Boundary singularities of solutions of semilinear elliptic equations with critical Hardy potentials

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equations with critical Hardy potentials

Konstantinos Gkikas, Laurent Véron

To cite this version:

Konstantinos Gkikas, Laurent Véron. Boundary singularities of solutions of semilinear elliptic equa- tions with critical Hardy potentials. 2015. �hal-01071467v4�

elliptic equations with critical Hardy potentials

Konstantinos T. Gkikas

Centro de Modelamiento Matemàtico Universidad de Chile, Santiago de Chile, Chile

email: kugkikas@gmail.com

Laurent Véron

Laboratoire de Mathématiques et Physique Théorique Université François Rabelais, Tours, FRANCE

email: veronl@univ-tours.fr

Abstract

We study the boundary behaviour of positive functionsusatisfying (E)−∆u−d^2κ(x)u+g(u) = 0 in a bounded domainΩofR^N, where0 < κ≤ ¹4,gis a continuous nonndecreasing function andd(.) is the distance function to∂Ω. We first construct the Martin kernel associated to the the linear operator L^κ =−∆−d^2κ(x)and give a general condition for solving equation (E) with any Radon measureµfor boundary data. Wheng(u) =|u|^q−1uwe show the existence of a critical exponentqc =qc(N, κ)>1 whith the following properties: when0 < q < qc any measure is eligible for solving (E) withµfor boundary data; if q ≥ qc, a necessary and sufficient condition is expressed in terms of the absolute continuity ofµwith respect to some Besov capacity. The same capacity characterizes the removable compact boundary sets. At end any positive solution (F)−∆u− d^2κ(x)u+|u|^q−1u = 0 withq > 1 admits a boundary trace which is a positive outer regular Borel measure. When1 < q < qc we prove that to any positive outer regular Borel measure we can associate a positive solutions of (F) with this boundary trace.

Contents

1 Introduction 2

2 The linear operatorLκ=−∆−_d^2κ(x) 8

2.1 Classical results on Hardy’s inequality and the operatorL^κ . . . 8

2.2 Preliminaries . . . 10

2.3 L^κ-harmonic measure . . . 18

2.4 The Poisson kernel ofL^κ . . . 24

1

3 The nonlinear problem with measures data 32 3.1 The linear boundary value problem withL¹data . . . 32 3.2 General nonlinearities . . . 36 3.3 The power case . . . 40

4 Isolated boundary singularities 50

4.1 The sphericalL^κ-harmonic problem . . . 51 4.2 The nonlinear eigenvalue problem . . . 53 4.3 Isolated boundary singularities . . . 55

5 The boundary trace of positive solutions 58

5.1 Construction of the boundary trace . . . 58 5.2 Subcritical case . . . 66

6 Appendix I: barriers and a priori estimates 74

6.1 Barriers . . . 74 6.2 A priori estimates . . . 78 6.3 Moser Iteration . . . 82 Key words: Semilinear elliptic equation; Hardy potentials; Harmonic measure; Singular integrals;

Besov capacities; Optimal lifting operator; Boundary singularities; Boundary trace.

MSC2010: Primary 35J66, 35J10. Secondary 31A15, 35H25, 28A12.

1 Introduction

LetΩbe a bounded smooth domain inR^N andd(x) = dist(x,Ω^c). In this article we study several aspects of the nonlinear boundary value associated to the equation

−∆u− κ

d²(x)u+|u|^p−1u= 0 in Ω, (1.1) wherep >1. The study of the boundary trace of solutions of(1.1)is a natural framework for a general study of several nonlinear problems where the nonlinearity, the geometric properties of the domain and the coefficientκinteract. On this point of view, the caseκ= 0 has been thoroughly treated by Mar- cus and Véron (e.g. [24], [25], [27], [26] and the synthesis presented in [28]). The associated linear Schrödinger operator

u7→ L^κu:=−∆u− κ

d²(x)u (1.2)

plays an important role in functional analysis because of the particular singularity of the potential V(x) := −_d^2κ(x). The case κ < 0 and more generally of nonnegative potentials has been studied by Ancona [3] who has shown the existence of a Martin kernel which allows a general representation formula of nonnegative solutions of

L^κu= 0 in Ω. (1.3)

Whenκ < ¹₄, Ancona proved thatL^κisweakly coerciveinH₀¹(Ω). Thus any positive solutionuof (1.3) admits a representation under the form

u(x) = Z

∂Ω

K_Lκ(x, ξ)dµ(ξ) in Ω, (1.4)

see [3, Remark p. 523]. Furthermore the kernelK_Lκ(x, ξ)with pole atξis unique up to a multiplication [3, Th 3]. Whenκ= ¹₄, thenL^κis no longer weakly coercive inH₀¹(Ω)and Ancona’s results cannot be applied.

Ancona’s representation theorem turned out to be the key ingredient of the full classification of positive solutions of

−∆u+u^q = 0 in Ω, (1.5)

which was obtained by Marcus [21]. In a more general setting, Véron and Yarur [34] constructed a capacitary theory associated to the linear equation

L^Vu:=−∆u+V(x)u= 0 in Ω, (1.6)

where the potentialV is nonnegative and singular near∂Ω. WhenV(x) := −d^2κ(x) withκ > 0,V is called a Hardy potential. There is a critical valueκ= ¹₄. Ifκ > ¹₄, no positive solution of(1.3)exists.

