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

TL;DR

This paper establishes the existence and uniqueness of p-harmonic functions with specific asymptotic behavior in conical domains generated by non-smooth spherical surfaces, extending understanding of such functions in irregular geometries.

Contribution

It proves the existence and uniqueness of p-harmonic functions of a specific form in non-smooth conical domains, generalizing previous results to irregular spherical boundaries.

Findings

Existence of p-harmonic functions in non-smooth cones.

Uniqueness of the exponent and normalized function under Lipschitz conditions.

Extension of classical results to irregular spherical domains.

Abstract

We prove the existence of p-harmonic functions under the form u(r, $\sigma$) = r --$\beta$ $\omega$($\sigma$) in any cone C S generated by a spherical domain S and vanishing on $\partial$C S. We prove the uniqueness of the exponent $\beta$ and of the normalized function $\omega$ under a Lipschitz condition on S.

Equations

$$\begin{array}[]{lll}-div^{\prime}\left(\left(\beta^{2}\omega^{2}+\left|\nabla^{\prime}\omega\right|^{2}\right)^{\frac{p-2}{2}}\nabla^{\prime}\omega\right)=(p-1)\beta(\beta-\beta_{0})\left(\beta^{2}\omega^{2}+\left|\nabla^{\prime}\omega\right|^{2}\right)^{\frac{p-2}{2}}\omega&\text{in }S\\ \phantom{-div^{\prime}\left(\left(\beta^{2}\omega^{2}+\left|\nabla^{\prime}\omega\right|^{2}\right)^{\frac{p-2}{2}}\nabla^{\prime}\right)}\omega=0&\text{in }\partial S\end{array}$$

$$\begin{array}[]{lll}\left|\ln\frac{\omega(\sigma_{1})}{\omega^{\prime}(\sigma_{1})}-\ln\frac{\omega(\sigma_{2})}{\omega^{\prime}(\sigma_{2})}\right|\leq c_{1}\left|\sigma_{1}-\sigma_{2}\right|^{\alpha}\quad\forall\,\sigma_{1},\sigma_{2}\in S,\end{array}$$

$${\Vert \omega \Vert }_{C^{\gamma }(S)}\le c_1{\Vert \omega \Vert }_{L^p(S)},$$

$${\Vert \omega \Vert }_{L^{\infty }(S)}\le c_1{\Vert \omega \Vert }_{L^p(S)},$$

$$\begin{array}[]{lll}\displaystyle(i)&{\displaystyle\int_{S}}\left(\beta^{2}\omega^{2}+\left|\nabla^{\prime}\omega\right|^{2}\right)^{\frac{p}{2}}dS\leq\left(\beta(p\beta-(p-1)\beta_{0})\right)^{\frac{p}{2}}{\displaystyle\int_{S}}\left|\omega\right|^{p}dS\quad\!\text{if }\,p\geq 2,\\[11.38109pt] (ii)&{\displaystyle\int_{S}}\left(\beta^{2}\omega^{2}+\left|\nabla^{\prime}\omega\right|^{2}\right)^{\frac{p}{2}}dS\leq\beta^{p-1}(p\beta-(p-1)\beta_{0}){\displaystyle\int_{S}}\left|\omega\right|^{p}dS\quad\text{if }\,1<p<2.\end{array}$$

$$\begin{array}[]{lll}{\displaystyle\int_{S}}\left(\beta^{2}\omega^{2}+\left|\nabla^{\prime}\omega\right|^{2}\right)^{\frac{p-2}{2}}\langle\nabla^{\prime}\omega.\nabla^{\prime}\zeta\rangle dS=(p-1)\beta(\beta-\beta_{0}){\displaystyle\int_{S}}\left(\beta^{2}\omega^{2}+\left|\nabla^{\prime}\omega\right|^{2}\right)^{\frac{p-2}{2}}\omega\zeta dS\\[11.38109pt] \phantom{---}\leq c_{2}{\displaystyle\int_{S}}\left|\nabla^{\prime}\omega\right|^{p-2}\omega^{2}\min\{\left|\omega\right|,k\}^{\alpha-1}dS+c_{2}\beta^{p}{\displaystyle\int_{S}}\left|\omega\right|^{p}\min\{\left|\omega\right|,k\}^{\alpha-1}dS\\[11.38109pt] \phantom{---}\leq c_{2}\left({\displaystyle\int_{S}}\left|\omega\right|^{p}\min\{\left|\omega\right|,k\}^{\alpha-1}dS\right)^{\frac{p-2}{p}}\left({\displaystyle\int_{S}}\left|\nabla^{\prime}\omega\right|^{p}\min\{\left|\omega\right|,k\}^{\alpha-1}dS\right)^{\frac{2}{p}}\\[11.38109pt] \phantom{---}+c_{2}\beta^{p}{\displaystyle\int_{S}}\left|\omega\right|^{p}\min\{\left|\omega\right|,k\}^{\alpha-1}dS,\end{array}$$

