Schrödinger operators with Leray-Hardy potential singular on the boundary

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singular on the boundary

Huyuan Chen, Laurent Veron

To cite this version:

Huyuan Chen, Laurent Veron. Schrödinger operators with Leray-Hardy potential singular on the boundary. Journal of Differential Equations, Elsevier, In press. �hal-02157156v3�

singular on the boundary

Huyuan Chen^∗ Laurent V´eron ^†

Abstract

We study the kernel function of the operator u 7→ L_µu = −∆u+ _|x|^µ₂u in a bounded smooth domain Ω ⊂ R^N+ such that 0 ∈ ∂Ω, whereµ ≥ −^N₄² is a constant. We show the existence of a Poisson kernel vanishing at 0 and a singular kernel with a singularity at 0. We prove the existence and uniqueness of weak solutions of Lµu= 0 in Ω with boundary data ν+kδ₀, whereν is a Radon measure on∂Ω\ {0},k∈Rand show that this boundary data corresponds in a unique way to the boundary trace of positive solution of Lµu= 0 in Ω.

Key Words: Hardy Potential, Harnack inequality, Limit set, Radon Measure.

MSC2010: 35J75, 35B44.

Contents

1 Introduction 2

2 The half-space setting 7

3 The Poisson kernel 8

3.1 Construction of the Poisson kernel when µ >0 . . . 12 3.2 Construction of the Poisson kernel when µ₁ ≤µ <0 . . . 14

4 The singular kernel 16

4.1 Classification of Boundary isolated singularities . . . 17 4.2 Existence and uniqueness . . . 25

5 The Dirichlet problem 27

∗Department of Mathematics, Jiangxi Normal University, Nanchang 330022, China. E-mail: chen- [email protected]

†Laboratoire de Math´ematiques et Physique Th´eorique, Universit´e de Tours, 37200 Tours, France. E-mail:

[email protected]

1

1 Introduction

We denote by L_µ the Schr¨odinger operator defined in a domain Ω⊂R^N by L_µu:=−∆u+ µ

|x|²u,

whereµis a real constant andN ≥2. This operator which is associated to the Hardy inequality has been thoroughly studied in the last thirty years. When the singular point 0 belongs to Ω, it appears a critical value

µ₀ =−

N −2 2

2

and the range of the parameterµin which the operator is bounded from below is [µ₀,∞). This is linked to the Hardy inequality

Z

Ω

|∇ζ|²+µ₀ Z

Ω

ζ²

|x|²dx≥0 for all ζ ∈C₀^∞(Ω).

Furthermore this inequality is never achieved if Ω is bounded, in which case a remainder was shown to exist by Br´ezis and V´azquez [4]. When λ is a Radon measure in Ω, the associated Dirichlet problem

L_µu=λ in Ω, u= 0 on ∂Ω

is studied in its full generality in [8] and [9] thanks to the introduction of a notion of very weak solution associated to some specific weight. Thanks to this new formulation an extensive treatment of the associated semilinear problem

L_µu+g(u) =λ in Ω, u= 0 on ∂Ω,

where g:R→Ris a continuous nondecreasing function, is developed in [9].

In this article we assume that the singular point of the potential lies on the boundary of the domain Ω, and we are mainly interested in the two problems:

1- To define a notion of very weak solutionfor the problem L_µu= 0 in Ω,

u=ν on ∂Ω, (1.1)

where ν is a Radon measure on ∂Ω, and more generaly on ∂Ω\ {0};

2- To prove the existence of a boundary trace for any positiveL_µ-harmonic function, i.e. solution of L_µu= 0 in Ω and to connect it to the problem (1.1).

The model example is Ω =R^N₊ :={x= (x⁰, xN)∈R^N−1×R: xN >0} although it is not a bounded domain. There exists a critical value

µ≥µ1 :=−N²

4 . (1.2)

This value is fundamental for the operator L_µ to be bounded from below since there holds, Z

R^N+

|∇ζ|²+µ1

Z

R^N+

ζ²

|x|²dx≥0 for allζ ∈C₀^∞(R^N+). (1.3) The analysis of the model case is explicit. Let (r, σ) ∈R+×S^N−1+ be the spherical coordinates inR^N+, then, if (1.2) is satisfied, there exist two different types of positiveL_µ-harmonic functions vanishing on∂R^N₊ \ {0},

γ_µ(r, σ) =r^α+ψ₁(σ) and φ_µ(r, σ) =

( r^α−ψ₁(σ) ifµ > µ₁, r^−N−22 ln(r⁻¹)ψ₁(σ) ifµ=µ₁,

(1.4) where ψ1(σ) = ^x_|x|^N generates ker(−∆⁰+ (N −1)I) in H₀¹(S^N−1+ ), and where

α+:=α+(µ) = 2−N

2 +

r

µ+N²

4 and α−:=α−(µ) = 2−N

2 −

r

µ+N²

4 . (1.5) Putdγ_µ(x) =γ_µ(x)dx. We define the γ_µ-dual operator L^∗_µ ofL_µby

L^∗_µζ =−∆ζ− 2

γ_µh∇γ_µ,∇ζi for all ζ ∈C²(R^N+), (1.6) and we prove that φµis, in some sense, the fundamental solution of

L_µu= 0 in R^N+, u=δ₀ on ∂R^N+, since it satisfies

Z

R^N+

φ_µL^∗_µζdγ_µ(x) =c_µζ(0) for all ζ ∈C_c(R^N₊)∩C^1,1(R^N+)

such that ρL^∗_µζ ∈L^∞(R^N₊), wherec_µ>0 is a normalized constant andρ(x) = dist(x, ∂Ω). Here ρ(x) =x_N when Ω =R^N+.