When0 < κ≤ ¹4, there exist positive solutions, and the geometry of the domain plays a fundamental role in the study of the mere linear equation(1.3). We define the constantcΩby

cΩ= inf

v∈H0¹(Ω)\{0}

Z

Ω|∇v|²dx Z

Ω v² d²(x)dx

. (1.7)

It is known that0 < cΩ ≤ ¹4, and ifΩis convex thencΩ = ¹₄ (see [22]). When0 < κ≤ ¹4,which is always assumed in the sequeland−∆d≥0in the sense of distributions, it is possible to define the first eigenvalueλκof the operatorL^κ. If we define the two fundamental exponentsα+andα₋by

α+= 1 +√

1−4κ and α₋ = 1−√

1−4κ, (1.8)

then the first eigenvalue is achieved by an eigenfunctionφκwhich satisfiesφκ(x)≈d^α2+(x)asd(x)→ 0. Similarly, the Green kernelG_Lκassociated toLκinherits this type of boundary behaviour since there holds

1 Cκ

min

( 1

|x−y|^N−2,d^α2+(x)d^α2+(y)

|x−y|^N+α+−2 )

≤G_Lκ(x, y)≤Cκmin

( 1

|x−y|^N−2,d^α2+(x)d^α2+(y)

|x−y|^N+α+−2 )

. (1.9) We show thatLκsatisfies the maximum principle in the sense that ifu∈H_loc¹ ∩C(Ω)is a subsolution, i.e.Lκu≤0, such that

(i) lim sup

x→y

u(x)

d^α−(x) ≤0 if0< κ < ¹₄, (ii) lim sup

x→y

p u(x)

d(x)|lnd(x)| ≤0 ifκ= ¹₄,

(1.10)

for ally∈∂Ω, thenu≤0. This result has to be compared with the result on the the existence of positive sub-harmonic functions inΩgiven in [4, Theorem 2. 3] which is associated to the maximum principle in neighborhood of∂Ωstated in [4, Lemma 2. 4].

Ifξ∈∂Ωandr >0, we set∆r(ξ) =∂Ω∩Br(ξ). We prove that a positive solution ofL^κu= 0 which vanishes on a part of the boundary in the sense that

(i) lim

x→y

u(x)

d^α−(x) = 0 ∀y∈∆r(ξ) if0< κ < ¹₄, (ii) lim

x→y

p u(x)

d(x)|lnd(x)| = 0 ∀y∈∆r(ξ) ifκ= ¹₄,

(1.11)

satisfies

u(x)

φκ(x) ≤C1 u(y)

φκ(y) ∀x, y∈∆^r₂(ξ), (1.12)

for someC1=C1(Ω, κ)>0.

For anyh∈C(∂Ω)we construct the unique solutionv:=vhof the Dirichlet problem L^κv= 0 inΩ

v=h on∂Ω, (1.13)

noting that the boundary datahis achieved in the sense that

xlim→y

u(x)

d^α−(x) =h(y)if 0< κ < 1

4 and lim

x→y

p u(x)

d(x)|lnd(x)| =h(y)if κ=1 4.

Using this construction and estimates (1.10) we show the existence of theL^κ-measure, which is a bounded Borel measureω^xwith the property that for anyh∈C(∂Ω), the above functionvhsatisfies

vh(x) = Z

∂Ω

h(y)dω^x(y). (1.14)

Because of Harnack inequality, the measuresω^xandω^zare mutually absolutely continuous forx, z∈Ω, and for anyx∈Ωwe can define the Radon-Nikodym derivative

K(x, y) := dω^x

dω^x0(y) forω^x0-almosty∈∂Ω. (1.15) There existsr0:=r0(Ω)such that for anyx∈Ωverifyingd(x)≤r0, there exists a uniqueξ=ξx∈∂Ω with the property thatd(x) =|x−ξx|. If we denote byΩ^′_r0the set ofx∈Ωsuch that0< d(x)< r0, the mappingΠfromΩ^′_r0to[0, r0]×∂Ωdefined byΠ(x) = (d(x), ξx)is aC¹diffeomorphism. Ifξ∈∂Ω and0≤r≤r0, we setxr(ξ) = Π⁻¹(r, ξ). LetW be defined inΩby

W(x) =

( d^α2−(x) if κ < ¹₄,

pd(x)|lnd(x)| if κ= ¹₄. (1.16)

We prove that theLκ-harmonic measure can be equivalently defined by ω^x(E) = inf

ψ:ψ∈C+(Ω), L^κ-superharmonic inΩand s.t. lim inf

x→E

ψ(x) W(x) ≥1

, (1.17) on compact setsE⊂∂Ωand then extended by regularity to Borel subsets of∂Ω.

TheLκ-harmonic measure is connected to the Green kernel ofLκby the following estimates Theorem AThere existsC3:=C3(Ω)>0such that for anyr∈(0, r0]andξ∈∂Ω, there holds

1

C3r^N+α2−−2G_Lκ(xr(ξ), x) ≤ω^x(∆r(ξ))

≤C3r^N+α2−−2G_Lκ(xr(ξ), x) ∀x∈Ω\B4r(ξ), (1.18)

if0< κ < ¹₄, and

1

C3r^N−2+12|lnd(x)|G_L1

4(xr(ξ), x)

≤ω^x(∆r(ξ))

≤C3r^N−2+12|lnd(x)|G_L1 4

(xr(ξ), x) ∀x∈Ω\B4r(ξ), (1.19) whenκ=¹₄. As a consequenceω^xhas the doubling property. The previous estimates allow to construct a kernel function ofL^κ inΩ, prove its uniqueness up to an homothety. When normalized, the kernel function denoted byK_Lκ is the Martin kernel, defined by

K_Lκ(x, ξ) = lim

x→ξ

G_Lκ(x, y)

G_Lκ(x, x0) ∀ξ∈∂Ω, (1.20) for somex0∈Ω. Thank to this kernel we can represent any positiveL^κ-harmonic functionuby mean of a Poisson type formula which endows the form

u(x) = Z

∂Ω

K_Lκ(x, ξ)dµ(ξ). (1.21)

for some unique positive Radon measureµon∂Ω. The measureµis called the boundary trace ofu.