$${\int }_S{\left({\beta }^2{\omega }^2+{\left|{\nabla }^{\prime }\omega \right|}^2\right)}^{\frac{p-2}{2}}⟨{\nabla }^{\prime }\omega .{\nabla }^{\prime }\zeta ⟩dS\ge c_3(p){\int }_S{\left|{\nabla }^{\prime }\omega \right|}^p\min \{\left|\omega \right|,k{\}}^{\alpha -1}dS,$$

$$\begin{array}[]{lll}{\displaystyle\int_{S}}\left|\nabla^{\prime}\omega\right|^{p}\min\{\left|\omega\right|,k\}^{\alpha-1}dS\leq c_{4}{\displaystyle\int_{S}}\left|\omega\right|^{p}\min\{\left|\omega\right|,k\}^{\alpha-1}dS,\end{array}$$

$$\begin{array}[]{lll}{\displaystyle\int_{S}}\left|\nabla^{\prime}j(\omega)\right|^{p}dS\leq c_{4}{\displaystyle\int_{S}}\left|j(\omega)\right|^{p}dS,\end{array}$$

$$\begin{array}[]{lll}{\displaystyle\int_{S}}\left(\beta^{2}\omega^{2}+\left|\nabla^{\prime}\omega\right|^{2}\right)^{\frac{p-2}{2}}\langle\nabla^{\prime}\omega.\nabla^{\prime}\zeta\rangle dS={\displaystyle\int_{S}}\left(\beta^{2}\omega^{2}+\left|\nabla^{\prime}\omega\right|^{2}\right)^{\frac{p-2}{2}}\left|\nabla^{\prime}\omega\right|^{2}\min\{\left|\omega\right|,k\}^{\alpha-1}dS\\[11.38109pt] \phantom{{\displaystyle\int_{S}}\left(\beta^{2}\omega^{2}+\left|\nabla^{\prime}\omega\right|^{2}\right)^{\frac{p-2}{2}}\nabla^{\prime}\omega.}+(\alpha-1){\displaystyle\int_{S\cap\{\left|\omega\right|<k\}}}\left(\beta^{2}\omega^{2}+\left|\nabla^{\prime}\omega\right|^{2}\right)^{\frac{p-2}{2}}\left|\nabla^{\prime}\omega\right|^{2}\left|\omega\right|^{\alpha-1}dS.\end{array}$$

$$\begin{array}[]{lll}{\displaystyle\int_{S}}\left(\beta^{2}\omega^{2}+\left|\nabla^{\prime}\omega\right|^{2}\right)^{\frac{p-2}{2}}\left|\nabla^{\prime}\omega\right|^{2}\min\{\left|\omega\right|,k\}^{\alpha-1}dS={\displaystyle\int_{S}}\left(\beta^{2}\omega^{2}+\left|\nabla^{\prime}\omega\right|^{2}\right)^{\frac{p}{2}}\min\{\left|\omega\right|,k\}^{\alpha-1}dS\\[11.38109pt] \phantom{-------}-\beta^{2}{\displaystyle\int_{S}}\left(\beta^{2}\omega^{2}+\left|\nabla^{\prime}\omega\right|^{2}\right)^{\frac{p-2}{2}}\min\{\left|\omega\right|,k\}^{\alpha-1}\omega^{2}dS\\[11.38109pt] \phantom{-----}\geq{\displaystyle\int_{S}}\left|\nabla^{\prime}\omega\right|^{p}\min\{\left|\omega\right|,k\}^{\alpha-1}dS-\beta^{2}{\displaystyle\int_{S}}\left(\beta^{2}\omega^{2}+\left|\nabla^{\prime}\omega\right|^{2}\right)^{\frac{p-2}{2}}\min\{\left|\omega\right|,k\}^{\alpha-1}\omega^{2}dS,\end{array}$$

