B1 ). As in the proof of [12, Lemma 2.2], for any x0 ∈ Ω there exists r0 > 0 such that Br0 (x0 ) ⊂ Ω and for t > 0 small enough wt,x0 (x) = t(r02 − |x − x0 |2 ) is a subsolution of (2.20), hence of (2.22). Therefore there exists ηθ solution of L∗µ ηθ = |x|−θ

in Ω∗ ,

ηθ = 0

on ∂Ω.

(2.23)

Furthermore θ 7→ ηθ is increasing and bounded from above by y1 , hence it converges to a function η1 which satisfies (2.23). Then Z  Z  −θ |un − um ||x| + |g(un ) − g(um )|ηθ dγµ (x) ≤ |νn − νm |ηθ dγµ . (2.24) Ω



12

Leray-Hardy equations with absorption

Letting θ → 1, we obtain as a complement of (2.21)  Z  Z |un − um | + |g(un ) − g(um )|η1 dγµ (x) ≤ |νn − νm |η1 dγµ . |x| Ω Ω

(2.25)

Hence {un } is a Cauchy sequence in L1 (Ω, |x|−1 dγµ ) with limit u and {g(un )} is a Cauchy sequence in L1 (Ω, ρdγµ ) with limit g(u). Then (2.18) holds. As for (1.31) it is a consequence of (2.19) and (1.32) is proved similarly. 

3

The subcritical case

In this section as well as in the next one we always assume that N ≥ 3 and µ ≥ µ0 , or N = 2 and µ > 0, since the case N = 2, µ = 0, which necessitates specific tools, has already been completely treated in [26]. We recall that the set M(Ω∗ ; Γµ ) of Radon measures is defined in the introduction as the set of measures in Ω∗ satisfying (1.25), and any positive measure ν ∈ M(Ω∗ ; Γµ ) is naturaly extended by formula (1.26) as a positive measure in Ω. The space M(Ω; Γµ ) is the space of measures ν on C0 (Ω) such that ν = νbΩ∗ +kδ0 , (3.1) where νbΩ∗ ∈ M(Ω∗ ; Γµ ).

3.1

The linear equation

Lemma 3.1 If ν ∈ M(Ω; Γµ ), then there exists a unique weak solution u ∈ L1 (Ω, |x|−1 dγµ ) to ( Lµ u = ν in Ω, (3.2) u = 0 on ∂Ω. This solution is denoted by Gµ [ν], and this defines the Green operator of Lµ in Ω with homogeneous Dirichlet conditions. Proof. By linearity and using the result of [12] on fundamental solution, we can assume that k = 0 and ν ≥ 0. Let {νn } ⊂ L1 (Ω, ρdγµ ) be a sequence such that νn ≥ 0 and Z Z ξΓµ νn dx → ξd(Γµ ν) for all ξ ∈ Xµ (Ω), Ω



and by Proposition 2.1, we may let un be the unique, nonnegative weak solution of ( Lµ un = νn in Ω, un = 0 with n ∈ N. There holds Z Ω

un L∗µ ξdγµ (x)

on ∂Ω,

(3.3)

Z =

ξνn Γµ dx Ω

for all ξ ∈ Xµ (Ω).

(3.4)

13

Leray-Hardy equations with absorption

Then un ≥ 0 and using the function η1 defined in the proof of Theorem A for test function, we have Z Z un c dγµ = η1 Γµ νn dx ≤ ckνkM(Ω,Γµ ) , (3.5) Ω |x| Ω 1 dγµ (x)). which implies that {un } is bounded in L1 (Ω, |x| For any  > 0 sufficiently small, set the test function ξ in {ζ ∈ Xµ (Ω) : ζ = 0 in B }, then we have that Z Z ξνn Γµ dx for all ξ ∈ Xµ (Ω). un L∗µ ξdγµ (x) = (3.6) Ω\B (0)

Ω\B (0)

¯ ⊂ O0 ⊂ O ¯ 0 ⊂ Ω \ B (0), there exists c > 0 Therefore, for any open sets O, O0 verifying O independent of n such that kun kL1 (O0 ) ≤ ckνkM(Ω,Γµ ) . Note that in Ω \ B , the operator L∗µ is uniformly elliptic and the measure dγµ is equivalent to the N-dimensional Lebesgue measure dx, then [30, Corollary 2.8] could be applied to obtain that for some c > 0 independent of n but dependent of O0 , kun kW 1,q (O) ≤ ckun kL1 (O0 ) + k˜ νn kL1 (Ω,dγµ ) ≤ ckνkM(Ω,Γµ ) . 1,q (Ω \ {0}). That is, {un } is uniformly bounded in Wloc As a consequence, since  is arbitrary, there exist a subsequence, still denoted by {un }n and a function u such that un → u a.e. in Ω.

Meanwhile, we deduce from Fatou’s lemma, Z Z u dγµ ≤ c η1 Γµ dν. Ω |x| Ω

(3.7)

Next we claim that un → u in L1 (Ω, |x|−1 dγµ ). Let ω ⊂ Ω be a Borel set and ψω be the solution of ( L∗µ ψω = |x|−1 χω in Ω, (3.8) ψω = 0 on ∂Ω. Then ψω ≤ η1 , thus it is uniformly bounded. Assuming that Ω ⊂ B1 , clearly ψω is bounded from above by the solution Ψω of ( L∗µ Ψω = |x|−1 χω in B1 , (3.9) Ψω = 0 on ∂B1 , and by standard rearrangement, supB1 Ψω ≤ supB1 Ψrω , where Ψrω solves ( L∗µ Ψrω = |x|−1 B(|ω|) in B1 , Ψrω = 0

on ∂B1 ,

(3.10)

14

Leray-Hardy equations with absorption

where (|ω|) = This implies



|ω| |B1

1

N

. Then Ψrω is radially decreasing and lim|ω|→0 Ψrω = 0, uniformly on B1 . lim ψω (x) = 0

|ω|→0

uniformly in B1 .