FurthermoreK_Lκsatisfies the following two-side estimates

Theorem BThere existsC3:=C3(Ω, κ)>0such that for any(x, ξ)∈Ω×∂Ωthere holds 1

C3

d^α2+

|x−ξ|^N+α+−2 ≤K_Lκ(x, ξ)≤C3

d^α2+

|x−ξ|^N+α+−2. (1.22) In the sections 3-6 of this paper we develop the study of the semilinear equation (E) and emphasize the particular case of equation(1.1). With the help of the previous estimates we adapt the approach developed in [16] to prove the existence of weak solutions to the nonlinear boundary value problem

−∆u− κ

d²(x)u+g(u) =ν in Ω

u=µ in ∂Ω, (1.23)

wheregis a continuous nondecreasing function such thatg(0)≥0andνandµare Radon measures on Ωand∂Ωrespectively . We define the classX_κ(Ω)of test functions by

X_κ(Ω) =n

η∈L²(Ω)s.t.∇(d^−α2+η)∈L²_φk(Ω)andφ⁻_κ¹Lκη ∈L^∞(Ω) o

, (1.24) and we prove

Theorem CAssumegsatisfies Z _∞

1

(g(s) +|g(−s)|)s⁻²

N−1+α+ 2 N−2+α+

2 ds <∞. (1.25)

Then for any Radon measuresνonΩand such thatR

Ωφκd|µ|<∞andµon∂Ωthere exists a unique u∈L¹_φκ(Ω)such thatg(u)∈L¹_φκ(Ω)which satisfies

Z

Ω

(uLκη+g(u)η)dx= Z

Ω

(ηdν+K_L

κ[µ]Lκηdx) ∀η∈X_κ(Ω). (1.26)

Wheng(r) =|r|^q−1rthe critical value isqc = ^N+

α+ 2

N+^α₂⁺−2and (1.25) is satisfied for0≤q < qc(the subcritical range). In this range of values ofq, existence and uniqueness of a solution to

−∆u− κ

d²(x)u+|u|^q−1u= 0 in Ω

u=µ in ∂Ω, (1.27)

has been recently obtained by Marcus and Nguyen [23]. However, whenq ≥ qc not all the Radon measures are eligible for solving problem (1.27).

We prove the following result in the statement of whichC^RN−1

2−^2+α2q′⁺,q^′ denotes the Besov capacity associated to the Besov spaceB²⁻

2+α+

2q′ ,q^′(R^N−1).

Theorem DAssumeq≥ qc andµis a positive Radon measure on∂Ω. Then problem (1.27) admits a weak solution if and only ifµvanishes on Borel setsE⊂∂Ωsuch thatC^RN−1

2−^2+α2q′⁺,q^′(E) = 0.

Note that a special case of this result is proved in [23] whenµ=δafor a boundary point andq≥qc. In that caseδadoes not vanish on{a}althoughC^RN−1

2−^2+α2q′⁺,q^′({a}) = 0.

This capacity plays a fundamental for characterizing the removable compact boundary sets which can only exist in thesupercritical rangeq≥qc.

Theorem EAssumeq≥qcandK⊂∂Ωis compact. Then any functionu∈C(Ω\K)which satisfies

−∆u− κ

d²(x)u+|u|^q−1u= 0 in Ω

u= 0 in ∂Ω\K, (1.28)

is identically zero if and only ifC^RN−1

2−^2+α2q′⁺,q^′(K) = 0.

The proof of Theorems D and E is delicate and based upon the use of theoptimal lifting operator which has been introduced in [24] and the kernels estimates of [27, Appendix].

We show that any positive solutionuof (1.1) admits a boundary trace in the class of outer regular positive Borel measures, not necessarily locally bounded, and more precisely we prove that the following dichotomy holds:

Theorem FLetube a positive solution of (1.1) inΩanda∈∂Ω. Then (i) either for anyǫ >0

δ→lim0

Z

Σδ∩Bǫ(a)

udω^x_Ω⁰′

δ =∞, (1.29)

whereΩ^′_δ={x∈Ω :d(x)> δ},Σδ=∂Ω^′_δandω_Ω^x0′ δ

is the harmonic measure inΩ^′_δ,

(ii) or there existǫ0>0and a positive Radon measureλon∂Ω∩Bǫ0(a)such that for anyZ ∈C(Ω) with support inΩ∪(∂Ω∩Bǫ0(a)), there holds

δlim→0

Z

Σδ∩Bǫ(a)

Zudω^x_Ω⁰′ δ =

Z

∂Ω∩Bǫ(a)

Zdλ. (1.30)

The set of pointsa∈∂Ωsuch that (i) (resp. (ii)) holds is closed (resp. relatively open) and denoted byS^u(respR^u). There exists a unique Radon measureµuonR^u such that, for anyZ ∈ C(Ω)with support inΩ∪ R^uthere holds

δlim→0

Z

Σδ

Zudω^x_Ω⁰′ δ =

Z

Ru

Zdµu. (1.31)

The couple(S^u, µu)is calledthe boundary trace ofuand denoted byT r∂Ω(u). A notion of normalized boundary trace of positivemoderate solutionsof (1.1), i.e. solutions such thatu∈L^q(φκ), is developed in [23]. It is proved therein that there exists a boundary traceµ≈({∅}, µu), and that the corresponding representation ofuvia the Martin and Green kernels holds.