WhenR^N₊ is replaced by a bounded domain Ω satisfying the condition (C-1) 0∈∂Ω , Ω⊂R^N+ and hx,ni=O(|x|²) for all x∈∂Ω,

where n=n_x is the outward normal vector at x, inequality (1.3) holds but it is never achieved in the Hilbert space H₀¹(Ω). Note that the last condition in (C-1) holds if Ω is a C² domain. It is proved in [5] that under the assumption (C-1) there exists a remainder under the following form:

Z

Ω

|∇ζ|²+µ1

Z

Ω

ζ²

|x|²dx≥ 1 4 Z

Ω

ζ²

|x|²ln²(|x|R⁻¹_Ω )dx for all ζ ∈C_c^∞(Ω), (1.7) where R_Ω= max

z∈Ω |z|. Then there holds

`^Ω_µ := inf Z

Ω

|∇v|²+ µ

|x|²v²

dx:v∈C_c¹(Ω), Z

Ω

v²dx= 1

>0.

This first eigenvalue is achieved in H₀¹(Ω) if µ > µ₁, or in the space H(Ω) which is the closure of C_c¹(Ω) for the norm

v7→ kvk_H(Ω) :=

s Z

Ω

|∇v|²+ µ1

|x|²v²

dx,

when µ=µ₁. In the sequel we set H_µ(Ω) =

( H₀¹(Ω) if µ > µ1, H(Ω) if µ=µ₁.

Moreover, under the assumption (C-1) the imbedding of H_µ(Ω) in L²(Ω) is compact (see e.g.

[6]). We denote byγ_µ^Ω the positive eigenfunction, its satisfies ( L_µγ_µ^Ω =`^Ω_µγ_µ^Ω in Ω,

γ_µ^Ω = 0 on ∂Ω\ {0}. (1.8) We prove in Appendix that there exist cj =cj(Ω, µ)>0, j= 1,2, such that

(i) γ_µ^Ω(x) =c₁ρ(x)|x|^α+−1(1 +o(1)) as x→0, (ii) |∇γ^Ω_µ(x)| ≤c2

γ_µ^Ω(x)

ρ(x) for all x∈Ω.

(1.9)

This function will play the role of a weight function for replacing γ_µ. Next we construct the Poisson kernel K^Ω_µ of L_µ in Ω×∂Ω. Whenµ≥0 this construction can be made by truncation as in [18], considering for >0 andλ∈M₊(∂Ω) the solutionu of







−∆u+ µ

max{²,|x|²}u= 0 in Ω, u=λ on ∂Ω.

By a more elaborate method, we also construct the Poisson kernel when µ₁ ≤ µ < 0. It is important to notice that when µ >0 the kernel has the property that

K_µ^Ω(x,0) = 0 for all x∈Ω\ {0} (1.10) by [18, Theorem A.1]. Because of (1.10) it is clear that the Poisson kernel cannot be the tool for describing all the positive L_µ-harmonic functions. Our first concern in this article is to clarify the Poisson kernel ofL_µ.

We first characterize the positiveL_µ-harmonic functions which are singular at 0.

Theorem ALet Ωbe aC² bounded domain such that 0∈∂Ωandµ≥µ1. Ifuis a nonnegative L_µ-harmonic function in Ω vanishing on B_r0(0)∩(∂Ω\ {0}) for some r₀>0, then there exists k≥0 such that

x→0lim

u(x)

ρ(x)|x|^α−−1 =k,

if µ > µ₁ and

x→0lim

|x|^N2u(x)

ρ(x) ln|x| =−k, if µ=µ₁.

Actually the above convergences hold in a stronger way. In order to prove that such solutions truly exist we construct the kernel function φ^Ω_µ (see [13] for the denomination) which is the analogue in a bounded domain of the explicit singular solution φ_µ defined in R^N₊.

Theorem B Let Ω be a C² bounded domain satisfying (C-1) and µ ≥µ₁. Then there exists a positive L_µ-harmonic function in Ω, which vanishes on ∂Ω\ {0}, which satisfies

φ^Ω_µ(x) =ρ(x)|x|^α−−1(1 +o(1)) as x→0, (1.11) if µ > µ₁, and

φ^Ω_µ1(x) =ρ(x)|x|^−N2(|ln|x||+ 1)(1 +o(1)) as x→0, (1.12) if µ=µ1.

As in the model case, we define theγ_µ^Ω-dual operator ofL_µby L^∗_µζ =−∆ζ− 2

γ_µ^Ωh∇γ^Ω_µ,∇ζi+`^Ω_µζ for all ζ ∈C^1,1(Ω).

The following commutation formula holds

L_µ(γ_µ^Ωζ) =γ_µ^ΩL^∗_µζ. (1.13) Corollary CLetΩbe aC² bounded domain satisfying(C-1)andµ≥µ₁. Thenφ^Ω_µ is the unique function belonging to L¹(Ω, ρ⁻¹dγ^Ω_µ), which satisfies

Z

Ω

uL^∗_µζdγ_µ^Ω=kc_µζ(0) for all ζ ∈Xµ(Ω), (1.14) where dγ_µ^Ω =γ^Ω_µdx, here and in the sequel the test function space

Xµ(Ω) =

ζ ∈C(Ω) : γ^Ω_µζ ∈Hµ(Ω)and ρL^∗_µζ ∈L^∞(Ω) .

