The spherical p-harmonic eigenvalue problem in non-smooth domains

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non-smooth domains

Konstantinos Gkikas, Laurent Véron

To cite this version:

Konstantinos Gkikas, Laurent Véron. The spherical p-harmonic eigenvalue problem in non-smooth

domains. Journal of Functional Analysis, Elsevier, 2018, 274, pp.1155-1176. �hal-01501604�

in non-smooth domains

Konstantinos Gkikas

Laurent Véron

Abstract

We prove the existence of p-harmonic functions under the formu(r, σ) =r^−βω(σ)in any coneC_S generated by a spherical domainSand vanishing on∂C_S. We prove the uniqueness of the exponentβ and of the normalized functionωunder a Lipschitz condition onS.

2010 Mathematics Subject Classification.35J72; 35J92.

Key words.p-Laplacian operator; polar sets; Harnack inequality; boundary Harnack inequality;p-Martin boundary.

Contents

1 Introduction 1

2 Existence 2

2.1 Estimates . . . 2

2.2 Approximations from inside . . . 5

2.3 Approximations from outside . . . 9

3 Uniqueness 10 3.1 Uniqueness of exponentβ . . . 10

3.2 Uniqueness of eigenfunction . . . 11

3.2.1 The convex case . . . 11

3.2.2 Proof of Theorem E . . . 12

1 Introduction

Letp >1,Sa domain of the unit sphereS^N−1ofR^N andCS :={(r, σ) :r > 0, σ∈S}the positive cone generated byS. If one looks forp-harmonic functions inCS under the formu(x) = u(r, σ) = r^−βω(σ)vanishing on∂C_S\ {0}, thenωsatisfies thesphericalp-harmonic eigenvalue problemonS

−div⁰

β²ω²+|∇⁰ω|^2p−2₂

∇⁰ω

= (p−1)β(β−β0)

β²ω²+|∇⁰ω|^2p−2₂

ω inS

ω= 0 in∂S

(1.1)

∗[email protected]

†[email protected]

1

withβ₀ = ^N_p−1^−p and werediv⁰ and∇⁰ denote the divergence operator and the covariant gradient on S^N−1endowed with the metric induced by its isometric inbedding intoR^N. Separable solutions play a key role for describing the boundary behaviour and the singularities of solutions of a large variety of quasilinear equations. WhenN = 2the equation is completely integrable and has been solved by Kroll in the regular caseβ <0and Kichenassamy and Véron in the the singular caseβ >0. In higher dimension, Tolksdorff [15] proved the following:

Theorem AIfS is a smooth spherical domain, there exist two couples(βS, ωS)and (β_S⁰, ω_S⁰)where βS >0andβ_S⁰ <0,ωS andω_S⁰ are positiveC²(S)-functions vanishing on∂Swhich solve(1.1)with (β, ω) = (βS, ωS) or(β, ω) = (β_S⁰, ω⁰_S). FurthermoreβS andβ_S⁰ are unique, andωS and ω_S⁰ are unique up to an homothety.

A more general and transparent proof has been obtained by Porretta and Véron [13], but always in the case of a smooth spherical domain. The aim of this article is to extend Theorem A to a general spherical domain. If we consider an increasing sequence of smooth domains{Sk}such thatSk ⊂Sk ⊂Sk+1

and∪kSk=Swe prove the following:

Theorem BAssume thatS^cis not polar. Then the sequence of theβS_k>0from Theorem A is decreasing and converges toβS >0. There existsωS ∈W₀^1,p(S)∩L^∞(S)weak solution of(1.1)withβ =βS. FurthermoreβS>0is the largest exponentβsuch that(1.1)admits a positive solutionωS ∈W₀^1,p(S).

Under a mild assumption onSit is possible to approximate it by a decreasing sequence of smooth domainsS_k⁰ such thatS_k⁰ ⊂S⁰_k ⊂S_k−1⁰ and∩kS_k⁰ =S

Theorem CAssume thatS =

o

S. Then the sequenceβ_S⁰

k > 0is increasing and converges toβˆS >0 and there existsωˆS ∈W₀^1,p(S)∩L^∞(S)weak solution of(1.1)withβ = ˆβS. FurthermoreβˆS is the smallest exponentβsuch that(1.1)admits a positive solutionωS∈W₀^1,p(S).

We prove the uniqueness of the exponentβ, under a Lipschitz assumption onS.

Theorem D Assume thatS is a Lipschitz domain, then βS = ˆβS and if ω and ω⁰ are two positive solutions of(1.1)inW₀^1,p(S), there exists a constantc >0such thatc⁻¹ω⁰ ≤ω≤cω⁰.