If1 < q < qc we denote byukδa positive solution of (1.1) withµ = kδa for somea ∈ ∂Ωand k≥0. Then there existslimk→∞ukδa=u_∞,aand we prove the following:

Theorem GAssume1< q < qcanda∈∂Ω. Ifuis a positive solution of (1.1) such thata∈ S^u, then u≥u_∞,a.

In order to go further in the study of boundary singularities, we construct separable solutions of (1.1) inR^N₊ ={x= (x^′, xN) :xN >0}={(r, σ)∈R₊×S₊^N−1}which vanish on∂R^N₊ \ {0}under the formu(r, σ) =r^−q−21ω(σ), wherer >0,σ∈S₊^N−1. They are solutions of

−∆_SN−1ω−ℓq,Nω− κ

e_N.σω+|ω|^q−1ω= 0 inS₊^N−1

ω= 0 in∂S₊^N−1, (1.32)

where∆_SN−1 is the Laplace-Beltrami operator,e_N the unit vector pointing toward the North pole and ℓq,N is a positive constant. We prove that if1 < q < qc, then problem (1.32) admits a unique positive solution ωκ while no such solution exists if q ≥ qc. To this phenomenon is associated a result of classification of the positive solutions of (1.1) inΩwhich vanishes of∂Ω\ {0}(here we assume that 0∈∂Ωand that the tangent hyperplane to∂Ωat0is{x:x.e_N = 0},and that there existsr0 >0such thatBr0(r0e_N)⊂Ω, Br0(r0e_N)⊂ {x:x.e_N ≥0}andd(r0e_N) =|r0e_N|=r0).

Theorem HAssume1 < q < qcand letu∈ C(Ω\ {a}be a solution of (1.1) inΩwhich vanishes of

∂Ω\ {a}. Then (i) eitheru=u_∞,aand

lim_r→0r^q−12 u(r, .) =ωκ (1.33) locally uniformly inS₊^N−1,

(ii) or there existsk≥0such thatu=ukδaand

u(x) =kK_Lκ(x, a)(1 +o1)) asx→0. (1.34) If1< q < qcwe prove that to any couple(F, µ)whereF is a closed subset of∂Ωandµa positive Radon measure onR=∂Ω\F, we can associate a positive solutionuof (1.1) inΩwith the property thatT r∂Ω(u) = (F, µ). The construction is based upon the existence of barrier functions which allow to prove local a priori estimate that is satisfied by any positive solution with boundary trace(F,0). The delicate proof of the existence of these barrier is presented in Appendix I. A priori estimates which follow from the barrier method are presented in Appendix II. In Appendix III we develop some regularity results based upon Moser’s iterative scheme adapted to the framework of the Hardy operator.

The results presented here are announced in [15].

2 The linear operator L

= −∆ −

Throughout this articlecj(j=1,2,...) denote positive constants the value of which may change from one occurrence to another. The notationκis reserved to the value of the coefficient of the Hardy potential.

2.1 Classical results on Hardy’s inequality and the operator L

We first recall some known results concerning Hardy’s inequalities and the associated eigenvalue prob- lem (see [11], [14]).

1- The constant cΩ defined in (1.7) has value in(0,¹₄]. If Ωis convex or if the functiondis super- harmonic thencΩ= ¹₄. Moreover this equality is verified if and only if there exists no minimizer to the problem (1.7) [22]. For anyκ∈(0,¹₄]there exists

inf Z

Ω

|∇u|²− κ d²u²

dx: Z

Ω

u²dx= 1

=λκ>−∞. (2.1)

Furthermoreλκ>0ifκ < cΩor ifk≤ ¹4anddis a superharmonic function inΩ.(see [8]).

2- If0< κ < ¹₄the minimizerφκof (2.1) belongs to the spaceH₀¹(Ω)and it satisfies

φκ≈d^α2+(x), (2.2)

whereα+(as well asα₋) are defined by (1.8).

3- Ifκ= ¹₄, there is no minimizer inH₀¹(Ω), but there exists a non-negative functionφ¹₄ ∈ H_loc¹ (Ω) such that

φ¹

4 ≈d¹²(x), (2.3)

and it solves

−∆u− 1

4d²u=λku in Ω in the sense of distributions. In addition, the functionψ¹

4 =d⁻¹²φ¹

4 belongs toH₀¹(Ω;d(x)dx).

4- LetH₀¹(Ω, d^α(x)dx)denote the closure ofC₀^∞(Ω)functions under the norm

||u||²H₀¹(Ω,d^α(x)dx)= Z

Ω|∇u|²d^α(x)dx+ Z

Ω|u|²d^α(x)dx. (2.4) Ifα≥1there holds [14, Th. 2.11]

H₀¹(Ω, d^α(x)dx) =H¹(Ω, d^α(x)dx) ∀α≥1. (2.5)

5- Let0< κ≤cΩ. LetH_κ(Ω)be the subset of functions ofH_loc¹ (Ω)satisfying Z

Ω

|∇φ|²− κ d²φ²

dx <∞. (2.6)

Then the mapping

φ7→

Z

Ω

|∇φ|²− κ d²φ²

dx ¹₂

(2.7)

is a norm onH_κ(Ω). The closureW_κ(Ω)ofC₀^∞(Ω)intoH_κ(Ω)satisfies W_κ(Ω) =H₀¹(Ω) ∀0< κ < cΩ and W1

4(Ω)⊂W₀^1,q(Ω) ifλ¹

4 >0 ∀1≤q <2, (2.8) see [5, Th B]. As a consequenceW_κ(Ω)is compactly imbedded intoL^r(Ω)for anyr∈[1,2^∗).