(3.11)

Using (3.4) with ξ = ψω , Z Z Z un dγµ (x) = νn Γµ ψω dx ≤ sup ψω νn Γµ dx → 0 as |ω| → 0. Ω ω |x| ω ω Therefore {un } is uniformly integrable for the measure |x|−1 dγµ . Letting n → ∞ in (3.4) implies the claim. 

3.2

Dirac masses

We assume that g : R → R is a continuous nondecreasing function such that rg(r) ≥ 0 for all r ∈ R. The next lemma dealing with problem ( Lµ u + g(u) = kδ0 in Ω, (3.12) u=0 on ∂Ω, is an extension of [22, Theorem 3.1, Theorem 3.2]. Actually it was quoted without demonstration in this article as Remark 3.1 and Remark 3.2 and we give here their proof. Notice also that when N ≥ 3 and µ = µ0 we give a more complete result that [22, Theorem 3.2]. Lemma 3.2 Let k ∈ R and g : R → R be a continuous nondecreasing function such that rg(r) ≥ 0 for all r ∈ R. Then problem (3.12) admits a unique solution u := ukδ0 if one of the following conditions is satisfied: (i) N ≥ 2, µ > µ0 and g satisfies (1.21); (ii) N ≥ 3, µ = µ0 and g satisfies (1.35). Proof. Without loss of generality we assume BR ⊂ Ω ⊂ B1 for some R ∈ (0, 1). (i) The case µ > µ0 . It follows from [22, Theorem 3.1] that for any k ∈ R there exists a radial ∗ ) satisfying function vk,1 (resp. vk,R ) defined in B1∗ (resp. BR Lµ v + g(v) = 0

in B1∗

∗ (resp. in BR ),

(3.13)

vanishing respectively on ∂B1 and ∂BR and satisfying vk,1 (x) vk,R (x) k = lim = . x→0 Φµ (x) x→0 Φµ (x) cµ lim

(3.14)

Furthermore g(vk,1 ) ∈ L1 (B1 , dγµ ) (resp. g(vk,R ) ∈ L1 (BR , dγµ )). Assume that k > 0, then ∗ and the extension of v 0 ≤ vk,R ≤ vk,1 in BR ˜k,R by 0 in Ω∗ is a subsolution of (3.13) in Ω∗ and

15

Leray-Hardy equations with absorption

it is still smaller than vk,1 in Ω∗ . By the well known method on super and subsolutions (see e.g. [32, Theorem 1.4.6]), there exists a function u in Ω∗ satisfying v˜k,R ≤ u ≤ vk,1 in Ω∗ and  Lµ u + g(u) = 0 in Ω∗ ,     u=0 on ∂Ω, (3.15)  k u(x)   = .  lim x→0 Φµ (x) cµ By standard methods in the study of isolated singularities (see e.g. [22], [29], and [15] and [16] for various extensions) k x lim |x|1−τ− ∇u(x) = τ− . (3.16) x→0 cµ |x| For any  > 0 and ξ ∈ Xµ (Ω), Z 0= (Lµ u + g(u))Γµ ξdx Ω\B Z k uL∗µ ξdγµ (x) + (τ− − τ+ ) |S N −1 |ξ(0)(1 + o(1)). = cµ Ω\B Using (1.15), we obtain Z

uL∗µ ξdγµ (x) = kξ(0).

(3.17)



(ii)The case µ = µ0 . In [22, Theorem 3.2] it is proved that if for some b > 0 there holds Z ∞   N −2 I := g bt N +2 ln t t−2 dt < ∞, (3.18) 1

then there exists a solution of (1.22) satisfying (1.19) with γ =

(N +2)b . 2

Actually we claim that N −2

the finiteness of this integral is independent of the value of b. To see that, set s = t N +2 , then Z 2N N +2 ∞ I= g (βs ln s) s− N −2 ds, N −2 1 with β =

N +2 N −2 b.

Set τ = βs ln s, then 

ln ln s ln β ln s 1 + + ln s ln s

 =⇒ ln s = ln τ (1 + o(1))

as s → ∞.

We infer that for  > 0 there exists s > 2 and τ = s ln s such that Z ∞ 2N g (βs ln s) s− N −2 ds N +2 N +2 (1 − )β N −2 ≤ Z ∞s ≤ (1 + )β N −2 , N +2 2N g (τ ) (ln τ ) N −2 τ − N −2 dτ τ

(3.19)

16

Leray-Hardy equations with absorption

which implies the claim. Next we prove as in case (i) the existence of vk,1 (resp. vk,R ) defined ∗ ) satisfying in B1∗ (resp. BR in B1∗

Lµ0 v + g(v) = 0

∗ (resp. in BR ),

(3.20)

vanishing respectively on ∂B1 and ∂BR and satisfying lim

x→0

vk,1 (x) vk,R (x) k . = lim = x→0 Φµ (x) Φµ (x) cµ0

We end the proof as above.

(3.21) 

Remark. It is important to notice that conditions (1.21) and (1.35) (or equivalently (1.23)) are also necessary for the existence of radial solutions in a ball, hence their are also necessary for the existence of non radial solutions of the Dirichlet problem (3.12).