Furthermore, if u is a nonnegative L_µ-harmonic function vanishing on ∂Ω\ {0}, there exists k≥0 such that u=kφ^Ω_µ.

We denote byM(Ω;γ_µ^Ω) the set of Radon measures ν in Ω such that sup

Z

Ω

ζd|λ|:ζ∈C_c(Ω),0≤ζ ≤γ_µ^Ω

:=

Z

Ω

γ_µ^Ωd|ν|<+∞.

If ν∈M₊(Ω;γ_µ^Ω) the measure γ^Ω_µν is a nonnegative bounded measure in Ω. Put β_µ^Ω(x) =−∂γ_µ^Ω(x)

∂nx

= lim

t→0⁺

γ_µ^Ω(x−tnx)

t = lim

t→0⁺

γ_µ^Ω(x−tnx)

ρ^∗(x−tnx)), ∀x∈∂Ω\ {0} (1.15)

and from (3.2) and (1.9), we have that

c1|x|^α+−1 ≤β_µ^Ω(x)≤c1c3|x|^α+−1 forx∈∂Ω\ {0}. (1.16) As a consequence, the following potential function plays an important role in defining our bound- ary data. Denote

β_µ(x) =|x|^α+−1 forx∈R^N\ {0}. (1.17) The set Radon measures λin∂Ω\ {0} such that

sup (Z

∂Ω\{0}

ζd|λ|:ζ∈Cc(∂Ω\ {0}),0≤ζ ≤βµ

) :=

Z

∂Ω\{0}

βµd|λ|<+∞

is denoted byM(∂Ω\ {0};β_µ). The extension ofλ∈M₊(∂Ω\ {0};β_µ) as a measureβ_µλin∂Ω is given by

Z

∂Ω

ζd(β_µλ) = sup Z

∂Ω

υβ_µdλ:υ∈C_c(∂Ω\ {0}),0≤υ≤ζ

for all ζ∈C_c(∂Ω), ζ≥0 and βµλ =βµλ+−βµλ− ifλ is a signed measure in M(∂Ω\ {0};βµ), and this defines the set M(∂Ω;β_µ) of all such extensions. The Dirac mass at 0 does not belong to M(∂Ω;β_µ), but it is the limit of sequences of measures in this space in the same way as it is a limit of measures in M₊(∂Ω\ {0};βµ). In the next result we prove the existence and uniqueness of a solution to

L_µu=ν in Ω,

u=λ+kδ₀ on ∂Ω. (1.18)

Thanks to (1.7) the Green kernel G^Ω_µ is easily constructible. If ν ∈ M₊(Ω;γ^Ω_µ) and λ ∈ M(∂Ω;β_µ) the following expressions are well defined

K^Ωµ[λ](x) = Z

∂Ω

K_µ^Ω(x, y)dλ(y) and G^Ωµ[ν](x) = Z

Ω

G^Ω_µ(x, y)dν(y).

Our main existence result is the following.

Theorem D Let Ω be a C² bounded domain satisfying (C-1) and µ≥ µ1. If ν ∈ M₊(Ω;γ_µ^Ω), λ∈M(∂Ω;βµ) and k∈R, the function

u=G^Ω_µ[ν] +K^Ω_µ[λ] +kφ^Ω_µ :=H^Ω_µ[(ν, λ, k)] (1.19) is the unique solution of (1.18) in the very weak sense that u∈L¹(Ω, ρ⁻¹dγ_µ^Ω) and

Z

Ω

uL^∗_µζdγ_µ^Ω = Z

Ω

ζd(γ_µ^Ων) + Z

∂Ω

ζd(β_µ^Ωλ) +kc_µζ(0) for all ζ ∈Xµ(Ω). (1.20)

In the next result we prove that all the positive L_µ-harmonic functions in Ω are described by formula (1.19) (with ν = 0).

Theorem E Let Ω be a C² bounded domain such that 0 ∈∂Ω satisfying (C-1), µ≥µ₁ and u be a nonnegativeL_µ-harmonic functions in Ω. Then there exist λ∈M(∂Ω;βµ) and k≥0, such that

u=K^Ωµ[λ] +kφ^Ω_µ =H^Ωµ[(0, λ, k)].

The couple (λ, kδ0) is called the boundary trace of u.

The rest of this paper is organized as follows. In Section 2, we introduce the distributional identity of L_µ harmonic function φ_µ in R^N+. Section 3 is devoted to build the Kato’s type inequalities, to construct Poisson kernel and related properties. Section 4 is addressed to classify the boundary isolated singular L_µ harmonic functions in a bounded domain, i.e. Theorem A and to show the existence and related distributional identity in a (C-1) domain: proofs of Theorem B and Corollary C. We classify the boundary trace for general L_µ harmonic functions and give the existence of L_µ harmonic functions with the boundary trace (λ, kδ0):

Theorem D and Theorem E respectively in Section 5. Finally, we show Estimates (1.9) and (3.2) in Appendix.