6- Letα >0andΩ⊂R^N be a bounded domain. There existsc^∗ >0depending on diam(Ω), N andα such that for anyv∈C₀^∞(Ω)

Z

Ω|v|^{2(N+α)N+α−2}d^αdx ^N+α−_N+α²

≤c^∗ Z

Ω|∇v|²d^αdx. (2.9)

For a proof see [14, Th. 2.9].

The boundary behaviour of the first eigenfunction yields a two-side similar estimate of the Green kernel for Schrödinger operators with a general Hardy type potentials [14, Corollary 1.9].

Proposition 2.1. Consider the operatorE:=−∆−V,inΩwhereV =V1+V2,with

|V1| ≤ 1

4d²(x) andV2∈L^p(Ω), p > N 2 . We also assume that

0< λ1:= inf

u∈H₀¹(Ω)

Z

Ω |∇u|²dx−V u² dx Z

Ω

u²dx

,

and that toλ1is associated a positive eigenfunctionφ1. If, for someα≥1andC1, C2>0, there holds c1d^α2(x)≤φ1(x)≤c2d^α2(x) ∀x∈Ω,

then the Green kernelG^Ω_Eassociated toEinΩsatisfies G^Ω_E(x, y)≈c3min

1

|x−y|^N−2, d^α2(x)d^α2(y)

|x−y|^N+α−2

. (2.10)

Next we define the setsΩδ,Ω^′_δandΣδby

Ωδ ={x∈Ω : d(x)< δ}, Ω^′_δ ={x∈Ω : d(x)> δ} and Σδ ={x∈Ω : d(x) =δ}. (2.11) Definition 2.2. LetG ⊂ Ωbe open and letH_c¹(G)denote the subspace ofH¹(G)of functions with compact support inG. A functionh∈W_loc^1,1(G)isL^κ-harmonic inGif

Z

G∇h.∇ψdx−κ Z

Ω

1

d²(x)hψdx= 0 ∀ψ∈H_c¹(G).

A functionh∈H_loc¹ (G)∩C(G)isL^κ-subharmonic inGif Z

G∇h.∇ψdx−κ Z

Ω

1

d²(x)hψdx≤0 ∀ψ∈H_c¹(G), ψ≥0.

We say thathis a localL^κ-subharmonic function if there existsδ >0such thath∈H_loc¹ (Ωδ)∩C(Ωδ) isL^κ-subharmonic inΩδ.Similarly, (local)L^κ-superharmonicshare defined with” ≥”in the above inequality.

Note thatLκ-harmonic functions areC² inGby standard elliptic equations regularity theory. The Phragmen-Lindelöf principle yields the following alternative [4, Theorem 2.6].

Proposition 2.3. Letκ ≤ ¹4. Ifhis a localL^κ-subharmonic function, then the following alternative holds:

(i) either for every local positiveL^κ-superharmonic functionh lim sup

d(x)→0

h(x)

h(x) >0, (2.12)

(ii) or for every local positiveL^κ-superharmonic functionh lim sup

d(x)→0

h(x)

h(x) <∞. (2.13)

Definition 2.4. If a localL^κ-subharmonic function hsatisfies (i) (resp. (ii)) it is called a largeL^κ- subharmonic (resp. a smallLκ-subharmonic).

The next statement is [4, Theorem 2.9].

Proposition 2.5. Lethbe a small localLκ-subharmonic ofLκ. (i) Ifκ < ¹₄, then the following alternative holds:

either lim sup

x→∂Ω

h(x)

(d(x))^α2− >0 or lim sup

x→∂Ω

h(x)

(d(x))^α2+ <∞. (ii) Ifκ=¹₄, then the following alternative holds:

either lim sup

x→∂Ω

h(x)

(d(x))¹² log(¹_d)>0 or lim sup

x→∂Ω

h(x)

(d(x))¹² <∞. Definition 2.6. Letf0∈L²_loc(Ω). We say that a functionu∈H_loc¹ (Ω)is a solution of

L^κu=f0 in Ω, (2.14)

if there holds Z

Ω∇u.∇ψdx−κ Z

Ω

1

d²(x)uψdx= Z

Ω

f0ψdx ∀ψ∈C₀^∞(Ω). (2.15)

2.2 Preliminaries

In this part we study some regularity properties of solutions of linear equations involvingL^κ. Lemma 2.7. (i) Ifα >1andd^−α2u∈H¹(Ω, d^α(x)dx),thenu∈H₀¹(Ω).

(ii) Ifα= 1andd⁻¹²u∈H¹(Ω, d(x)dx),thenu∈W₀^1,p(Ω), ∀p <2.

Proof. There existsβ0 > 0such thatd ∈ C²(Ωβ0)and setu = d^α2v. In the two cases (i)-(ii), our assumptions imply

u∈L²(Ω) and ∇u−α

2ud⁻¹∇d∈L²(Ω). (2.16)

(i) Sincev ∈ H¹(Ω, d^α(x)dx), by (2.5) there exists a sequencevn ∈ C₀^∞(Ω)such that vn → v in H¹(Ω, d^α(x)dx).Setun=d^αvn.Let0< β≤ ^β2⁰ andψβbe a cut of function such thatψβ = 0inΩ^′_β andψβ= 1inΩ^β

2.Thenun=d^α2(ψβvn+ (1−ψβ)vn).Thus it is enough to prove thateun=d^α2ψβvn

remains bounded inH¹(Ω)independently ofn. Setwn=ψβvn, then Z

Ω|∇eun|²dx= Z

Ωβ

|∇wn|²dx≤c4

Z

Ωβ

d^α|∇wn|²dx+ Z

Ωβ

d^α−2w_n²dx

! .