3.3

Measures in Ω∗

We consider now the problem (

Lµ u + g(u) = ν u=0

in Ω, on ∂Ω,

(3.22)

where ν ∈ M(Ω∗ ; Γµ ). Lemma 3.1 Let µ ≥ µ0 . Assume that g satisfies (1.8) if N ≥ 3 or the β± (g) defined by (1.9) satisfy β− (g) < 0 < β+ (g)Pif N = 2, and let ν ∈ M(Ω∗ ; Γµ ). If N = 2, we assume that ν can be decomposed as ν = νr + j αj δaj where νr has no atom, the αj satisfy (1.10) and {aj } ⊂ Ω∗ . Then problem (3.22) admits a unique weak solution. Proof. We assume first that ν ≥ 0 and let r0 = dist (x, ∂Ω). For 0 < σ < r0 , we set Ωσ = Ω\{B σ } and νσ = νχΩσ and for 0 <  < σ we consider the following problem in Ω     Lµ u + g(u) = νσ in Ω , u=0 on ∂Ω, (3.23)   u=0 on ∂B . Since 0 ∈ / Ω problem (3.23) admits a unique solution uνσ , which is smaller than Gµ [ν] and satisfies 0 ≤ uνσ , ≤ uνσ0 ,0 in Ω for all 0 < 0 ≤  and 0 < σ 0 ≤ σ. ∗  For any ξ ∈ C1,1 c (Ω ) and  small enough so that supp (ξ) ⊂ Ω , there holds Z Z  ∗ uνσ , Lµ ξ + g(uνσ , )ξ dγµ = ξΓµ dνσ . Ω



There exists uνσ = lim uνσ , and it satisfies the identity →0

Z Ω

uνσ L∗µ ξ



Z

+ g(uνσ )ξ dγµ =

ξΓµ dνσ Ω

∗ for all ξ ∈ C1,1 c (Ω ).

(3.24)

17

Leray-Hardy equations with absorption

As a consequence of the maximum principle and Lemma 3.1, there holds 0 ≤ uνσ ≤ Gµ [νσ ] ≤ Gµ [ν].

(3.25)

Since νσ vanishes in Bσ , Gµ [νσ ](x) ≤ cΦµ (x) in a neighborhood of 0, and uνσ is also bounded 0 0 by cΦµ in this neighborhood. This implies that Φ−1 µ (x)uνσ (x) → c as x → 0 for some c ≥ 0. 1,1 Next let ξ ∈ Cc (Ω),    ( 2−1 1 + cos 2π|x| if |x| ≤ σ2 , σ `n (r) = 0 if |x| > σ2 , and ξn = ξ`n . Then

Z

uνσ L∗µ ξn

Z



+ g(uνσ )ξn dγµ =



ξn Γµ dνσ .

(3.26)



When n → ∞, Z

Z ξn Γµ dνσ →

ξΓµ dνσ



and



Z

Z g(uσ )ξn dγµ →



g(uσ )ξdγµ . Ω

Now for the first inegral term in (3.26), we have Z Z uνσ L∗µ ξn dγµ = `n uσ L∗µ ξdγµ + In + IIn + IIIn , Ω

where



Z In = −

uσ ξ∆`n dγµ ,

Bσ 2

Z IIn = −2

uσ h∇ξ, ∇`n idγµ

Bσ 2

and

Z uσ h

IIIn = −τ+ Bσ

x , ∇`n idγµ . |x|2

2

Using the fact that ξ(x) → ξ(0) and ∇ξ(x) → ∇ξ(0) we easily infer that In , IIn and IIIn converge to 0 when n → ∞, the most complicated case being the case when µ = µ0 , which is the justification of introducing the explicit cut-off function `n . Therefore (3.24) is still valid if it is assumed that ξ ∈ Cc1,1 (Ω). This means that uνσ is a weak solution of ( Lµ u + g(u) = νσ in Ω, (3.27) u=0 on ∂Ω. Furthermore uνσ is unique and uνσ is a decreasing function of σ with limit u when σ → 0. Taking η1 as test function, we have Z Z Z  −1 c|x| uνσ + η1 g(uνσ ) dγµ = η1 d (γµ νσ ) ≤ η1 d (γµ ν) . Ω





18

Leray-Hardy equations with absorption

By using the monotone convergence theorem we infer that uνσ → u in L1 (Ω, |x|−1 dγµ ) and g(uνσ ) → g(uν ) in L1 (Ω, dγµ ). Hence u = uν is the weak solution of (3.22). Next we consider a signed measure ν = ν+ − ν− . We denote by uν+σ , , u−ν−σ , and uν σ , the σ , −ν σ and ν σ ,  respectively. Then solutions of (3.23) in Ω corresponding to ν+ − u−ν−σ , ≤ uν σ , ≤ uν+σ , .

(3.28)

The correspondence  7→ uν+σ , and  7→ u−ν−σ , are respectively increasing and decreasing. Furthermore uν σ , is locally bounded, hence by local compactness and up to a subsequence uν σ , converges a.e. in B to some function uν σ . Since u−ν−σ , → u−ν−σ and uν+σ , → uν+σ in L1 (Ω, |x|−1 dγµ ), it follows by Vitali’s theorem that uν σ , → uν σ in L1 (Ω, |x|−1 dγµ ). Similarly, using the monotonicity of g, g(uν σ , ) → g(uν σ ) in L1 (Ω, dγµ ). By local compactness, uν σ → u a.e. in Ω. Using the same argument of uniform integrability, we have that uν σ → u in L1 (Ω, |x|−1 dγµ ) and g(uν σ ) → g(u) in L1 (Ω, dγµ ) when σ → 0 and u satisfies Z Z  ∗ (3.29) uLµ ξ + g(u)ξ dγµ = ξd(dγµ ν) for any ξ ∈ Cc1,1 (Ω∗ ). Ω



Finally the singularity at 0 is removable by the same argument as above which implies that u solves (3.29) and thus u = uν is the weak solution of (3.22). 