The proof of Theorem C is based upon a sharp form of boundary Harnack inequality proved in [10],

ln_ω^ω(σ0(σ¹₁⁾)−ln_ω^ω(σ0(σ²₂⁾)

≤c1|σ1−σ2|^α ∀σ1, σ2∈S, (1.2) for somec₁ =c₁(N, p, S)>0andα∈(0,1). Actually we have a stronger result, much more delicate to obtain.

Theorem ELetSbe a Lipschitz subdomain ofS^N−1. Then two positive solutions of(1.1)inW₀^1,p(S) are proportional.

The proof is based upon a non trivial adaptation of a series of deep results of Lewis and Nyström [10] concerning thep-Martin boundary of domains.

AcknowledgementsThis article has been prepared with the support of the collaboration programs ECOS C14E08.

2 Existence

2.1 Estimates

Through this article we assume thatS^cis not polar, or equivalently that it has positivec^S_1,p^N−1-capacity.

Lemma 2.1. Assumep >1. Then any solutionω∈W₀^1,p(S)of(1.1)satisfies

kωk_Cγ(S)≤c₁kωk_Lp(S), (2.1)

ifp > N−1whereγ= 1−^N_p⁻¹ ifp > N−1and

kωk_L∞(S)≤c1kωk_Lp(S), (2.2)

if1< p≤N−1, wherec₁>0depends onp,N,β.

Proof. Multiplying the equation byωand using Hölder’s inequality, we derive (i)

Z

S

β²ω²+|∇⁰ω|^2p₂

dS≤(β(pβ−(p−1)β0))^p2 Z

S

|ω|^pdS if p≥2, (ii)

Z

S

β²ω²+|∇⁰ω|^2p₂

dS≤β^p−1(pβ−(p−1)β0) Z

S

|ω|^pdS if 1< p <2.

(2.3)

Notice that these inequalities hold for allp >1. Ifp > N−1 (2.1)follows by Morrey’inequality. Here after we assume1< p≤N−1. Letα≥1andk >0. Thenζ = min{|ω|, k}^α−1ωis an admissible test function, hence

1- Ifp≥2, Z

S

β²ω²+|∇⁰ω|^2p−2₂

h∇⁰ω.∇⁰ζidS= (p−1)β(β−β₀) Z

S

β²ω²+|∇⁰ω|^2p−2₂ ωζdS

≤c2

Z

S

|∇⁰ω|^p−2ω²min{|ω|, k}^α−1dS+c2β^p Z

S

|ω|^pmin{|ω|, k}^α−1dS

≤c₂ Z

S

|ω|^pmin{|ω|, k}^α−1dS

^p−2_p Z

S

|∇⁰ω|^pmin{|ω|, k}^α−1dS _p²

+c2β^p Z

S

|ω|^pmin{|ω|, k}^α−1dS,

(2.4)

wherec₂=c₂(N, p, β)>0. Since Z

S

β²ω²+|∇⁰ω|^2p−2₂

h∇⁰ω.∇⁰ζidS≥c3(p) Z

S

|∇⁰ω|^pmin{|ω|, k}^α−1dS, it implies that there existsc4=c4(N, p, β)such that

Z

S

|∇⁰ω|^pmin{|ω|, k}^α−1dS≤c₄ Z

S

|ω|^pmin{|ω|, k}^α−1dS, (2.5) which yields

Z

S

|∇⁰j(ω)|^pdS≤c4

Z

S

|j(ω)|^pdS, (2.6)

wherej(ω) = min{|ω|, k}^α−1p ω.

2- If1< p <2, then Z

S

β²ω²+|∇⁰ω|^2p−2₂

h∇⁰ω.∇⁰ζidS= Z

S

β²ω²+|∇⁰ω|^2p−2₂

|∇⁰ω|²min{|ω|, k}^α−1dS

+ (α−1) Z

S∩{|ω|<k}

β²ω²+|∇⁰ω|^2p−2₂

|∇⁰ω|²|ω|^α−1dS.

(2.7)

Since Z

S

β²ω²+|∇⁰ω|^2p−2₂

|∇⁰ω|²min{|ω|, k}^α−1dS= Z

S

β²ω²+|∇⁰ω|^2p₂

min{|ω|, k}^α−1dS

−β² Z

S

β²ω²+|∇⁰ω|^2p−2₂

min{|ω|, k}^α−1ω²dS

≥ Z

S

|∇⁰ω|^pmin{|ω|, k}^α−1dS−β² Z

S

β²ω²+|∇⁰ω|^2p−2₂

min{|ω|, k}^α−1ω²dS, we derive

Z

S

|∇⁰ω|^pmin{|ω|, k}^α−1dS≤β^p−1(pβ−(p−1)β₀) Z

S

|ω|^pmin{|ω|, k}^α−1dS, (2.8) which leads to(2.6). Lettingk→ ∞we infer by Fatou’s lemma,

Z

S

∇⁰|ω|^{α−1p +1}

p

dS≤c₄ Z

S

|ω|^α−1+pdS. (2.9)

Ifp < N−1we derive from Sobolev inequality and puttingq=α−1 +pands= _N−1−p^N−1 >1 Z

S

|ω|^sqdS ¹_s

≤c₅ Z

S

|ω|^qdS, (2.10)

andc₅>0depends onN,pandβ. Iterating this estimate by Moser’s method we derive(2.10).