In a forthcomming article [10] we develop our observations to study qualitative properties of the semilinear problem

L_µu+g(u) = 0 in Ω, u=λ on ∂Ω,

when the origin lies on the boundary of Ω and λis a Radon measure defined in proper way.

2 The half-space setting

LetR^N₊ :={x= (x⁰, xN)∈R^N−1×R: xN >0}, (r, σ)∈R+×S^N−1₊ be the spherical coordinates inR^N+ and ∆⁰ is the Laplace Beltrami operator onS^N−1. Then

L_µu=−∂_rru− N−1

r ∂ru− 1

r²∆⁰u+ µ r²u.

Ifu(r, σ) =r^αφ(σ) is a (separable) solution ofL_µu= 0 vanishing on∂R^N₊\ {0}, thenψ_k satisfies ( −∆⁰ψ_k=λ_kψ_k in S^N−1₊ :=S^N−1∩R^N+,

ψ_k= 0 in ∂S^N−1+ ≈S^N−2, where λ_k is a constant which necessarily belongs to the spectrum

σS^N−1+

(−∆⁰) ={λ_k=k(N +k−2) :k∈N^∗}, and α=αk+, αk− is a root of

α²+ (N−2)α−λ_k−µ= 0. (2.1)

The fundamental state corresponds to k= 1, in which case since λ1 =N −1, existence of real roots of (2.1) necessitates µ≥µ₁ =−^N₄² =µ₁ and we denote α_{1 +} =α₊ and α₁−=α−. Note that this value is connected to the boundary Hardy

Z

R^N+

|∇φ|²+µ1

Z

R^N+

φ²

|x|²dx≥0 for all φ∈C₀^∞(S₊^N−1).

If this condition is fulfilled, the two roots α₊ and α− corresponding to k= 1 and λ₁ are α+ = 2−N

2 +√

µ−µ1 and α−= 2−N

2 −√

µ−µ1. (2.2)

The corresponding positive separable solutions γµ and φµ of L_µu = 0 vanishing on ∂R^N+ \ {0}

are defined by (1.4). We set dγ_µ(x) =γ_µ(x)dxand define the operator L^∗_µ by (1.6).

Proposition 2.1 The function φµ belongs to L¹_loc(R^N+, ρ⁻¹dγµ). It satisfies Z

R^N+

φµL^∗_µζdγµ(x) =cµζ(0) for all ζ ∈Xµ(R^N+), (2.3) where

cµ= _N¹ ( √

µ−µ₁|S^N−1| if µ > µ₁,

1

2N|S^N−1| if µ=µ₁ (2.4)

and Xµ(R^N₊) = n

ζ ∈Cc(R^N₊) : ρL^∗_µζ ∈L^∞(R^N₊) o

.

Proof. Let ζ ∈ Xµ(R^N₊), > 0 and set B⁺ = B(0)∩R^N₊, (B⁺)^c = B^c(0)∩R^N₊ and Γ⁺ =

∂B(0)∩R^N+, by direct compute, we have that 0 =

Z

(^B+)^cζγµL_µφµdx

= Z

(^B+)^c

φ_µL^∗_µζdγ_µ(x) + Z

Γ⁺

−∂φ_µ

∂n ζγ_µ+

γ_µ∂ζ

∂n +ζ∂γ_µ

∂n

φ_µ

dS

= Z

(^B+)^c

φµL^∗_µζdγµ(x) +ζ(0)A() and n=−⁻¹x on Γ⁺ , and

A() =











−2√ µ−µ₁

Z

S^N−1+

x²_NdS+O() if µ > µ₁,

− Z

S^N−1+

x²_NdS+O() if µ=µ1,

(2.5)

together with the fact that Z

S^N−1+

x²_NdS = _2N¹ |S^N−1|, which implies (2.3)-(2.4).

3 The Poisson kernel

In this section we assume that Ω is a boundedC² domain included in B1 (which can always be assumed by scaling) and 0∈∂Ω.

We letσ^Ω_µ ∈H_µ(Ω) be the unique variational solution of L_µu= γ_µ^Ω

ρ^∗ in Ω and u= 0 on ∂Ω, (3.1)

where ρ^∗(x) = min{¹

l^Ω_µ, ρ}. We prove in Appendix that there isc₃ >1 such that

γ_µ^Ω ≤σ_µ^Ω ≤c₃γ_µ^Ω in Ω. (3.2)

Note that σ_µ^Ω∈C²(Ω\ {0}) is a positive classical solution of (3.1) with zero Dirichlet boundary condition on ∂Ω\ {0}, i.e. σ^Ω_µ = 0 on∂Ω\ {0}. Moreover, ∂σ^Ω_µ

∂n <0 on∂Ω\ {0}. We set η= σ_µ^Ω

γ_µ^Ω in Ω, which satisfies

L^∗_µη= 1

ρ in Ω. (3.3)

This function play a key role in the sequel. Clearly,η ∈C²(Ω\ {0}) and 1≤η ≤c2 in Ω\ {0}

by (3.2).

We start with the following identity of commutation valid for all λ ∈ M(∂Ω;β_µ) and ζ ∈ C^1,1(Ω)

− Z

∂Ω

∂(ζγ_µ^Ω)

∂n dλ= Z

∂Ω

ζd(β_µ^Ωλ) for all ζ ∈C^1,1(Ω), (3.4) where β_µ^Ω is defined in (1.15) with the asymptotic behavior (1.16) near the origin.

The following inequality extends the classical Kato inequality to our framework.