3.4

Proof of Theorem B

The idea is to glue altogether two solutions one with the Dirac mass and the other with the measure in Ω∗ , this is the reason why the weak ∆2 condition is introduced. Lemma 3.3 Let ν = νbΩ∗ +kδ0 ∈ M+ (Ω; Γµ ) and σ > 0. We assume that νbΩ∗ (B σ ) = 0. Then there exists a unique weak solution to (1.6). Proof. Set νσ = νbΩ∗ . It has support in Ωσ = Ω\B σ . For 0 <  < σ we consider the approximate problem in Ω = Ω \ B  ,  in Ω ,   Lµ u + g(u) = νσ  

u=0

on ∂Ω,

u = ukδ0

on ∂B ,

(3.30)

where ukδ0 is the solution of problem (3.12) obtained in Lemma 3.2. It follows from [30, Theorem 3.7] that problem (3.30) admits a unique weak solution denoted by Uνσ , , thanks to the fact that the operator is not singular in Ω . We recall that uνσ , is the solution of (3.23) and Gµ [δ0 ] the fundamental solution in Ω. Then max{ukδ0 , uνσ , } ≤ Uνσ , ≤ uνσ + kGµ [δ0 ]

in Ω .

(3.31)

Furthermore one has Uνσ , ≤ Uνσ ,0 in Ω , for 0 < 0 < . Since uνσ ≤ uν and both kGµ [δ0 ] and uν belong to L1 (Ω, |x|−1 dγµ ), then it follows by the monotone convergence theorem that Uνσ , converges in L1 (Ω, |x|−1 dγµ ) and almost everywhere to some function Uνσ ∈ L1 (Ω, |x|−1 dγµ ).

19

Leray-Hardy equations with absorption

Since Γµ is a supersolution for equation Lµ u + g(u) = 0 in Bσ , for 0 < 0 < σ there exists c8 := c8 (0 , σ) > 0 such that uνσ (x) ≤ c8 |x|τ+

for all x ∈ B0 .

For any δ > 0, there exists 0 such that uνσ (x) ≤ δGµ [δ0 ](x) in B0 . Hence uνσ + kGµ [δ0 ] ≤ (k + δ)Gµ [δ0 ] in B0 , which implies in B0 \ B  ,

g(Uνσ , ) ≤ g((k + δ)Gµ [δ0 ])

(3.32)

and Z

Z 1 k+δ τ τ N −1 τ− τ+ +N −1 dr − + | g( k+δ g( cµ |x| )|x| dx = |S cµ r )r 0 BZ1 Z ∞ ∞ −2+ τ2 −1−p∗µ − g(t)t g(t)t dt c9 = c9 Z

g((k + δ)Gµ [δ0 ])dγµ (x) ≤ Ω

=

k+δ cµ

k+δ cµ

< ∞. Now, using the local ∆2 -condition,with a0 = g(Uνσ , ) ≤ g(uνσ +

k τ− cµ 0 )

k τ− cµ 0 ,

we see that

≤ K(a0 ) g(uνσ ) + g(a0 )

in Ω0 .



(3.33)

From (3.32) and (3.33) we infer that g(Uνσ , ) is bounded in L1 (Ω , dγµ ) independently of . If ξ ∈ C01,1 (Ω∗ ), we have for  > 0 small enough so that supp (ξ) ⊂ Ω Z Z  ∗ Uνσ , Lµ ξ + g(Uνσ , )ξ dγµ = ξΓµ dνσ . Ω

Letting  → 0 we obtain that Z



Uνσ L∗µ ξ

Z



+ g(Uνσ )ξ dγµ =



ξΓµ dνσ .

(3.34)



Let ξ ∈ C01,1 (Ω). Let also ηn ∈ C 1,1 (RN ) be a nonnegative cut-off function such that 0 ≤ ηn ≤ 1, ηn ≡ 1 in B c2 , ηn ≡ 0 in B 1 , and choose ξηn for test function. Then n

n

Z

ηn Uνσ L∗µ ξ

Z



Z

+ g(Uνσ )ηn ξ dγµ −



Uνσ An dγµ = Ω

ξηn Γµ dνσ ,

(3.35)



with An = ξ∆ηn + 2h∇ηn , ∇ξi + 2τ+ ξh∇ηn , |x|x2 i. Clearly Z lim

n→∞ Ω

ηn Uνσ L∗µ ξ



Z

 Uνσ L∗µ ξ + g(Uνσ )ξ dγµ ,

+ g(Uνσ )ηn ξ dγµ = Ω

and

Z lim

n→∞ Ω

Z ξηn Γµ dνσ =

ξΓµ dνσ . Ω

(3.36)

20

Leray-Hardy equations with absorption

We take ηn (r) =

1 2

    

Then

− 12 cos nπ r −

1 n



if

1 n

≤ r ≤ n2 ,

0

if r < n1 ,

1

if r > n2 .

      nπ N − 1 + 2τ+ n2 π 2 1 1 + . An = cos nπ r − sin nπ r − 2 n 2 r n

Letting  → 0 in (3.31), we have Uνσ (x) = kGµ [δ0 ](x)(1 + o(1)) = Hence

k τ− |x| (1 + o(1)) cµ

as x → 0.

√ 2k|S N −1 | µ − µ0 Uνσ An dγµ = = k. cµ Ω

Z lim

n→∞

(3.37)

This implies that Uνσ is the solution of (1.6) with ν replaced by νσ + kδ0 .