Ifp=N−1we have for1≤m < p−1andm^∗=^m(N−1)_N−1−m c6

Z

S

|ω|^{(α−1p +1)m∗}dS _m^pm∗

≤ Z

S

∇⁰|ω|^{α−1p +1}

m

dS _m^p

≤ |S|^mp−1c4

Z

S

|ω|^α−1+pdS, andc6=c6(N, p), hence

Z

S

|ω|^tqdS ¹_t

≤c5

Z

S

|ω|^qdS, (2.11)

witht=_p(N^m(N_−1−m)⁻¹⁾ =_N−1−m^m . The proof follows again by Moser’s iterative scheme.

Proposition 2.2. LetS₁andS₂be two subdomains ofS^N−1such thatS₁⊂S₁⊂S₂andS₂not polar.

Letβj >0, j=1,2, such that there exist positive solutionsωj∈W₀^1,p(Sj)solutions of

−div⁰

β_j²ω²_j+|∇⁰ω_j|^2p−2₂

∇⁰ω_j

= (p−1)β_j(β_j−β₀)

β_j²ω_j²+|∇⁰ω_j|^2p−2₂

ω_j inS_j

ωj= 0 in∂Sj.

(2.12) Thenβ1≥β2.

Proof. Setuj(r, σ) =r^−βjωj(σ),CS_j = (0,∞)×SJ and assumeβ1 < β2. By Harnack inequality ω2≥c >0onS1, thus

u2(r, σ)≥cr^−β2 a.e. inCS₁. For >0there existr>0such that

u₂(x)≥u₁(x) ∀x∈C_S1∩B_r.

Letδ >0, there existsR_δ >0such that

u1(x)≤δ ∀x∈CS₁∩B_R^c_δ. Henceζ = (u1−u2−δ)+ ∈W₀^1,p(Q^r_S^,Rδ

1 ), whereQ^r_S^,Rδ

1 = {x ∈ CS₁ : r <|x| < Rδ}. This implies

0 = Z

Q^r,Rδ_S

1

|∇u1|^p−2∇u1− |∇(u1)|^p−2∇(u1).∇ζ dx

= Z

Q^r,Rδ_S

1 ∩{u1−u2≥δ}

|∇u1|^p−2∇u1− |∇(u1)|^p−2∇(u1).∇(u1−u₂) dx.

Therefore∇(u1−u₂−δ)₊= 0a.e. inQ^r_S^,Rδ

1 , which leads tou₁−u₂≤δin the same set. Letting δ →0 yieldsRδ → ∞, thus we obtainu1 ≤u2inCS₁ ∩B^c_r henceu1 ≤ 0inCS₁, contradiction.

2.2 Approximations from inside

Proof of Theorem B. Let{Sk}be an increasing sequence of smooth domains such thatSk⊂Sk ⊂Sk+1. We denote by{(βS_k, ω_k)}the corresponding sequence of solutions of(1.1)withβ=β_Skandω=ω_k. The sequence{βSk}is uniquely determined by [15], it admits a limitβ:=β_S, and theω_kare the unique positive solutions such that

Z

S_k

|ωk|dS= 1.

Ifp≥2, we have Z

S_k

|∇⁰ωk|^pdS≤ Z

S_k

β²_Skω²_k+|∇⁰ωk|^2p−2₂

|∇⁰ωk|²dS

= (p−1)β_Sk(β_Sk−β₀) Z

S_k

β²_S

kω²_k+|∇⁰ω_k|^2p−2₂ ω_k²dS

≤2

(p−4)+

2 (p−1)βS_k(βS_k−β0) Z

Sk

β_S^p−2

k ω_k^p+|∇⁰ωk|^p−2ω_k² dS

≤c7(N, p, βS_k) Z

S_k

ω^p_kdS+1 2 Z

S_k

|∇⁰ωk|^pdS.

SinceβS_k≤β1, we derive

Z

S_k

|∇⁰ω_k|^pdS≤c₈, (2.13)

from the normalization assumption withc8= 2c7(N, p, β1).