Note thatα−2>−1.Now Z

Ωβ

d^α−2w_n²dx= 1 α−1

Z

Ωβ

w²_ndiv(d^α−1∇d)dx− 1 α−1

Z

Ωβ

(d^α−1(∆d)w²_ndx.

Now since|∆d(x)|< c5, ∀x∈Ωβ0,we have

1 α−1

Z

Ωβ

d^α−1(∆d)w²_ndx

≤c5β₀^α−1 α−1

Z

Ωβ

w_n²dx.

Also

Z

Ωβ

w²_ndiv(d^α−1∇d)dx = 2

Z

Ωβ

wnd^α2d^α2−1∇d.∇wndx

≤c6

Z

Ωβ

d^α|∇wn|²dx+δ Z

Ωβ

d^α−2w_n²dx,

wherec6=c6(δ)>0. The result follows in this case, if we chooseδsmall enough and then letn→ ∞. (ii) By the same calculations we have

Z

Ω

d^−p2|wn|^pdx≤c7

Z

Ωβ

d^p2|∇wn|^pdx≤c7

Z

Ω

d(x)dx ^p₂ Z

Ωβ

d|∇wn|²dx.

In the following statement we prove regularity up to the boundary for the function_φ^u

κ.

Proposition 2.8. Letf0∈L²(Ω).Then there exists a uniqueu∈H_loc¹ (Ω)such thatφ⁻_κ¹u∈H¹(Ω, d^α+(x)dx), satisfying (2.14). Furthermore, iff1 := _φ^f0

κ ∈L^q(Ω, φ²_κdx), q > ^N+α₂ ⁺, then there exists0 < β <1 such that

sup

x,y∈Ω, x6=y|x−y|^−β u(x)

φκ(x)− u(y) φκ(y)

< c8||f1||L^q(Ω,φ²_κdx). (2.17) Proof. If there exists a solutionu, thenψ= _φ^u

κ satisfies

−φ⁻_κ²div(φ²_κ∇ψ) +λκψ=φ⁻_κ¹f0, (2.18) and we recall thatφκ(x)≈d^α2+(x). We endow the spaceH¹(Ω, φ²_κdx)with the inner product

ha, bi= Z

Ω

(∇a.∇b+λκab)φ²_κdx.

By a solutionψof (2.18) we mean thatψ∈H₀¹(Ω, φ²_κdx)satisfies h∇ψ,∇ζi=

Z

Ω∇ψ.∇ζ φ²_κdx+λκ

Z

Ω

ψζφ²_κdx= Z

Ω

f0ζφκdx ∀ζ∈H₀¹(Ω, φ²_κdx). (2.19) By Riesz’s representation theorem we derive the existence and uniqueness of the solution in this space.

SinceH¹(Ω, φ²_κdx) =H₀¹(Ω, φ²_κdx)by [14, Th 2.11], any weak solutionuof (2.14) such thatφ⁻_κ¹u∈ H¹(Ω, φ²_κdx)is obtained by the above method.

Finally iff0∈L^q(Ω, φ²_κdx),whereq > ^N+α₂ ⁺,thanks to (2.9) we can prove the estimate

||ψ||L^∞(Ω)≤c8||f0||L^q(Ω,φ²_κdx), (2.20) wherec8=c8(Ω, κ, q).Then we can apply the Moser iteration (see subsection 6.3) to derive the Hölder regularity up to the boundary.

In the next results we make more precise the rate of convergence of a solution of (2.14) to its boundary value.

Proposition 2.9. Assumeκ < ¹₄. Iff0∈L²(Ω)andh∈H¹(Ω)there exists a unique weak solutionu of (2.14) belonging toH_loc¹ (Ω)and such thatd^−α2+(u−d^α2−h)∈H¹(Ω, d^α+(x)dx).Furthermore, if f1:= _φ^f0_κ ∈L^q(Ω, φ²_κdx), q > ^n+α₂ andh∈C²(Ω),then there exists0< β <1with the property that

x∈Ω, x→y∈∂Ωlim u(x)

(d(x))^α2− =h(y) ∀y∈∂Ω, uniformly with respect toy,

u d^α2−

L^∞(Ω)

≤c9

||h||C²(Ω)+||f1||L^q(Ω,φ²_κdx)

,

and

sup

x,y∈Ω, x6=y|x−y|^−β

u(x)

(d(x))^α2− − u(y) (d(y))^α2−

≤c10

||h||C²(Ω)+||f1||^Lq(Ω,φ2κdx)

, (2.21) withc9andc10depending onΩ, N, q, andκ.

Remark. By Lemma 2.7 we already know thatu−d^α2−h∈H₀¹(Ω).

Proof. Letβ ≤β0 andη ∈C²(Ω)be a function such thatη =d^α2−(x)inΩβandη(x)> c > 0,if x∈Ω^′_β. We setu=φκv+ηh.Thenvis a weak solution of

−div(φ²_κ∇v)

φ²_κ +λκv= 1 φκ

f0+ (∆η+κη

d²)h+ 2∇η.∇h+η∆h

, (2.22)

in the sense that Z

Ω∇v.∇ψ φ²_κdx+λκ

Z

Ω

v ψ φ²_κdx= Z

Ω

f0+ (∆η+κη

d²)h+ 2∇η.∇h ψ φκdx

− Z

Ω∇h.∇(ηψ φκ)dx ∀ψ∈C₀^∞(Ω). (2.23)

Letψ∈C₀^∞(Ωβ). By an argument similar to the one in the proof of Lemma 2.7 we have Z

Ω

ψ²dx= Z

Ωβ

ψ²dx= Z

Ωβ

div(d∇d)|ψ|²dx− Z

Ωβ

d∆d|ψ|²dx,

which implies Z

Ωβ

ψ²dx≤c^′₁₀ Z

Ωβ

d²|∇ψ|²dx≤c11

Z

Ωβ

d^α+|∇ψ|²dx. (2.24)

Now

Z

Ωβ

(∆η+κη

d²)h+ 2∇η.∇h ψ φκdx

≤c12

Z

Ωβ

ψ²dx, and

Z

Ωβ

∇h.∇(ηψ φκ)dx ≤c13

Z

Ωβ

|∇h|²dx+ Z Z

Ωβ

d^α+|∇ψ|²dx+ Z Z

Ωβ

ψ²dx

! .