Lemma 3.4 Let ν = νbΩ∗ +kδ0 ∈ M+ (Ω; Γµ ). Then there exists a unique weak solution to (1.6). Proof. Following the notations of Lemma 3.3, we set νσ = χBσ νbΩ∗ and denote by Uνσ the solution of ( Lµ u + g(u) = νσ + kδ0 in Ω, (3.38) u=0 on ∂Ω. It is a positive function and there holds max{ukδ0 , uνσ } ≤ Uνσ ≤ uνσ + kGµ [δ0 ]

in Ω.

(3.39)

Since the mapping σ 7→ Uνσ is decreasing, then there exists U = lim Uνσ and U satisfies (3.39). σ→0

As a consequence Uνσ → U in L1 (Ω, |x|−1 dγµ ) as σ → 0. We take η1 for test function in the weak formulation of (3.39), then Z Z  |x|−1 Uνσ + η1 g(Uνσ ) dγµ = η1 Γµ dνσ + kη1 (0). Ω



By the monotone convergence theorem we obtain the identity Z Z Z  −1 ∗ |x| U + η1 g(U ) dγµ = η1 d(γµ νbΩ ) + kη1 (0) = η1 d(γµ ν), Ω





and the fact that g(Uνσ ) → g(U ) in L1 (Ω, ρdγµ ). Going to the limit as σ → 0 in the weak formulation of (3.38), we infer that U = uν is the solution of (1.6).  Proof of Theorem B. Assume ν = νbΩ∗ +kδ0 ∈ M(Ω; Γµ ) satisfies k > 0 and let ν+ = ν+ bΩ∗ +kδ0 and ν− = ν− bΩ∗ the positive and the negative part of ν. We denote by uν+ and u−ν− the weak solutions of (1.6) with respective data ν+ and −ν− . For 0 <  < σ such that B σ ⊂ Ω, we set

21

Leray-Hardy equations with absorption

νσ = χBσ νbΩ∗ , with positive and negative part νσ+ and νσ− and denote by Uνσ+ , , U−νσ− , and Uνσ , the respective solutions of  in Ω ,   Lµ u + g(u) = νσ+ u=0 on ∂Ω, (3.40)   u = ukδ0 on ∂B , ( Lµ u + g(u) = −νσ− in Ω , (3.41) u=0 on ∂Ω ∪ ∂B , and

  Lµ u + g(u) = νσ u=0  u = ukδ0

in Ω , on ∂Ω, on ∂B ,

(3.42)

Then U−νσ− , ≤ Uνσ , ≤ Uνσ+ , .

(3.43)

Furthermore Uνσ+ , satisfies (3.31) and, in coherence with the notations of Lemma 3.1 with νσ replaced by −νσ− , u−νσ− ≤ U−νσ− , = u−νσ− , . (3.44) By compactness, {Uνσ ,j }j converges almost everywhere in Ω to some function U for some sequence {j } converging to 0. Moreover Uνσ ,j converges to Uνσ in L1 (Ω, |x|−1 dγµ ) because Uνσ+ , → uνσ+ +kδ0 and u−νσ− , → u−νσ− in L1 (Ω, |x|−1 dγµ ) by Lemma 3.1 and (3.43) holds. Similarly g(Uνσ ,j ) converges to g(U ) in L1 (Ω, ρdγµ ). This implies that U satisfies Z Z  ∗ U Lµ ξ + g(U )ξ dγµ = ξΓµ dνσ for all ξ ∈ C01,1 (Ω∗ ). Ω



C01,1 (Ω),

In order to use test functions in inequality (derived from (3.43)) and the

we proceed as in the proof of Lemma 3.3, using the

u−νσ− ≤ Uνσ ≤ uνσ+ +kδ0 .

(3.45)

By (3.33), uνσ+ +kδ0 (x) = kGµ [δ0 ](x)(1 + o(1)) when x → 0 and u−νσ− = o(Gµ [δ0 ]) near 0. This implies Uνσ (x) = kGµ [δ0 ](x)(1 + o(1)) as x → 0 and we conclude as in the proof of Lemma 3.3 that u = uνσ +kδ0 . At end we let σ → 0. Up to a sequence {σj } converging to 0 such that uνσj +kδ0 → U almost everywhere and u−ν− ≤ U ≤ uν+ +kδ0 . (3.46) Since by Lemma 3.4, uνσ+ +kδ0 → uν+ +kδ0 in L1 (Ω, |x|−1 dγµ ) and g(uνσ+ +kδ0 ) → g(uν+ +kδ0 ) in L1 (Ω, ρdγµ ), we infer that the convergences of uνσj +kδ0 → U and g(uνσj +kδ0 ) → g(U ) occur respectively in the same space, therefore U = uν+kδ0 , it is the weak solution of (1.6).  Remark. In the course of the proof we have used the following result which is independent of any assumption on g except for the monotonicity: If {νn } ⊂ M+ (Ω; Γµ ) is an increasing sequence of g-good measures converging to a measure ν ∈ M+ (Ω; Γµ ), then ν is a g-good measure, {uνn } converges to uν in L1 (Ω, |x|−1 dγµ ) and {g(uνn )} converges to g(uν ) in L1 (Ω, ρdγµ ).

22

Leray-Hardy equations with absorption

3.5

Proof of Theorem C

The construction of a solution is essentially similar to the one of Theorem B, the only modifications lies in Lemma 3.3. Estimate (3.31) remains valid with ukδ0 (x) =

2−N k |x| 2 |S N −1 |

ln |x|−1 (1 + o(1)) = kGµ [δ0 ](x)(1 + o(1))

as x → 0.