Lemma 3.1 Assume N ≥3and µ≥µ1 or N = 2 andµ > µ1. Then for any f ∈L¹(Ω, γ_µ^Ωdx), h∈L¹(∂Ω, β_µ^Ωdx), there exists a unique weak solution u to

L_µu=f in Ω,

u=h on ∂Ω (3.5)

in the sense that

Z

Ω

uL^∗_µζdγ_µ^Ω= Z

Ω

f ζdγ_µ^Ω+ Z

∂Ω

hζdβ^Ω_µ for all ζ ∈Xµ(Ω). (3.6) Furthermore, for any ζ ∈Xµ(Ω), ζ ≥0, there holds

Z

Ω

|u|L^∗_µζdγ_µ^Ω ≤ Z

Ω

sgn(u)f ζdγ_µ^Ω+ Z

∂Ω

|h|ζdβ_µ^Ω (3.7)

and Z

Ω

u⁺L^∗_µζdγ^Ω_µ ≤ Z

Ω

sgn₊(u)f ζdγ^Ω_µ + Z

∂Ω

h⁺ζdβ_µ^Ω. (3.8)

Proof. Uniqueness. Assume that u is a weak solution of (3.5) with f = h= 0. Then for any ζ ∈Xµ(Ω) there holds

Z

Ω

uL^∗_µζdγ_µ^Ω = 0.

Let φ∈C_c^∞(Ω), andυ∈H_µ(Ω) be the variational solution of L_µυ= γ_µ^Ω

ρ φ and u∈Hµ(Ω).

Then υ ∈ C^∞(Ω), |υ| ≤ kφk_L^∞σ^Ω_µ; the equation is satisfied everywhere and in the sense of distributions in Ω. Clearly w:= (γ_µ^Ω)⁻¹υ belongs to C^∞(Ω) and satisfies

L^∗_µ:= 1 ρφ.

Thus,

Z

Ω

u

ρφdγ_µ^Ω= 0.

Since φis arbitrary, we have that u= 0.

Existence and estimates. We proceed by approximation as in [8, Prop. 2.1]. We assume that {(f_n, h_n)} ⊂ C₀¹(Ω)×C₀¹(∂Ω\ {0}) is a sequence which converges to (f, h) in L¹(Ω, γ_µ^Ωdx)× L¹(∂Ω, β_µ^Ωdx). We set V(x) = |x|⁻², denote by K^Ω₀ the Poisson potential of −∆ in Ω and consider the approximate problem

( L_µwn=fn−µVK^Ω₀[hn] in Ω,

w_n= 0 on ∂Ω. (3.9)

Near 0, we have VK^Ω0[hn](x) = O(^ρ(x)_|x|2), hence, if N ≥ 3, VK^Ω0[hn] ∈ L²(Ω). If N = 2, the function x 7→ ^ρ(x)_|x|2 belongs to the Lorentz space L^2,∞(Ω) which is the dual of L^2,1(Ω).

Since Hµ(Ω) ⊂ L^2,1(Ω) by (1.7), it follows that H_µ⁰(Ω) ⊂ L^2,∞(Ω). Hence, by Lax-Milgram theorem there exists a uniquewn∈Hµ(Ω) such that (3.9) holds in the variational sense. Then u_n=w_n+K^Ω₀[h_n], which has the same regularity as w_n, satisfies

L_µun=fn in Ω,

u_n=h_n on ∂Ω. (3.10)

For σ >0, we set

mσ(t) =

( |t| − ^σ₂ if |t| ≥σ,

t²

2σ if |t|< σ.

Then mσ is convex, |m⁰_σ(t)| ≤ 1 and m⁰_σ(t) →sign0(t) as σ ↓ 0. Let ζ ∈ C^1,1(Ω), ζ ≥ 0. We have that

Z

Ω

h∇w_n,∇(ζm⁰_σ(u_n)γ_µ^Ω)i+µV w_nζm⁰_σ(u_n)γ_µ^Ω dx

= Z

Ω

ζm⁰_σ(un)γ_µ^Ωfndx−µ Z

Ω

VK^Ω0[hn]ζm⁰_σ(un)γ_µ^Ωdx=:R(σ).

By the factu_n=w_n+K^Ω₀[h_n], we have that R(σ) =

Z

Ω

h∇u_n,∇(ζm⁰_σ(un)γ_µ^Ω)idx− Z

Ω

h∇K^Ω0[hn],∇(ζm⁰_σ(un)γ_µ^Ω)idx +µ

Z

Ω

V u_nζm⁰_σ(u_n)γ_µ^Ωdx−µ Z

Ω

VK^Ω₀[h_n]ζm⁰_σ(u_n)γ_µ^Ωdx

= Z

Ω

|∇u_n|²m⁰⁰_σ(un)ζγ^Ω_µdx+ Z

Ω

h∇m_σ(un),∇(ζγ_µ^Ω)idx+µ Z

Ω

V unζm⁰_σ(un)γ_µ^Ωdx

−µ Z

Ω

VK^Ω0[h_n]ζm⁰_σ(u_n)γ_µ^Ωdx

≥ − Z

Ω

m_σ(u_n)∆(ζγ^Ω_µ)dx+ Z

∂Ω

m_σ(h_n)ζ∂γ_µ^Ω

∂n dS+µ Z

Ω

V u_nζm⁰_σ(u_n)γ_µ^Ωdx

−µ Z

Ω

VK^Ω₀[h_n]ζm⁰_σ(u_n)γ_µ^Ωdx

≥ Z

Ω

mσ(un)L^∗_µζdγ_µ^Ω− Z

∂Ω

mσ(hn)dβ_µ^Ω+µ Z

Ω

V (unm⁰_σ(un)−mσ(un))γ_µ^Ωζdx

−µ Z

Ω

VK^Ω₀[h_n]ζm⁰_σ(u_n)γ_µ^Ωdx, thus, we obtain that

Z

Ω

mσ(un)L^∗_µζdγ_µ^Ω+µ Z

Ω

V

unm⁰_σ(un)−mσ(un)