By (2.24) we can takeψ∈H¹(Ω, d^α+(x)dx)for test function. Thus we derive that there exists a weak solutionv∈H¹(Ω, d^α+(x)dx)of (2.23).

To prove (2.21) we first obtain that ifψ∈C₀^∞(Ωε) Z

Ω

ψdx=− Z

Ωε

d∇d.∇ψdx− Z

Ωε

d∆dψdx.

Since Z

Ω

(∆η+κη

d²)h+ 2∇η.∇h+η∆h ψ φκdx

≤c14||h||C²(Ω)

Z

Ω|ψ|dx

≤ 1 2

Z

Ωε

d^α+|∇ψ|²dx+c15(Ω, κ)||h||C²(Ω), we use again (2.9) and Moser’s iterative scheme as in Proposition 2.8, and we obtain

||v||L^∞(Ω)≤c9

||h||C²(Ω)+||f0||L^q(Ω,φ²_κdx)

,

wherec9 =c9(Ω, q, κ)>0. From inequality it follows thatvis Hölder continuous up to the boundary and the uniform convergence holds.

Proposition 2.10. Assumeκ= ¹₄. Iff0∈ L²(Ω)andh∈H¹(Ω),there exists a unique functionuin H_loc¹ (Ω)weak solution of

L¹₄u=f0

verifyingd⁻¹²(u−d¹²|logd|h)∈H¹(Ω, d(x)dx).Furthermore, iff1 := _φ^f0₁

4

∈L^q(Ω), q > ⁿ⁺¹₂ and h∈C²(Ω),then there exists0< β <1such that

x∈Ω, xlim→y∈∂Ω

u

d¹²|logd|(x) =h(y) ∀y∈∂Ω, uniformly with respect toy,

√ u

d|log_D^d

0| L^∞(Ω)

≤c16

||h||C²(Ω)+||f1||L^q(Ω,φ²₁

4

dx)

whereD0= 2 sup_x∈Ωd(x).Finally there holds

sup

x,y∈Ω, x6=y|x−y|^−β

p u(x)

d(x)|log^d(x)_D0 |− u(y) pd(y)|log^d(y)_D0 |

< c17

||h||C²(Ω)+||f1||L^q(Ω,φ²₁

4

dx)

. (2.25) Proof. Using again Lemma 2.7, we know thatu−d¹²|logd|h∈W₀^1,p(Ω), ∀p <2.The proof is very similar to the proof of Proposition 2.9. The only differences are we imposeη=d¹²|logd|inΩβand we use the fact that|logd| ∈L^p(Ω),∀p≥1.

In the next result we prove that the boundary Harnack inequality holds, provided the vanishing prop- erty of a solution is understood in a an appropriate way.

Proposition 2.11. Letδ >0be small enough,ξ∈∂Ωandu∈H_loc¹ (Bδ(ξ)∩Ω)∩C(Bδ(ξ)∩Ω)be a positiveL¹₄-harmonic function inBδ(ξ)∩Ωvanishing on∂Ω∩Bδ(ξ)in the sense that

dist^(x,Klim⁾→0

u(x)

d¹²(x)|logd(x)| = 0 ∀K⊂∂Ω∩Bδ(ξ), Kcompact. (2.26) Then there exists a constantc18=c18(N,Ω, κ)>0such that

u(x) φ¹

4(x) ≤c18

u(y) φ¹

4(y) ∀x, y ∈Ω∩B^δ

2(ξ).

Proof. We already know thatu∈C²(Ω). Letδ≤min(β0,¹₂)such thatBδ(ξ)∩Ω⊂Ωδ⊂Ωβ0. By [4, Lemma 2.8] there exists a positive supersolutionζ∈C²(Ωδ)of (1.3) inΩδwith the following behaviour

ζ(x)≈d¹²(x) log 1

d(x) 1 +c19

log 1

d(x) ₋β!

, for someβ∈(0,1)andc19=c19(Ω)>0. Setv=ζ⁻¹u, then it satisfies

−ζ⁻²div(ζ²∇v)≤0 in Bδ(ξ)∩Ω. (2.27) Letη ∈ C₀^∞(Bδ(ξ))such that0 ≤η ≤1 andη = 1inB^3δ

4 (ξ).We setvs = η²(v−s)+ Since by assumptionvshas compact support inBδ(ξ)∩Ω, we can use it as a test function in (2.27) and we get

Z

Bδ(ξ)∩Ω

ζ²∇v.∇vsdx= Z

Bδ(ξ)∩Ω

ζ²∇(v−s)+.∇vsdx≤0, (2.28) which yields Z

Bδ(ξ)∩Ω|∇(v−s)+|²ζ²η²dx≤4 Z

Bδ(ξ)∩Ω|∇η|²(v−s)²₊ζ²dx.

Lettings→0we derive Z

Bδ(ξ)∩Ω|∇v|²ζ²η²dx≤4 Z

Bδ(ξ)∩Ω|∇η|²v²ζ²dx.