If1< p <2, we have Z

Sk

|∇⁰ωk|^pdS≤ Z

Sk

β_S²

kω_k²+|∇⁰ωk|^2p₂ dS

≤βS_k(pβS_k+ (p−1)β0) Z

Sk

β²_S

kω²_k+|∇⁰ωk|^2p−2₂ ω_k²dS

≤β_k^p−1(pβS_k+ (p−1)β0) Z

S_k

ω_k^pdS,

and we obtain(2.13)withc₈=β₁^p−1(pβ₁+ (p−1)β₀).

Next we extend ωk by 0 inS_k^c. Then there existsω ∈ W₀^1,p(S) such that ωk * ω weakly in W₀^1,p(S), up to subsequence that we still denote{ω_k}, andω_k→ωinL^p(S).

Step 1: We claim that∇⁰ωkconverges to∇⁰ωlocally inL^p(S).

Leta∈Sandr >0such thatB4r(a)⊂S. Then fork≥k0,B2r(a)⊂Sk. Letζ∈C₀^∞(B2r(a))such that0≤ζ≤1,ζ= 1inB_r(a). For test function we chooseη_k =ζ(ω−ω_k), then

Z

S_k

β²_Skω²_k+|∇⁰ωk|^2p−2₂

h∇⁰ωk.∇⁰ηkidS= (p−1)βS_k(βS_k−β0) Z

S_k

β_S²_kω²_k+|∇⁰ωk|^2p−2₂

ωkηkdS.

By the above inequality, we have Z

B_2r(a)

β²ω²+|∇⁰ω|^2p−2₂

∇⁰ω− β_S²

kω²_k+|∇⁰ω_k|^2p−2₂

∇⁰ω_k.∇⁰η_k

dS

= Z

B2r(a)

β²ω²+|∇⁰ω|^2p−2₂

h∇⁰ω.∇⁰ηkidS

−(p−1)βS_k(βS_k−β0) Z

Sk

β_S²

kω_k²+|∇⁰ωk|^2p−2₂

ωkηkdS.

Using the weak convergence of the gradient, we have

k→∞lim Z

B_2r(a)

β²ω²+|∇⁰ω|^2p−2₂

h∇⁰ω.∇⁰ηkidS= 0.

Sinceωkis uniformly bounded inW₀^1,p(S)andωk→ωinL^p(S), we have

k→∞lim Z

B2r(a)

β_S²

kω_k²+|∇⁰ωk|^2p−2₂

ωkηkdS= 0, and

k→∞lim Z

B2r(a)

(ω−ωk)

β²ω²+|∇⁰ω|^2p−2₂

∇⁰ω−

β_S²_kω_k²+|∇⁰ωk|^2p−2₂

∇⁰ωk.∇⁰ζ

dS= 0.

Combining the above relations we infer

k→∞lim Z

B2r(a)

ζ

β²ω²+|∇⁰ω|^2p−2₂

∇⁰ω−

β²_Skω²_k+|∇⁰ωk|^2p−2₂

∇⁰ωk.∇⁰(ω−ωk)

dS= 0.

(2.14) Next we write

Z

B2r(a)

ζ

β²ω²+|∇⁰ω|^2p−2₂

∇⁰ω− β_S²

kω_k²+|∇⁰ωk|^2p−2₂

∇⁰ωk.∇⁰(ω−ωk)

dS

=1 2 Z

B2r(a)

ζ

β²ω²+|∇⁰ω|^2p−2₂ +

β_S²

kω_k²+|∇⁰ωk|^2p−2₂

|∇⁰(ω−ωk)|²dS

+1 2 Z

B_2r(a)

ζ

β²ω²+|∇⁰ω|^2p−2₂

−

β_S²_kω_k²+|∇⁰ωk|^2p−2₂

×

|∇⁰ω|²+β²ω²−β²_S

kω²_k− |∇⁰ωk|² dS

−1 2 Z

B_2r(a)

ζ

β²ω²+|∇⁰ω|^2p−2₂

− β_S²

kω_k²+|∇⁰ωk|^2p−2₂

β²ω²−β²_S

kω_k² dS.

(2.15)

Ifp≥2, we have from(2.4), Z

B_2r(a)

ζ

β²ω²+|∇⁰ω|^2p−2₂

∇⁰ω− β_S²

kω_k²+|∇⁰ω_k|^2p−2₂

∇⁰ω_k.∇⁰(ω−ω_k)

dS

≥1 2 Z

B2r(a)

ζ

|∇⁰ω|^p−2+|∇⁰ωk|^p−2

|∇⁰(ω−ωk)|²dS

−1 2 Z

B_2r(a)

ζ

β²ω²+|∇⁰ω|^2p−2₂

− β_S²

kω_k²+|∇⁰ωk|^2p−2₂

β²ω²−β²_S

kω_k² dS

≥min{2⁻¹,2^2−p} Z

B_2r(a)

ζ|∇⁰(ω−ω_k)|^pdS

−1 2 Z

B2r(a)

ζ

β²ω²+|∇⁰ω|^2p−2₂

− β_S²

kω_k²+|∇⁰ωk|^2p−2₂

β²ω²−β²_S

kω_k² dS.