ζdγ^Ω_µ

≤ Z

Ω

ζm⁰_σ(u_n)f_ndγ_µ^Ω+ Z

∂Ω

m_σ(h_n)ζdβ_µ^Ω.

(3.11)

Since m_σ is convex, u_nm⁰_σ(u_n)−m_σ(u_n)≥0. Hence for µ≥0, we can let σ→ 0 in (3.11) and obtain (3.7).

Forµ∈[µ₁,0), we note that

0≤unm⁰_σ(un)−mσ(un)≤ |u_n|²

2σ² χ_{|un|≤σ}

and

0≤ Z

Ω

V unm⁰_σ(un)−mσ(un)

ζdγ_µ^Ω ≤ kζk_L^∞ 2

Z

{|un|≤σ}

|x|^−1−N2+

√µ−µ₁

dx. (3.12)

Hence if N ≥3, or N = 2 and µ > µ1 =−1, the right-hand side of (3.12) tends to 0 as σ →0 and then we obtain (3.7). The proof of (3.8) is similar.

Applying estimate (3.7) tou_n−u_m, we obtain for all ζ ∈Xµ(Ω),ζ ≥0, Z

Ω

|u_n−u_m|L^∗_µζdγ_µ^Ω≤ Z

Ω

|f_n−f_m|ζdγ_µ^Ω+ Z

∂Ω

|h_n−h_m|ζdβ_µ^Ω.

For test function, we take η, the solution of (3.3), then Z

Ω

|u_n−u_m| ρ dγ_µ^Ω≤

Z

Ω

|f_n−fm|dσ_µ^Ω+ Z

∂Ω

|h_n−hm|d(ηβ_µ^Ω).

Therefore{u_n}is a cauchy sequence inL¹(Ω, ρ⁻¹dγ^Ω_µ) with limitu. Sinceu_n satisfies (3.10), we let ngo to infty in

Z

Ω

unL^∗_µζ dγ^Ω_µ = Z

Ω

ζσ^Ω_µfndx+ Z

∂Ω

ζβ_µ^ΩhndS

and obtain (3.6).

Lemma 3.2 Assume λ∈M(∂Ω;βµ) and ζ ∈Xµ(Ω), then there holds Z

Ω

L^∗_µζ− µ

|x|²ζ

K^Ω₀[λ]dγ_µ^Ω = Z

∂Ω

ζd(β_µ^Ωλ).

Proof. Note thatd(β_µ^Ωλ) is equivalent tod(β_µλ) by (1.16). By (1.13) we have almost everywhere in Ω,

L^∗_µζ− µ

|x|²ζ

K^Ω0[λ]γ_µ^Ω =

L_µ(γ_µ^Ωζ)− µ

|x|²γ_µ^Ωζ

K^Ω0[λ] =−∆(γ_µ^Ωζ)K^Ω0[λ].

If we assume that λvanishes in a neighborood of 0 we derive from (3.4)

− Z

Ω

∆(γ_µ^Ωζ)K^Ω₀[λ]dx=− Z

∂Ω

∂(ζγ_µ^Ω)

∂n dλ= Z

∂Ω

ζd(β_µ^Ωλ).

Sinceγ_µ^Ω

L^∗_µζ− µ

|x|²ζ

is bounded, we obtain the result first ifλis nonnegative by considering the sequence{χ

Bcλ} and letting→0, and then for anyλ=λ⁺−λ⁻. We observe also that the existence of the Green kernel follows from Lax-Milgram theorem which gives the existence of a unique variational solution in Hµ(Ω) to







−∆u+ µ

|x|²u=f in Ω, u= 0 on ∂Ω.

We denote by G^Ω_µ the Green kernel and by G^Ωµ the corresponding Green operator defined by G^Ωµ[f](x) =R

ΩG^Ω_µ(x, y)f(y)dy.

3.1 Construction of the Poisson kernel when µ >0

For the sake of completeness, we recall the construction in [18]. For > 0, we set V(x) = max{⁻²,|x|⁻²} and V₀(x) =V(x) =|x|⁻², and if λ∈M(∂Ω), let u be the solution of

−∆u+µVu= 0 in Ω, u=λ on ∂Ω.

Then

u(x) = Z

∂Ω

K_µ,^Ω (x, y)dλ(y) =K^Ωµ,[λ].

We obtain by the maximum principle,

K_µ,^Ω ≤K_µ^Ω0,⁰ ≤K₀^Ω for all µ≥µ⁰ ≥0 and ⁰≥ >0, where K^Ω is the usual Poisson kernel in Ω and there exists

K_µ^Ω(x, y) = lim

→0K_µ,^Ω (x, y) for all (x, y)∈Ω×∂Ω.