$$\begin{array}[]{lll}{\displaystyle\int_{S}}\left|\nabla^{\prime}\omega\right|^{p}\min\{\left|\omega\right|,k\}^{\alpha-1}dS\leq\beta^{p-1}(p\beta-(p-1)\beta_{0}){\displaystyle\int_{S}}\left|\omega\right|^{p}\min\{\left|\omega\right|,k\}^{\alpha-1}dS,\end{array}$$

$$\begin{array}[]{lll}{\displaystyle\int_{S}}\left|\nabla^{\prime}\left|\omega\right|^{\frac{\alpha-1}{p}+1}\right|^{p}dS\leq c_{4}{\displaystyle\int_{S}}\left|\omega\right|^{\alpha-1+p}dS.\end{array}$$

$$\begin{array}[]{lll}\left({\displaystyle\int_{S}}\left|\omega\right|^{sq}dS\right)^{\frac{1}{s}}\leq c_{5}{\displaystyle\int_{S}}\left|\omega\right|^{q}dS,\end{array}$$

$$c_6{\left({\int }_S{\left|\omega \right|}^{(\frac{\alpha -1}{p}+1)m^{\ast }}dS\right)}^{\frac{pm}{m^{\ast }}}\le {\left({\int }_S{\left|{\nabla }^{\prime }{\left|\omega \right|}^{\frac{\alpha -1}{p}+1}\right|}^{m}dS\right)}^{\frac{p}{m}}\le {\left|S\right|}^{\frac{p}{m}-1}{c}_4{\int }_S{\left|\omega \right|}^{\alpha -1+p}dS,$$

$$\begin{array}[]{lll}\left({\displaystyle\int_{S}}\left|\omega\right|^{tq}dS\right)^{\frac{1}{t}}\leq c_{5}{\displaystyle\int_{S}}\left|\omega\right|^{q}dS,\end{array}$$

$$\begin{array}[]{lll}-div^{\prime}\left(\left(\beta_{j}^{2}\omega_{j}^{2}+\left|\nabla^{\prime}\omega_{j}\right|^{2}\right)^{\frac{p-2}{2}}\nabla^{\prime}\omega_{j}\right)=(p-1)\beta_{j}(\beta_{j}-\beta_{0})\left(\beta_{j}^{2}\omega_{j}^{2}+\left|\nabla^{\prime}\omega_{j}\right|^{2}\right)^{\frac{p-2}{2}}\omega_{j}&\text{in }S_{j}\\ \phantom{-div^{\prime}\left(\left(\beta^{2}\omega_{j}^{2}+\left|\nabla^{\prime}\omega_{j}\right|^{2}\right)^{\frac{p-2}{2}}\nabla^{\prime}\right)}\omega_{j}=0&\text{in }\partial S_{j}.\end{array}$$

$$u_2(r,\sigma )\ge c{r}^{-{\beta }_2}\text{a.e. in }{C}_{S_1}.$$

$$ϵu_2(x)\ge u_1(x)\forall x\in C_{S_1}\cap {\overline{B}}_{r_{ϵ}}.$$

$$u_1(x)\le \delta \forall x\in C_{S_1}\cap B_{R_{\delta }}^c.$$

$$\begin{array}[]{lll}0={\displaystyle\int_{Q^{r_{\epsilon},R_{\delta}}_{S_{1}}}}\big{\langle}\left|\nabla u_{1}\right|^{p-2}\nabla u_{1}-\left|\nabla(\epsilon u_{1})\right|^{p-2}\nabla(\epsilon u_{1}).\nabla\zeta\big{\rangle}dx\\[11.38109pt] \phantom{0}={\displaystyle\int_{Q^{r_{\epsilon},R_{\delta}}_{S_{1}}\cap\{u_{1}-\epsilon u_{2}\geq\delta\}}}\big{\langle}\left|\nabla u_{1}\right|^{p-2}\nabla u_{1}-\left|\nabla(\epsilon u_{1})\right|^{p-2}\nabla(\epsilon u_{1}).\nabla(u_{1}-u_{2})\big{\rangle}dx.\end{array}$$