(2.16) Sinceωk →ωinL^p(S),βS_k→βandωk, ωare uniformly bounded inW₀^1,p(S), we derive

Z

B2r(a)

ζ

β²ω²+|∇⁰ω|^2p−2₂

− β_S²

kω²_k+|∇⁰ωk|^2p−2₂

β²ω²−β_S²

kω_k²

dS→0 ask→ ∞. Jointly with(2.14)we infer that

k→∞lim Z

B_r(a)

|∇⁰(ω−ωk)|^pdS= 0. (2.17)

If1< p <2, then Z

B2r(a)

ζ

β²ω²+|∇⁰ω|^2p−2₂

∇⁰ω− β²_S

kω²_k+|∇⁰ωk|^2p−2₂

∇⁰ωk.∇⁰(ω−ωk)

dS

= Z

B2r(a)

ζ

β²_S

kω²_k+|∇⁰ω|^2p−2₂

∇⁰ω− β_S²

kω_k²+|∇⁰ωk|^2p−2₂

∇⁰ωk.∇⁰(ω−ωk)

dS

+ Z

B_2r(a)

ζ

β²ω²+|∇⁰ω|^2p−2₂

− β_S²

kω_k²+|∇⁰ω|^2p−2₂

∇⁰ω.∇⁰(ω−ω_k)

dS.

(2.18) Up to extracting a subsequence, we have thatω_k →ω a.e. inS and that there existsΦ∈L¹(S)such that

|ωk|^p+|ω|^p≤Φ a.e. inS and ∀k≥1. (2.19) Since

β²_Skω²_k+|∇⁰ω|^2p−2₂

|∇ω| ≤

β_S²_kω²_k+|∇⁰ω|^2p−1₂

≤β_S^p−1

k ω^p−1_k +|∇⁰ω|^p−1, and

β²ω²+|∇⁰ω|^2p−2₂

|∇ω| ≤β^p−1ω^p−1+|∇⁰ω|^p−1, we derive that

β²ω²+|∇⁰ω|^2p−2₂

−

β_S²_kω²_k+|∇⁰ω|^2p−2₂

|∇⁰ω| ≤2

β^p−1Φ^p−1+|∇⁰ω|^p−1 , which implies that

ζ

β²ω²+|∇⁰ω|^2p−2₂

−

β_S²_kω_k²+|∇⁰ω|^2p−2₂

∇⁰ω→0 inL^p0(S)

wherep⁰is the conjugate ofp, and finally Z

B2r(a)

ζh

β²ω²+|∇⁰ω|^2p−2₂

− β²_S

kω²_k+|∇⁰ω|^2p−2₂

∇⁰ω.∇⁰(ω−ωk)idS→0 ask→ ∞.

(2.20) For the last term on the right-hand side of(2.18), we have, forγ∈R+andA,B∈R^N,

γ+|B|^2p−2₂ B−

γ+|A|^2p−2₂ A=

Z 1 0

d dt

γ+|tB+ (1−t)A|^2p−2₂

(tB+ (1−t)A)

dt

= Z 1

0

γ+|tB+ (1−t)A|^2p−2₂ dt

(B−A)

+ (p−2) Z 1

0

γ+|tB+ (1−t)A|^2p−4₂

htB+ (1−t)A.B−Ai(tB+ (1−t)A)dt.

This implies h

γ+|B|^2p−2₂ B−

γ+|A|^2p−2₂

A.B−Ai= Z 1

0

γ+|tB+ (1−t)A|^2p−2₂ dt

|B−A|²

+ (p−2) Z 1

0

γ+|tB+ (1−t)A|^2p−4₂

htB+ (1−t)A.B−Ai²dt.

We observe that Z 1

0

γ+|tB+ (1−t)A|^2p−4₂

htB+ (1−t)A.B−Ai²dt

≤ |B−A|² Z 1

0

γ+|tB+ (1−t)A|^2p−2₂ dt, and since1< p <2, we finally obtain

h

γ+|B|^2p−2₂ B−

γ+|A|^2p−2₂

A.B−Ai

≥(p−1) Z 1

0

γ+|tB+ (1−t)A|^2p−2₂ dt

|B−A|²

≥(p−1)|B−A|²

γ+ 1 +|B|²+|A|^2p−2₂ .

(2.21)

We plug this estimate into(2.18)withγ=β²_kω_k²,A=∇⁰ωandB=∇⁰ω_k, then Z

B2r(a)

ζh β²_S

kω²_k+|∇⁰ω|^2p−2₂

∇⁰ω− β_S²

kω_k²+|∇⁰ωk|^2p−2₂

∇⁰ωk.∇⁰(ω−ωk)idS

≥ Z

B_2r(a)

ζ|∇⁰(ω−ωk)|²

β_k²ω²_k+ 1 +|∇⁰ωk|²+|∇⁰ω|^2p−2₂ dS.