(3.47)

2−N

Since uνσ (x) ≤ c|x| 2 , (3.32) holds with δ > 0 arbitrarily small. Next Z Z   2−N −1 |x| 2−N 2 2 dx g |Sk+δ g((k + δ)Gµ [δ0 ])dγµ (x) ≤ |x| ln |x| N −1 | B1 Ω Z 1   2−N −1 r N 2 2 dr = |S N −1 | g |Sk+δ ln r r N −1 | Z ∞ 0 2N = c10 g(t ln t)t− N −2 < ∞, c0

by (3.19) and (1.35). The end of the proof for Theorem C is similar to the one of Theorem B.  Proof of Corollary D. If g(r) = gp (r) = |r|p−1 r, p > 1, the existence of a solution with ν = kδ0 is a direct consequence of conditions (1.34) and (1.35). If k = 0 and νbΩ∗ 6= 0, the existence is  ensured if (1.8) holds, hence p < NN−2 . Assertion (iii) follows.

4 4.1

The supercritical case Reduced measures

The notion of reduced measures introduced by Brezis, Marcus and Ponce [8] turned out to be a useful tool in the construction of solutions in a measure framework. We will develop only the aspect needed for the proof of theorem E. If k ∈ N∗ , we set ( min{g(r), g(k)} if r ≥ 0, gk (r) = (4.1) max{g(r), g(−k)} if r > 0. Since gk satisfies (1.34) and (1.35), for any ν ∈ M+ (Ω; Γµ ) there exists a unique weak solution u = uν,k of ( Lµ u + gk (u) = ν in Ω, (4.2) u=0 on ∂Ω. Furthermore, from the proof of Lemma 3.4 and Kato’s type estimates Proposition 2.1 we have that 0 ≤ uν+ ,k0 ≤ uν+ ,k for all k 0 ≥ k > 0. (4.3) Proposition 4.1 Let ν ∈ M+ (Ω; Γµ ). Then the sequence of weak solutions {uν,k } of ( Lµ u + gk (u) = ν in Ω, u=0

on ∂Ω,

(4.4)

23

Leray-Hardy equations with absorption

decreases and converges, when k → ∞, to some nonnegative function u, and there exists a measure ν ∗ ∈ M+ (Ω; Γµ ) such that 0 ≤ ν ∗ ≤ ν and u = uν ∗ . Proof. The proof is similar to the one of [8, Prop. 4.1]. Observe that uν,k ↓ u∗ and the sequence {uν,k } is uniformly integrable in L1 (Ω, |x|−1 dγµ ). By Fatou’s lemma u satisfies Z Z  ∗ ∗ ∗ (4.5) u Lµ ξ + g(u )ξ dγµ (x) ≤ ξd(Γµ ν) for all ξ ∈ Xµ(Ω), ξ ≥ 0. Ω



Hence u∗ is a subsolution of (1.6) and by construction it is the largest of all nonnegative subsolutions. The mapping Z  u∗ L∗µ ξ + g(u∗ )ξ dγµ (x) for all ξ ∈ Cc∞ (Ω), ξ 7→ Ω

is a positive distribution, hence a measure ν ∗ , called the reduced measure of ν. It satisfies 0 ≤ ν ∗ ≤ ν and u∗ = uν ∗ .  Lemma 4.2 Let ν, ν 0 ∈ M+ (Ω; Γµ ). If ν 0 ≤ ν and ν = ν ∗ , then ν 0 = ν 0∗ . Proof. Let uν 0 ,k be the weak solution of the truncated equation ( Lµ u + gk (u) = ν 0 in Ω, u=0

(4.6)

on ∂Ω.

Then 0 ≤ uν 0 ,k ≤ uν,k . By Proposition 4.1, we know that uν,k ↓ uν ∗ = uν and uν 0 ,k ↓ u0∗ a.e. in L1 (Ω, |x|−1 dγµ ) and then Lµ (uν,k − uν ) + gk (uν,k ) − gk (uν ) = g(uν ) − gk (uν ), from what follows, by Proposition 2.1, Z Z Z (uν,k − uν ))|x|−1 dγµ + |gk (uν,k ) − gk (uν )|η1 dγµ ≤ |g(uν ) − gk (uν )|η1 dγµ . Ω





By the increasing monotonicity of mapping k 7→ gk (uν ), we have gk (uν ) → g(uν ) in L1 (Ω, ρdγµ ) as k → +∞, hence Z Z |gk (uν,k ) − g(uν )|η1 dγµ ≤ 2 |g(uν ) − gk (uν )|η1 dγµ → 0 as n → ∞. Ω



Because gk (uν 0 ,k ) ≤ gk (uν,k ) it follows by Vitali’s convergence theorem that gk (uν 0 ,k ) → g(u0∗ ) in L1 (Ω, ρdγµ ). Using the weak formulation of (4.6), we infer that u0∗ verifies Z Z  0∗ ∗ 0∗ u Lµ ξ + g(u )ξ dγµ = ξd(γµ ν 0 ) for all ξ ∈ Xµ (Ω). Ω

This yields

u0∗

= uν 0 .

The next result follows from Lemma 4.2.