$${\int }_{S_k}\left|{\omega }_k\right|dS=1.$$

$$\begin{array}[]{lll}{\displaystyle\int_{S_{k}}}\left|\nabla^{\prime}\omega_{k}\right|^{p}dS\leq{\displaystyle\int_{S_{k}}}\left(\beta_{S_{k}}^{2}\omega_{k}^{2}+\left|\nabla^{\prime}\omega_{k}\right|^{2}\right)^{\frac{p-2}{2}}\left|\nabla^{\prime}\omega_{k}\right|^{2}dS\\[11.38109pt] \phantom{{\displaystyle\int_{S_{k}}}\left|\nabla^{\prime}\omega_{k}\right|^{p}dS}=(p-1)\beta_{S_{k}}(\beta_{S_{k}}-\beta_{0}){\displaystyle\int_{S_{k}}}\left(\beta_{S_{k}}^{2}\omega_{k}^{2}+\left|\nabla^{\prime}\omega_{k}\right|^{2}\right)^{\frac{p-2}{2}}\omega_{k}^{2}dS\\[11.38109pt] \phantom{{\displaystyle\int_{S_{k}}}\left|\nabla^{\prime}\omega_{k}\right|^{p}dS}\leq 2^{\frac{(p-4)_{+}}{2}}(p-1)\beta_{S_{k}}(\beta_{S_{k}}-\beta_{0}){\displaystyle\int_{S_{k}}}\left(\beta_{S_{k}}^{p-2}\omega_{k}^{p}+\left|\nabla^{\prime}\omega_{k}\right|^{p-2}\omega_{k}^{2}\right)dS\\[11.38109pt] \phantom{{\displaystyle\int_{S_{k}}}\left|\nabla^{\prime}\omega_{k}\right|^{p}dS}\leq c_{7}(N,p,\beta_{S_{k}}){\displaystyle\int_{S_{k}}}\omega_{k}^{p}dS+{\displaystyle\frac{1}{2}}{\displaystyle\int_{S_{k}}}\left|\nabla^{\prime}\omega_{k}\right|^{p}dS.\end{array}$$

$$\begin{array}[]{lll}{\displaystyle\int_{S_{k}}}\left|\nabla^{\prime}\omega_{k}\right|^{p}dS\leq c_{8},\end{array}$$

$$\begin{array}[]{lll}{\displaystyle\int_{S_{k}}}\left|\nabla^{\prime}\omega_{k}\right|^{p}dS\leq{\displaystyle\int_{S_{k}}}\left(\beta_{S_{k}}^{2}\omega_{k}^{2}+\left|\nabla^{\prime}\omega_{k}\right|^{2}\right)^{\frac{p}{2}}dS\\[11.38109pt] \phantom{{\displaystyle\int_{S_{k}}}\left|\nabla^{\prime}\omega_{k}\right|^{p}dS}\leq\beta_{S_{k}}(p\beta_{S_{k}}+(p-1)\beta_{0}){\displaystyle\int_{S_{k}}}\left(\beta_{S_{k}}^{2}\omega_{k}^{2}+\left|\nabla^{\prime}\omega_{k}\right|^{2}\right)^{\frac{p-2}{2}}\omega_{k}^{2}dS\\[11.38109pt] \phantom{{\displaystyle\int_{S_{k}}}\left|\nabla^{\prime}\omega_{k}\right|^{p}dS}\leq\beta^{p-1}_{k}(p\beta_{S_{k}}+(p-1)\beta_{0}){\displaystyle\int_{S_{k}}}\omega_{k}^{p}dS,\end{array}$$

$${\int }_{S_k}{\left({\beta }_{S_k}^2{\omega }_k^2+{\left|{\nabla }^{\prime }{\omega }_k\right|}^2\right)}^{\frac{p-2}{2}}⟨{\nabla }^{\prime }{\omega }_k.{\nabla }^{\prime }{\eta }_k⟩dS=(p-1){\beta }_{S_k}({\beta }_{S_k}-{\beta }_0){\int }_{S_k}{\left({\beta }_{S_k}^2{\omega }_k^2+{\left|{\nabla }^{\prime }{\omega }_k\right|}^2\right)}^{\frac{p-2}{2}}{\omega }_k{\eta }_{k}dS.$$

$$\begin{array}[]{lll}{\displaystyle\int_{B_{2r}(a)}}\bigg{\langle}\left(\beta^{2}\omega^{2}+\left|\nabla^{\prime}\omega\right|^{2}\right)^{\frac{p-2}{2}}\nabla^{\prime}\omega-\left(\beta_{S_{k}}^{2}\omega_{k}^{2}+\left|\nabla^{\prime}\omega_{k}\right|^{2}\right)^{\frac{p-2}{2}}\nabla^{\prime}\omega_{k}.\nabla^{\prime}\eta_{k}\bigg{\rangle}dS\\[11.38109pt] \phantom{------------}={\displaystyle\int_{B_{2r}(a)}}\left(\beta^{2}\omega^{2}+\left|\nabla^{\prime}\omega\right|^{2}\right)^{\frac{p-2}{2}}\langle\nabla^{\prime}\omega.\nabla^{\prime}\eta_{k}\rangle dS\\[11.38109pt] \phantom{------------}-(p-1)\beta_{S_{k}}(\beta_{S_{k}}-\beta_{0}){\displaystyle\int_{S_{k}}}\left(\beta_{S_{k}}^{2}\omega_{k}^{2}+\left|\nabla^{\prime}\omega_{k}\right|^{2}\right)^{\frac{p-2}{2}}\omega_{k}\eta_{k}dS.\end{array}$$