(2.22)

Setφ(.) =β_k²ω²_k+ 1 +|∇⁰ωk|²+|∇⁰ω|², then Z

B_r(a)

|∇⁰ω− ∇⁰ω_k|^pdS= Z

B_r(a)

|∇⁰ω− ∇⁰ω_k|^pφ^p(p−2)4 φ^−p(p−2)4 dS

≤ Z

Br(a)

|∇⁰ω− ∇⁰ωk|²φ^p−22 dS

!^p₂ Z

Br(a)

φ^p2dS

!^2−p₂ .

(2.23)

Jointly with(2.14)and(2.22)we conclude that(2.17). Step 1 follows by a standard covering argument.

Step 2: We claim thatω_kconverges toωinW₀^1,p(S).

Up to a subsequence that we denote again by{k}, we can assume thatωk →ωand∇⁰ωk → ∇⁰ωa.e.

inS. Letζ ∈C₀^∞(S), then there existsk∈Nsuch that the supportKofζis a compact subset ofSk

for allk≥k. If1< p <2,

β_S²_kω_k²+|∇⁰ωk|^2p−2₂

|∇⁰ωk| ≤ |∇⁰ωk|^p−1,

which bounded inL^p0(K), then uniformly integrable inKand by Vitali’s convergence theorem β_S²

kω²_k+|∇⁰ω_k|^2p−2₂

∇⁰ω_k →

β²ω²+|∇⁰ω|^2p−2₂

∇⁰ω, inL¹_loc(S). Similarly

β²_Skω_k²+|∇⁰ωk|^2p−2₂

ωk →

β²ω²+|∇⁰ω|^2p−2₂ ω, inL¹_loc(S). Ifp≥2

β_S²_kω_k²+|∇⁰ωk|^2p−2₂

|∇⁰ωk| ≤c

|ωk|^p−1+|∇⁰ωk|^p−1 ,

and we conclude again by Vitali’s convergence theorem that the previous convergences hold. Since Z

S_k

β²_S

kω²_k+|∇⁰ω_k|^2p−2₂

h∇⁰ω_k.∇⁰ζidS= (p−1)β_Sk(β_Sk−β₀) Z

S_k

β²_S

kω²_k+|∇⁰ω_k|^2p−2₂ ω_kζdS

we conclude thatωis a weak solution of(1.1)withβ=βS.

2.3 Approximations from outside

Proof of Theorem C. SinceS^chas a non-empty interior, the existence of a sequence{ω⁰_k}corresponding to solutions of(1.1)inS_k⁰ withβ =βS_k⁰ is the consequence of [13]. The fact that{βS_k⁰}is increasing follows from Proposition 2.2. We denote byβˆ:= ˆβSits limit, and it is smaller or equal toβS. Estimates (2.4)are valid withS⁰_k,ω_k⁰ andβS_k⁰ instead ofS,ωandβ. If we extendω⁰_kby0inS_k^0cthese estimates are valid with S^N−1 instead ofS⁰_k. Then up to a subsequence the existsω ∈ W^1,p(S^N−1) and a subsequence stil denoted by{k}such thatω_k⁰ * ωweakly inW^1,p(S^N−1), strongly inL^p(S^N−1)and a.e. inS^N−1. Furthermore, as in the proof of Theorem A, for any compact setK ⊂S,∇⁰ω_k⁰ → ∇⁰ω⁰ inL^p(K). This is sufficient to assert thatωis a weak solution of

−div⁰

βˆ²ω⁰²+|∇⁰ω⁰|^2p−2₂

∇⁰ω⁰

= (p−1) ˆβ( ˆβ−β₀)

βˆ²ω²+|∇⁰ω⁰|^2p−2₂

ω⁰ inS.

Moreoverω⁰b_S⁰

k belongs toW₀^1,p(S_k⁰)for allk. Sinceω_k⁰ = 0inS_k^c and converges a.e. toω, this last function vanishes a.e. in∪kS_k^c = (∩kSk)^c =S^c. Thereforeωvanishes a.e. inS^cand since it is quasi continuous, it vanishes,(1−p)- quasi everywhere inS^c. From Netrusov’s theorem (see [1, Th 10.1.1]- (iii)) there exists a sequence{η_n} ⊂ C₀^∞(S)which converges toω inW^1,p(S), thusω ∈ W₀^1,p(S).