24

Leray-Hardy equations with absorption

Lemma 4.3 Assume that ν = νbΩ∗ +kδ0 ∈ M+ (Ω; Γµ ), then ν ∗ = ν ∗ bΩ∗ +k ∗ δ0 ∈ M+ (Ω; Γµ ) with ν ∗ bΩ∗ ≤ νbΩ∗ and k ∗ ≤ k. More precisely, (i) If µ > µ0 and g satisfies (1.34), then k = k ∗ . (ii) If µ = µ0 and g satisfies (1.35), then k = k ∗ . (ii) If µ > µ0 (resp. µ = µ0 ) and g does not satisfy (1.21) (resp. (1.35)), then k ∗ = 0. The next result is useful in applications. Corollary 4.1 If ν ∈ M+ (Ω; Γµ ), then ν ∗ is the largest g-good measure smaller or equal to ν. Proof. Let λ ∈ M+ (Ω; Γµ ) be a g-good measure, λ ≤ ν. Then λ∗ = λ ≤ ν ∗ . Since ν ∗ is a g-good measure the result follows.  Proof of Theorem E. Assume that ν ≥ 0. By Lemma 4.2 and Remark at the end of Section 3.5 the following assertions are equivalent: (i) ν is gp -good. (ii) For any σ > 0, νσ = χBσc ν is gp -good. If νσ is good, then uνσ satisfies −∆uνσ + upνσ = νσ −

µ uν |x|2 σ

in D0 (Ω∗ )

(4.7)

and since uνσ (x) ≤ c|x|τ+ if |x| ≤ σ2 (4.7) holds in D0 (Ω). This implies that u ∈ Lp (Ω) and |x|−2 uνσ ∈ Lα (B σ2 ) for any α < (2−τN+ )+ . Using [1] the measure νσ is absolutely continuous with respect to the c2,p0 -Bessel capacity. If E ⊂ Ω is a Borel set such that c2,p0 (E) = 0, then c2,p0 (E ∩ Bσc ) = 0, hence ν(E ∩ Bσc ) = νσ (E ∩ Bσc ) = 0. By the monotone convergence theorem ν(E) = 0. Conversely, if ν is nonnegative and absolutely continuous with respect to the c2,p0 -Bessel capacity, then so is νσ = χBσc ν. For 0 ≤  ≤ σ2 we consider the problem  µ p    −∆u + |x|2 u + u = νσ u=0    u=0

in Ω := Ω \ B , on ∂B ,

(4.8)

on ∂Ω.

Since |x|µ2 is bounded in Ω and νσ is absolutely continuous with respect to the c2,p0 capacity there exists a solution uνσ , thanks to [1], unique by monotonicity. Now the mapping  7→ uνσ , is decreasing. We use the method developed in Lemma 3.1, when  → 0, we know that uνσ , increase to some uσ which is dominated by G[νσ ] and satisfies ( µ −∆u + 2 u + up = νσ in Ω∗ , |x| (4.9) u=0 on ∂Ω. Because uσ ≤ G[νσ ] and νσ = 0 in Bσ , there holds u(x) ≤ c011 Γµ (x) in B σ2 , and then uσ is a solution in Ω and u = uνσ . Letting σ → 0, we conclude as in Lemma 3.1 that uνσ converges to

25

Leray-Hardy equations with absorption

uν which is the weak solution of ( µ −∆u + 2 u + up = ν |x| u=0

in Ω,

(4.10)

on ∂Ω.

If ν is a signed measure absolutely continuous with respect to the c2,p0 -capacity, so are ν+ and ν− . Hence there exists solutions uν+ and uν− . For 0 <  < σ2 we construct uνσ , with the property that −u−ν− σ , ≤ uνσ , ≤ uν+ σ , , we let  → 0 and deduce the existence of uνσ which is eventually the weak solution of ( µ −∆u + 2 u + |u|p−1 u = νσ in Ω∗ , |x| (4.11) u=0 on ∂Ω, and satisfies −u−ν− σ ≤ uνσ ≤ uν+ σ . Letting σ → 0 we obtain that u = lim uνσ satisfies σ→0

(

−∆u +

µ u + |u|p−1 u = ν |x|2 u=0

in Ω∗ ,

(4.12)

on ∂Ω.

Hence u = uν and ν is a good solution.



Proof of Theorem F. Part 1. Without loss of generality we can assume that Ω is a bounded smooth domain. Let K ⊂ Ω be compact. If 0 ∈ K and p < p∗µ there exists a solution ukδ0 , hence K is not removable. If 0 ∈ / K and c2,p0 (K) > 0, there exists a capacitary measure νK ∈ W −2,p (Ω) ∩ M+ (Ω) with support in K. This measure is gp -good by Theorem E, hence K is not removable. Part 2. Conversely we first assume that 0 ∈ / K. Then there exists a subdomain D ⊂ Ω such ¯ and K ⊂ D. Hence a solution u of (1.37) is also a solution of that 0 ∈ /D −∆u +

µ u + |u|p−1 u = 0 |x|2

in D \ K,

¯ By [1, Theorem 3.1] it can be extended as a and the coefficient |x|µ2 is uniformly bounded in D. C 2 solution of the same equation in Ω0 . Hence, if c2,p0 (K) = 0 the set K is removable. If 0 ∈ K we have to assume at least p ≥ p∗µ in order that 0 is removable and p ≥ p0 in order there exists non-empty set with zero c2,p0 -capacity. Let ζ ∈ C01,1 (Ω) with 0 ≤ ζ ≤ 1, vanishing in a compact neighborhood D of K. Since 0 ∈ / Ω \ D, we first consider the case where u is nonnegative and satisfies in the usual sense −∆u +

µ u + up = 0 |x|2

in Ω \ D.

0

Taking ζ 2p for test function, we get −2p

0

Z uζ Ω

2p0 −1

Z Z Z 0 uζ 2p 0 2p0 −2 2 dx + ζ 2p up dx = 0. ∆ζdx − 2p (2p − 1) uζ |∇ζ| dx + µ 2 Ω Ω |x| Ω 0

0

26

Leray-Hardy equations with absorption

There holds

Z Z  1 Z  10 p p 2p0 p p0 p0 uζ 2p0 −1 ∆ζdx ≤ ζ u dx |∆ζ| ζ dx , Ω



Z 0≤



2p0 −2



Z

2

|∇ζ| dx ≤

ζ



 1 Z p

2p0 p

|∇ζ|

u dx



2p0

0

uζ 2p dx ≤ 0≤ 2 Ω |x|

Z ζ

p

dx

,



and Z

 10

2p0 p

1

0

ζ 2p dx 2p0 Ω |x|

Z

p

u dx



! 10 p

.