Therefore we infer, firstly by monotone convergence ifλ≥0, and then for anyλ∈M(∂Ω), that

→0limu(x) =u(x) = Z

∂Ω

K_µ^Ω(x, y)dλ(y) for all x∈Ω. (3.13) Since V is finite inB^c∩Ω, for anyx∈Ω, K_µ^Ω(x, y)>0 for ally∈∂Ω\ {0}. LettingG^Ω₀ be the Green kernel in Ω, there holds

u(x) +µ Z

Ω

G^Ω₀(x, y)V(y)u(y)dy= Z

∂Ω

K₀^Ω(x, y)dλ(y).

If λ≥0, we have by Fatou’s lemma, Z

Ω

G^Ω₀(x, y)V(y)u(y)dy≤lim inf

→0

Z

Ω

G^Ω₀(x, y)V(y)u(y)dy. (3.14) Combined with (3.13) it yields

u(x) +µ Z

Ω

G^Ω₀(x, y)V(y)u(y)dy≤ Z

∂Ω

K₀^Ω(x, y)dλ(y) for all x∈Ω.

Since the function u+µG^Ω₀[V u] is nonnegative and harmonic in Ω, it admits a boundary trace which is a nonnegative Radon measure λ^∗ and there holds

u(x) +µ Z

Ω

G^Ω₀(x, y)V(y)u(y)dy= Z

∂Ω

K₀^Ω(x, y)dλ^∗(y) for all x∈Ω. (3.15) Because of (3.14), we have that 0≤λ^∗≤λand the measureλ^∗is the reduced measure associated toλ. Since (3.15) is equivalent to

u(x) = Z

∂Ω

K_µ^Ω(x, y)dλ^∗(y),

there holds Z

∂Ω

K_µ^Ω(x, y)d(λ−λ^∗)(y) = 0.

This implies that λ=λ^∗ in∂Ω\ {0}. With the notations of [18], we recall that Sing_V(Ω) :=

y ∈∂Ω : ∃x₀ ∈Ω s.t. K_µ^Ω(x₀, y) = 0

⊂Z_V :=

y∈∂Ω : Z

Ω

K₀^Ω(x, y)V(x)ρ(x)dx=∞

.

Actually, ify∈ Sing_V(Ω),K_µ^Ω(x₀, y) = 0 for anyx₀∈Ω by Harnack inequality. Clearly 0∈Z_V and if y 6= 0 the integral term in the definition of ZV is finite. Hence Sing_V(Ω)⊂ Z_V ={0}.

Since for any truncated cone C_0,δ bΩ with vertex 0, there holds Z

C0,δ

V(x) dx

|x−y|^N−2 =∞,

it follows by Ancona’s result [18, Theorem A1] that 0∈ Sing_V(Ω). Finally K_µ^Ω(x,0) = 0 for all x∈Ω.

3.2 Construction of the Poisson kernel when µ₁ ≤µ < 0

For >0 and λ∈C(∂Ω),λ≥0 we denote by w=w_,λ the variational solution inHµ(Ω) of ( −∆w+µVw=−µVK^Ω₀[λ] in Ω,

w= 0 on ∂Ω.

Then w_,λ≥0 and u=u_,λ :=w+K^Ω₀[λ] satisfies

( −∆u+µVu= 0 in Ω,

u=λ on ∂Ω. (3.16)

Since −∆u_,λ+µV u_,λ≤0, there holds from Lemma 3.1 Z

Ω

u,λL^∗_µζdγ_µ^Ω≤ Z

∂Ω

λζdβ_µ^Ω forζ ∈Xµ(Ω), ζ ≥0 and in particular

Z

Ω

u_,λ ρ dγ_µ^Ω≤

Z

∂Ω

λd(ηβ_µ^Ω). (3.17)

If > ⁰>0 andλ⁰> λ >0 we have

−∆u_,λ

u_,λ +∆u⁰_,λ⁰

u⁰_,λ⁰ =µ(V⁰ −V)≤0.

Since Z

Ω

−∆u_,λ u,λ

+∆u⁰_,λ⁰ u⁰_,λ⁰

(u²_,λ−u²0,λ⁰)+dx

= Z

{u_,λ≥u0,λ0}

∇u_,λ− u_,λ

u⁰_,λ⁰∇u⁰_,λ⁰

2

+

∇u⁰_,λ⁰−u⁰_,λ⁰ u_,λ ∇u_,λ

2! dx,

we deduce that the function x7→ ^u_u^0,λ0

,λ (x) is constant on the set{x:u_,λ(x)> u⁰_,λ⁰(x)}. If this set is non-empty we get a contradiction since it is strictly included in Ω. Therefore the mapping

(, λ)7→u_,λ is decreasing in and increasing in λ.

Next we can assume that λ ∈ M₊(∂Ω) vanishes in B_δ∩∂Ω and that {λ_n} ⊂ C(∂Ω) is a sequence of functions which converge to λ in the weak sense of measures. We denote by u_,λn the solution of (3.16) with λreplaced by λ_n. Since µ <0,α₊ <1, ρ⁻¹γ_µ^Ω ∼ |x|^α+−1 ≥R^αω⁺⁻¹, where R_ω = max{|z|:z∈Ω}. Hence

Z

Ω

u,λndx≤c8

Z

∂Ω

λnd(ηβ_µ^Ω)≤c9kλk_M(∂Ω).