3 Uniqueness

3.1 Uniqueness of exponent β

Proof of Theorem D. IfS is Lipschitz,CS is also Lipschitz. We fixz ∈ S ≈ S^N−1∩∂CS and we apply [10, Th 2] inGz =CS ∩B¹

2(z)to two separablep-harmonic functionsu(r, σ) = r^−βω(σ)and u⁰(r, σ) =r^−β0ω⁰(σ). There existγ∈(0,¹₂),c₁₀>0andα∈(0,1)such that

ln u(y1)

u⁰(y1)−ln u(y2) u⁰(y2)

≤c10|y1−y2|^α ∀y1, y2∈CS∩Bγ(z). (3.24) Assume|y1|=|y2|= 1, then

ln ω(y1)

ω⁰(y₁)−ln ω(y2) ω⁰(y₂)

≤c10|y1−y2|^α ∀y1, y2∈S∩Bγ(z). (3.25) We denote by`(x, y)the geodesic distance onS<sup>N−1</sup>and by`(x, K)the geodesic distance from a point x∈S^N−1to a subsetK. Since the setSγ ={σ∈S :`(σ, ∂S)≤^γ₂}can be covered by a finite number of balls with center on∂S, we infer that

ln ω(y1)

ω⁰(y₁)−ln ω(y2) ω⁰(y₂)

≤c₁₁ ∀y₁, y₂∈S_γ. (3.26) InS\S^γ

2

we can use Harnack inequality to obtain

−c12≤lnω(y1)

ω(y2) ≤c12 ∀y1, y2∈S\S^γ

2

s.t.`(y1, y2)≤ ^γ₂. (3.27) Hence there exists a constantc₁₃>0such that(3.27)holds for anyy₁, y₂∈S\Sγ

2

, withc₁₂replaced byc13. Furthermore ω⁰ satisfies the same inequality in S\Sγ

2

. Combining the two inequalities we obtain

−2c13≤lnω(y1)

ω(y2)−lnω⁰(y1)

ω⁰(y2)≤2c13 ∀y1, y2∈S\S^γ

2

. (3.28)

Combining this estimate with(3.25)we derive that it holds for ally₁, y₂∈S. This implies e^−2c13 ω(y2)

ω⁰(y2) ≤ ω(y1)

ω⁰(y1) ≤e^2c13ω(y2)

ω⁰(y2) ∀y1, y2∈S. (3.29) Assume now that there exist two exponentsβ > β⁰ > 0 such thatr^−βω(.)andr^−β0ω⁰(.)arep- harmonic and positive in the coneCSand vanishes on∂CS. Putθ=_β^β0,η=ω^0θand

T(η) =−div⁰

β²η²+|∇⁰η|^2p−2₂

∇⁰η

−(p−1)β(β−β0)

β²η²+|∇⁰η|^2p−2₂ η, then

T(η) =−θ^p−2

β⁰²ω⁰²+|∇⁰ω⁰|^2p−2₂

(β−β⁰)ω⁰²+ (p−1)θ(θ−1)|∇⁰ω⁰|²

≤0.

Up to multiplyingω⁰ byλ, we can assume thatη ≤ ωand that the graphs of η andω are tangent in S. Sinceω⁰ ≤ cω,η = o(ω)near∂S. Hence there existsσ₀ ∈ S such thatω(σ₀) = η(σ₀)and the

coincidence set ofηandωis a compact subset ofS. We putw=ω−η, since∇ω(σ₀) =∇η(σ₀)we proceed as in [14, Th 4.1] (see also [4] in the flat case) and derive thatwsatisfies, in a system of local coordinates(σ₁, ..., σ_N−1)nearσ₀,

Lw:=−X

`,j

∂

∂σ_`

Aj,`

∂w

∂σ_j

+X

j

Cj

∂w

∂σ_`+Cw≥0,

where the matrix(A_j,`)is smooth, symmetric and positive nearσ₀and theC_jandCare bounded. Hence wis locally zero. By a standard argument of connectedness, this implies that the zero set ofwmust be

empty, contradiction. Henceβ=β⁰.

3.2 Uniqueness of eigenfunction

The proof is based upon a delicate adaptation of the characterisation of thep-Martin boundary obtained in [10], but we first give a proof in the convex case.

3.2.1 The convex case

Theorem 3.1. AssumeS is a convex spherical subdomain. Then two positive solutions of(1.1)are proportional.

Proof. We recall that a domainSis (geodesically) convex if a minimal geodesic joining two points of S is contained in S. IfS ⊂ S^N−1 is convex, the cone CS is convex too. SinceS is convex, it is Lipschitz and by Theorem D,βS = ˆβS :=β. Letωandω⁰be two positive solutions of(1.1)satisfying sup_Sω= sup_Sω⁰ = 1. We denote byuω(x) =|x|^−βω(.)anduω⁰(x) =|x|^−βω⁰(.)the corresponding separablep-harmonic functions defined inC_S. If0< a < b, we setC_S^a,b =C_S∩(B_b\B_a). Then for 0< <1we denote byuthe unique function which satisfies

−∆pu= 0 inC_S^,1 u=^−βω inCS∩∂B

u= 0 in (C_S∩∂B₁)∪ ∂C_S∩(B₁\B) .