Since

|∇(v−s)+|²ζ²η²↑ |∇v|²ζ²η² ass→0,

and convergence of∇(v−s)+to∇vholds a.e. inΩ, it follows by the monotone convergence theorem

s→lim0

Z

Bδ(ξ)∩Ω|∇(v−(v−s)+)|²ζ²η²dx= 0. (2.29) Finallyζvs→η²ζvinH¹(Bδ(ξ)∩Ω), which yields in particularη²u=η²ζv∈H₀¹(Bδ(ξ)∩Ω).

Step 2. By [4, Lemma 2.8] there exists a positive subsolutionh ∈ C²(Ωδ) of (1.3) inΩδ with the following behaviour

h(x)≈d¹²(x) log 1

d(x) 1−c20

log 1

d(x) −β!

,

whereβ ∈ (0,1)andc20 =c20(Ω) >0. Setw=h⁻¹uandws =η²(w−s)+. Thenws→ η²win H¹(Bδ(ξ)∩Ω)by Step 1. Putus=hws, thus, for0< s, s^′, we have

Z

Bδ(ξ)∩Ω|∇(us−us^′)|²dx−1 4

Z

Bδ(ξ)

|us−us^′|² d²(x) dx=

Z

Bδ(ξ)∩Ω

h²|∇(ws−ws^′)|²dx (2.30) +

Z

Bδ(ξ)∩Ω|∇h|²|ws−ws^′|²dx+ Z

Bδ(ξ)∩Ω

h∇h.∇(us−us^′)²dx−1 4

Z

Bδ(ξ)∩Ω

h²|ws−ws^′|² d²(x) dx

≤ Z

Bδ(ξ)∩Ω

h²|∇(ws−ws^′)|²dx,

where, in the last inequality, we have performed by parts integration and then used the fact thathis a subsolution. Thus we have by (2.29) that

s,slim^′→0

Z

Bδ(ξ)|∇(us−us^′)|²dx−1 4

Z

Bδ(ξ)

|us−us^′|²

d²(x) dx= 0. (2.31)

Step 3.LetW(Ω)denote the closure ofC₀^∞(Ω)in the space of functionsφsatisfying

||φ||²H :=

Z

Ω|∇Φ|²dx−1 4

Z

Ω

|Φ|²

d²(x)dx <∞. Thusη²u∈W(Ω), which implies

ηu φ¹

4

∈H₀¹(Ω, d(x)dx).

Next we setv˜=φ⁻1¹

4 u; then˜v∈H¹(B^3δ

4(ξ), d(x)dx)and it satisfies

−φ⁻1² 4

div(φ²1

4∇v) +˜ λ¹

4v˜= 0.

Put˜v^∗(x, t) =e^tλ14v, then˜ v˜^∗satisfies

˜ v_t^∗−φ⁻₁²

4

div(φ²1

4∇˜v^∗) = 0 (2.32)

in the weak sense of [14, Definition 2.9]. By [14, Theorem 1.5],v˜^∗satisfies a Harnack inequality up to the boundary ofΩin the sense that

ess supn

˜

v^∗(y,t) : (y,t)∈ B^r2(ξ)×[^r₄²,^r₂²]o

≤C ess infn

˜

v^∗(y,t) : (y,t)∈ B2^r(ξ)×[^3r₄²,r²]o (2.33)

whereB^r2(ξ)is a Lipschitz deformed Euclidean ball (see [14, p. 244] and Definition 6.6). Sinceris bounded andv˜satisfies the same estimate up to a constant depending on Ωand finally there exists a constantc18=c18(Ω)>0such that

v(x)≤c18v(y) ∀x, y∈B^δ

2(ξ).

The result follows.

In the caseκ < ¹₄, the result holds with minor modifications.

Proposition 2.12. Letδ >0be small enough,ξ∈Ω,0< κ < ¹₄andu∈H_loc¹ (Bδ(ξ)∩Ω)∩C(Bδ(ξ)∩ Ω)be a nonnegativeL^κ-harmonic inBδ(ξ)vanishing on∂Ω∩Bδ(ξ)in the sense that

dist^(x,K)lim →0

u(x)

(d(x))^α2− = 0 ∀K⊂∂Ω∩Bδ(ξ), Kcompact. (2.34) Then there existsc21=c21(Ω, κ)>0such that

u(x) φκ(x) ≤c21

u(y)

φκ(y) ∀x, y∈Ω∩B^δ

2(ξ).

Proof. As in the previous proof we apply [4, Lemma 2.8], we consider a super-solutionζ ≈d^α−(1 + c19d^β)and a sub-h≈d^α−(1−c20d^β)whereβ ∈(0,√

1−4κ). Thus ηu

φκ ∈H₀¹(Ω, d^α+(x)dx),

whereηis a cut-off function adapted toBr(ξ). The function˜v=φ⁻_κ¹usatisfies

−φ⁻_κ²div(φ²_κ∇v) +˜ λκ˜v= 0, andv˜∈H₀¹(B^3δ

4 (ξ), d^α+(x)dx). Then the proof follows as in the previous Proposition.

Proposition 2.13. Letu∈H_loc¹ (Ω)∩C(Ω)be aL¹₄-subharmonic function such that lim sup

d(x)→0

u(x)

d¹²(x)|logd(x)| ≤0.

Thenu≤0.

Proof. We setv = max(u,0)and we proceed as in the Step 1 of the proof of Proposition 2.11 with η= 1. The result follows by lettings→0.

Similarly we have

Proposition 2.14. Letu∈H_loc¹ (Ω)∩C(Ω)be aLκ-subharmonic function such that lim sup

d(x)→0

u(x) (d(x))^α2− ≤0.

Thenu≤0.