Hence u_,λn and Vu_,λn are uniformly bounded in L¹(Ω). From standard regularity estimates the sequence {u_,λn}_n∈N is bounded in the Lorentz spaces L^{N−1N ,∞}(Ω) and weakly relatively compact in L¹(Ω) (see e.g. [12]). This implies that, up to a subsequence, u,λn converges in L¹(Ω) and a.e. in Ω to a weak solution u_,λ of

( −∆u+µVu= 0 in Ω, u=λ on ∂Ω, that is a function which satisfies

Z

Ω

(−u_,λ∆ζ+µVu,λζ)dx=− Z

∂Ω

∂ζ

∂ndλ for all ζ ∈C₀^1,1(Ω). (3.18) Furthermore (3.17) holds (with the same notation). For test functionζ in (3.18), we takeζ =θ1

be the solution of

−∆θ₁ = 1 in Ω, θ₁ = 0 on ∂Ω.

Then Z

Ω

u,λdx=−µ Z

Ω

Vu,λθ1dx− Z

∂Ω

∂θ1

∂ndλ.

By the monotone convergence theorem we obtain that Vu_,λ →V u_λ inL¹(Ω, θ₁dx) by letting

→0 and Z

Ω

uλdx=−µ Z

Ω

V uλθ1dx− Z

∂Ω

∂θ1

∂ndλ.

Hence Z

Ω

(−∆ζ+µV ζ)u_λdx=− Z

∂Ω

∂ζ

∂ndλ for all ζ ∈C₀^1,1(Ω).

We also have Z

Ω

u_,λnL^∗_µζdγ_µ^Ω=µ Z

Ω

(V −V)u_,λnζdγ_µ^Ω+ Z

∂Ω

ζλ_ndβ^Ω_µ for all ζ ∈Xµ(Ω).

Since u_,λn converges in L¹(Ω) we obtain ifζ ≥0, Z

Ω

u_,λL^∗_µζdγ_µ^Ω =µ Z

Ω

(V −V)u_,λζdγ_µ^Ω+ Z

∂Ω

ζd(λβ_µ^Ω)≤ Z

∂Ω

ζd(λβ^Ω_µ)

by using the fact that µ(V −V)≤0. When→0,u_,λ increases and converges to some u_λ in L¹(Ω, ρ⁻¹dγ_µ^Ω), which satisfies

Z

Ω

u_λL^∗_µζdγ_µ^Ω≤ Z

∂Ω

ζd(λβ_µ^Ω) for all ζ ∈Xµ(Ω), ζ ≥0.

For δ >0 denote byζ_δ the solution of L^∗_µζ_δ =χ_Ω

δL^∗_µζ where Ω_δ={x∈Ω :ρ(x)> δ}.

Asζ ∈Xµ(Ω),|L^∗_µζ| ≤c10ρ, henceζ_δ ∈Xµ(Ω),|ζ_δ| ≤c10ηandζ_δ→ζ whenδ→0. Furthermore, sincec₁₁|x|is a supersolution for c₁₁>0 large enough, η_δ(x)≤c₁₁|x|. Hence

Z

Ωδ

u_,λL^∗_µζdγ_µ^Ω=µ Z

{|x|<}

1

|x|²u_,λζ_δdγ^Ω_µ + Z

∂Ω

ζ_δd(λβ_µ^Ω).

Because |x|⁻²u,λ|ζ_δ| ≤c11ρ⁻¹u,λ and u,λ →uλ inL¹(Ω, ρ⁻¹dγ_µ^Ω), we derive that

→0lim Z

{|x|<}

1

|x|²u_,λζ_δdγ_µ^Ω = 0, which implies

Z

Ω_δ

u_λL^∗_µζdγ_µ^Ω= Z

∂Ω

ζ_δd(λβ_µ^Ω).

Letting δ→0 we obtain by monotonicity Z

Ω

u_λL^∗_µζdγ_µ^Ω= Z

∂Ω

ζd(λβ_µ^Ω). (3.19)

Finally, ifλ∈M₊(∂Ω, β_µ) we replace it byλ_δ=χ_Bc

δ

λand denote by u_λδ the weak solution

of −∆u+µV u= 0 in Ω,

u=λδ on ∂Ω.

The mappingδ7→u_λδ is monotone. Hence, by the monotone convergence theoremu_λδ increases and converges to someu_λ inL¹(Ω, ρ⁻¹dγ^Ω_µ) and clearlyu_λ satisfies (3.19) for all ζ ∈Xµ(Ω).

4 The singular kernel

In this section we construct the singular kernel φ^Ω_µ and prove that it satisfies estimates (1.11)- (1.12) and it is associated to Dirac mass at 0. Up to a rotation we can assume that the inward normal direction to∂Ω at 0 iseN = (0⁰,1)∈R^N−1×R. Hence the tangent hyperplane to∂Ω at 0 is∂R^N+ =R^N−1. ForR >0 setB_R⁰ ={x⁰∈R^N−1 :|x⁰|< R} and DR=B_R⁰ ×(−R, R). Then there exist R > 0 and a C² function θ :B_R⁰ 7→ R such that ∂Ω∩D_R = {x = (x⁰, x_N) : x_N = θ(x⁰) for x⁰∈B_R⁰ }and Ω∩DR={x= (x⁰, xN) :θ(x⁰)< xN < R}. Furthermore ∇θ(0) = 0.