(3.30)

Then

(u_ω−1)₊≤u≤u_ω inC_S^,1. (3.31)

Furthermore7→ uis increasing. When ↓0,u ↑u0whereu0is positive andp-harmonic inC_S^1,0, vanishes on∂C_S^1,0\ {0}and satisfies(3.30)with= 0. In particular

r→0limr^βu0(r, σ) =ω(σ) locally uniformly inS. (3.32) We construct the same approximationu⁰inC_S^,1withω⁰instead ofω. Mutadis mutandis(3.31)holds andu⁰↑u⁰₀which is positive andp-harmonic inC_S¹, satisfies

(u_ω⁰−1)₊≤u⁰₀≤u_ω⁰ inC_S^1,0, and thus

r→0limr^βu⁰₀(r, σ) =ω⁰(σ) locally uniformly inS. (3.33) However, by [10, Th 4]u₀andu⁰₀are proportional. Combined with(3.32),(3.33)it implies the claim.

3.2.2 Proof of Theorem E

In what follows we borrow most of our construction from [10] that we adapt to the case of an infinite cone a make explicit for the sake of completeness. The nextnondegeneracy propertyof positivep-harmonic functions is proved in [10, Lemma 4.28].

Proposition 3.2. Let Ω ⊂ R^N be a bounded Lipschitz domain and 1 < p < ∞. Then there exist constantsρ >0,c14, c15>0depending respectively onΩ(forρ), andp,N and the Lipschitz normM of∂Ω(forc14andc15) with the property that for anyw∈∂Ωand any positivep-harmonic functionu inΩ, continuous inΩ∩B2ρ(w)and vanishing on∂Ω∩Bρ(w), one can findξ∈S^N−1, independent of u, such that

c⁻¹₁₄ u(y)

dist(y, ∂Ω)≤ h∇u(y), ξi ≤ |∇u(y)| ≤c14

u(y)

dist(y, ∂Ω), (3.34) for ally∈C_S∩Bρ|w|

c15

(w).

IfΩis replaced by a coneC_S, the nondegeneracy property still holds uniformly on∂C_S\ {0}.

Corollary 3.3. Let1< p <∞,S⊂S^N−1is a Lipschitz domain andC_Sthe cone generated byS.

(i) Then there exist constantsρ < ¹₂,c14, c15>0depending respectively onS(forρ), andp,Nand the Lipschitz normMof∂Sanddiam(S)(forc14andc15) with the property that for anyw∈∂CSand any positivep-harmonic functionuinCS, continuous inCS∩B_2ρ|w|(w)and vanishing on∂CS∩B_ρ|w|(w) continuous, one can findξ∈S^N−1, independent ofu, such that

c⁻¹₁₄ u(y)

dist(y, ∂CS)≤ h∇u(y), ξi ≤ |∇u(y)| ≤c₁₄ u(y)

dist(y, ∂CS), (3.35) for ally∈B ^ρ

c15

(w)∩CS.

(ii) Then there exist positive constantsκandc16, c17depending onS(forκ), andp,Nand the Lipschitz normM of∂Sanddiam(S)(forc16, c17such that for anya >0and any positivep-harmonic function uinC_S^a vanishing on∂CS∩B_a^c, there holds

c⁻¹₁₆ u(y)

dist(y, ∂CS) ≤ |∇u(y)| ≤c16

u(y)

dist(y, ∂CS) ∀y∈C_S^c17a s.t.dist(y, ∂CS)≤κ|y|. (3.36) Letω, ω⁰ ∈W₀^1,p(S)∩C(S)be positive solutions(1.1). Since_ω^ω0 is bounded from above and from below inS by positive constants, we can assume, as in the proof of Theorem D, thatω ≥ω⁰ inSand that the graphs ofωandω⁰are tangent. hence, ifω6=ω⁰, thenω > ω⁰ inSand there exists a sequence {σn}converging toσ0∈∂Sasn→ ∞such that

n→∞lim ω⁰(σ_n)

ω(σn) = 1.

We defineδ1= sup{δ >0 :δω < ω⁰}. Fort∈(δ1,1), we set φt= sup{ω⁰, tω} and ψt= inf

t δ1

ω⁰, ω

(3.37) We also set

v_φt(r, σ) =r^−βφ_t(σ) and v_ψt(r, σ) =r^−βψ_t(σ) ∀(r, σ)∈(0,∞)×S. (3.38)