By standard elliptic equations regularity estimates and Gagliardo-Nirenberg inequality [21] (and since 0 ≤ ζ ≤ 1), Z  10 p p0 p0 |∆ζ| ζ ≤ c11 kζkW 2,p0 Ω

and Z

2p0

|∇ζ|

 10 p

dx



≤ c12 kζkW 2,p0 .

Finally, if p > p0 := NN−2 , then 2p0 < N which implies that there exists c13 independent of ζ (with value in [0, 1]) such that ! 10 Z  10 0 p p ζ 2p dx ≤ := c13 . 0 dx 0 2p 2p Ω |x| B1 |x|

Z

Next we set Z X=

ζ

2p0 p

u dx

1

p

,



and we obtain if µ ≥ 0, if p ≥ p0  X p − 2p0 (2p0 − 1)c12 − p0 c12 kζkW 2,p0 X ≤ 0,

(4.13)

and if µ < 0 if p > p0 Xp −

  2p0 (2p0 − 1)c12 − p0 c12 kζkW 2,p0 − c13 µ X ≤ 0.

(4.14)

However, the condition p > p0 is ensured when µ < 0 since p ≥ p∗µ > p0 . We consider a sequence {ηn } ⊂ S(RN ) such that 0 ≤ ηn ≤ 1, ηn = 0 on a neighborhood of K and such that kηn kW 2,p0 → 0 when n → ∞. Such a sequence exists by the result in [24] since c2,p0 (K) = 0. Let ξ ∈ C0∞ (Ω) such that 0 ≤ ξ ≤ 1 and with value 1 in a neighborhood of K. We take 0 ζ := ζn = (1 − ηn )ξ in the above estimates. Letting n → ∞, then ζn → ξ in W 2,p and finally X

p−1

Z =

ξ Ω

 p−1 p  ≤ 2p0 (2p0 − 1)c12 − p0 c12 kξkW 2,p0 + c13 µ− , u dx

2p0 p

(4.15)

27

Leray-Hardy equations with absorption

under the condition that p > p0 if µ < 0, in which case there also holds Z 0 uζ 2p dx ≤ c13 X. 2 Ω |x|

(4.16)

However the condition p > p0 is not necessary in order the left-hand side of (4.16) be bounded, since we have Z 0  uζ 2p p 0 0 0 µ dx + X ≤ 2p (2p − 1)c − p c kζkW 2,p0 X, (4.17) 12 12 2 Ω |x| and X is bounded. Next we take ζ := ζn = (1 − ηn )ξ for test function in (1.37) and get Z Z Z uζn dx + ζn up dx = 0. − ((1 − ηn )∆ξ − ξ∆ηn − 2h∇ηn , ∇ξi) udx + µ 2 |x| Ω Ω Ω Since Z

Z uξ∆ηn dx ≤ Ω

and

p

1

u ξdx Ω

p

kηn kW 2,p0 → 0

as n → ∞,

Z Z 1 p p uh∇ηn , ∇ξidx ≤ k∇ξkL∞ kηn kW 1,p0 u |∇ξ|dx Ω

as n → ∞,



then we conclude that u satisfies Z Z Z uξ − u∆ξdx + µ dx + ξup dx = 0, 2 |x| Ω Ω Ω

(4.18)

which proves that u satisfies the equation in the sense of distributions. By standard regularity u is C 2 in Ω∗ , and by the maximum principle u(x) ≤ c14 Γµ (x) in Br0 ⊂ Ω. Integrating by part as in the proof of Lemma 3.2 we obtain that u satisfies Z  uL∗µ ξ + ξup dγµ (x) = 0 for every ξ ∈ Xµ (Ω). (4.19) Ω

Finally, if u is a signed solution, then |u| is a subsolution. For  > 0 we set K = {x ∈ RN : dist (x, K) ≤ }. If  is small enough K ⊂ Ω. Let v := v be the solution of  µ p  in Ω \ K ,  −∆v + 2 v + v = 0  |x| (4.20) v = |u|b∂K on ∂K ,    v = |u|b∂Ω on ∂Ω. Then |u| ≤ v . Furthermore, by Keller-Osserman estimate as in [22, Lemma 1.1], there holds v (x) ≤ c15 dist (x, K )

2 − p−1

for all x ∈ Ω \ K ,

(4.21)

where c14 > 0 depends on N , p and µ. Using local regularity theory and the Arzela-Ascoli ¯ \ K) theorem, there exists a sequence {n } converging to 0 an a function v ∈ C 2 (Ω \ K) ∩ C(Ω

Leray-Hardy equations with absorption

28

¯ \ K and in the C 2 (Ω \ K)-topology. This such that {vn } converges to v locally uniformly in Ω loc implies that v is a positive solution of (1.37) in Ω \ K. Hence it is a solution in Ω. This implies that u ∈ Lp (Ω) and |u(x)| ≤ v(x) ≤ c14 Γµ (x) in Ω∗ . We conclude as in the nonnegative case that u is a weak solution in Ω.  Acknowledgements: H. Chen is supported by NSF of China, No: 11726614, 11661045, by the Jiangxi Provincial Natural Science Foundation, No: 20161ACB20007, and by the Alexander von Humboldt Foundation.

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