$$\underset{k\to \infty }{\lim }{\int }_{B_{2r}(a)}{\left({\beta }^2{\omega }^2+{\left|{\nabla }^{\prime }\omega \right|}^2\right)}^{\frac{p-2}{2}}⟨{\nabla }^{\prime }\omega .{\nabla }^{\prime }{\eta }_k⟩dS=0.$$

$$\underset{k\to \infty }{\lim }{\int }_{B_{2r}(a)}{\left({\beta }_{S_k}^2{\omega }_k^2+{\left|{\nabla }^{\prime }{\omega }_k\right|}^2\right)}^{\frac{p-2}{2}}{\omega }_k{\eta }_{k}dS=0,$$

$$\displaystyle\lim_{k\to\infty}{\displaystyle\int_{B_{2r}(a)}}(\omega-\omega_{k})\bigg{\langle}\left(\beta^{2}\omega^{2}+\left|\nabla^{\prime}\omega\right|^{2}\right)^{\frac{p-2}{2}}\nabla^{\prime}\omega-\left(\beta_{S_{k}}^{2}\omega_{k}^{2}+\left|\nabla^{\prime}\omega_{k}\right|^{2}\right)^{\frac{p-2}{2}}\nabla^{\prime}\omega_{k}.\nabla^{\prime}\zeta\bigg{\rangle}dS=0.$$

Full text

2010 Mathematics Subject Classification. 35J72; 35J92 .

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

1. 1 Introduction

2. 2 Existence

3. 2.1 Estimates

4. 2.2 Approximations from inside

5. 2.3 Approximations from outside

6. 3 Uniqueness

7. 3.1 Uniqueness of exponent $\beta$

8. 3.2 Uniqueness of eigenfunction

9. 3.2.1 The convex case

10. 3.2.2 Proof of Theorem E

1 Introduction

Let $p>1$, $S$ a domain of the unit sphere $S^{N-1}$ of $\mathbb{R}^{N}$ and $C_{S}:=\{(r,\sigma):r>0,\sigma\in S\}$ the positive cone generated by $S$. If one looks for $p$-harmonic functions in $C_{S}$ under the form $u(x)=u(r,\sigma)=r^{-\beta}\omega(\sigma)$ vanishing on $\partial C_{S}\setminus\{0\}$, then $\omega$ satisfies the spherical $p$-harmonic eigenvalue problem on $S$

[TABLE]

with $\beta_{0}=\frac{N-p}{p-1}$ and were $div^{\prime}$ and $\nabla^{\prime}$ denote the divergence operator and the covariant gradient on $S^{N-1}$ endowed with the metric induced by its isometric inbedding into $\mathbb{R}^{N}$. Separable solutions play a key role for describing the boundary behaviour and the singularities of solutions of a large variety of quasilinear equations. When $N=2$ the equation is completely integrable and has been solved by Kroll in the regular case $\beta<0$ and Kichenassamy and Véron in the the singular case $\beta>0$. In higher dimension, Tolksdorff [15] proved the following:

**Theorem A **If $S$ is a smooth spherical domain, there exist two couples $(\beta_{S},\omega_{S})$ and $(\beta^{\prime}_{S},\omega^{\prime}_{S})$ where $\beta_{S}>0$ and $\beta^{\prime}_{S}<0$, $\omega_{S}$ and $\omega^{\prime}_{S}$ are positive $C^{2}(\overline{S})$-functions vanishing on $\partial S$ which solve $(\ref{I-1})$ with $(\beta,\omega)=(\beta_{S},\omega_{S})$ or $(\beta,\omega)=(\beta^{\prime}_{S},\omega^{\prime}_{S})$. Furthermore $\beta_{S}$ and $\beta^{\prime}_{S}$ are unique, and $\omega_{S}$ and $\omega^{\prime}_{S}$ 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 $\{S_{k}\}$ such that $S_{k}\subset\overline{S}_{k}\subset S_{k+1}$ and $\cup_{k}S_{k}=S$ we prove the following:

**Theorem B **Assume that $S^{c}$ is not polar. Then the sequence of the $\beta_{S_{k}}>0$ from Theorem A is decreasing and converges to $\beta_{S}>0$. There exists $\omega_{S}\in W^{1,p}_{0}(S)\cap L^{\infty}(S)$ weak solution of $(\ref{I-1})$ with $\beta=\beta_{S}$. Furthermore $\beta_{S}>0$ is the largest exponent $\beta$ such that $(\ref{I-1})$ admits a positive solution $\omega_{S}\in W^{1,p}_{0}(S)$.

Under a mild assumption on $S$ it is possible to approximate it by a decreasing sequence of smooth domains $S^{\prime}_{k}$ such that $S^{\prime}_{k}\subset\overline{S}^{\prime}_{k}\subset S^{\prime}_{k-1}$ and $\cap_{k}S^{\prime}_{k}=\overline{S}$

**Theorem C **Assume that $S=\overset{o}{\overline{S}}$. Then the sequence $\beta_{S^{\prime}_{k}}>0$ is increasing and converges to $\hat{\beta}_{S}>0$ and there exists $\hat{\omega}_{S}\in W^{1,p}_{0}(S)\cap L^{\infty}(S)$ weak solution of $(\ref{I-1})$ with $\beta=\hat{\beta}_{S}$. Furthermore $\hat{\beta}_{S}$ is the smallest exponent $\beta$ such that $(\ref{I-1})$ admits a positive solution $\omega_{S}\in W^{1,p}_{0}(S)$.

We prove the uniqueness of the exponent $\beta$, under a Lipschitz assumption on $S$.

**Theorem D **Assume that $S$ is a Lipschitz domain, then $\beta_{S}=\hat{\beta}_{S}$ and if $\omega$ and $\omega^{\prime}$ are two positive solutions of $(\ref{I-1})$ in $W^{1,p}_{0}(S)$, there exists a constant $c>0$ such that $c^{-1}\omega^{\prime}\leq\omega\leq c\omega^{\prime}$.

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

[TABLE]

for some $c_{1}=c_{1}(N,p,S)>0$ and $\alpha\in(0,1)$. Actually we have a stronger result, much more delicate to obtain.

**Theorem E **Let $S$ be a Lipschitz subdomain of $S^{N-1}$. Then two positive solutions of $(\ref{I-1})$ in $W^{1,p}_{0}(S)$ are proportional.

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

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

2 Existence

2.1 Estimates

Through this article we assume that $S^{c}$ is not polar, or equivalently that it has positive $c^{S^{N-1}}_{1,p}$-capacity.

Lemma 2.1**.**

Assume $p>1$. Then any solution $\omega\in W^{1,p}_{0}(S)$ of $(\ref{I-1})$ satisfies

[TABLE]

if $p>N-1$ where $\gamma=1-\frac{N-1}{p}$ if $p>N-1$ and

[TABLE]

if $1<p\leq N-1$, where $c_{1}>0$ depends on $p$, $N$, $\beta$.

Proof.  Multiplying the equation by $\omega$ and using Hölder’s inequality, we derive

[TABLE]

Notice that these inequalities hold for all $p>1$. If $p>N-1$ $(\ref{II-0})$ follows by Morrey’inequality. Here after we assume $1<p\leq N-1$. Let $\alpha\geq 1$ and $k>0$. Then $\zeta=\min\{\left|\omega\right|,k\}^{\alpha-1}\omega$ is an admissible test function, hence

1- If $p\geq 2$,

[TABLE]

where $c_{2}=c_{2}(N,p,\beta)>0$. Since

[TABLE]

it implies that there exists $c_{4}=c_{4}(N,p,\beta)$ such that

[TABLE]

which yields

[TABLE]

where $j(\omega)=\min\{\left|\omega\right|,k\}^{\frac{\alpha-1}{p}}\omega$.

2- If $1<p<2$, then

[TABLE]

Since

[TABLE]

we derive

[TABLE]

which leads to $(\ref{II-5})$. Letting $k\to\infty$ we infer by Fatou’s lemma,

[TABLE]

If $p<N-1$ we derive from Sobolev inequality and putting $q=\alpha-1+p$ and $s=\frac{N-1}{N-1-p}>1$

[TABLE]

and $c_{5}>0$ depends on $N$, $p$ and $\beta$. Iterating this estimate by Moser’s method we derive $(\ref{II-9})$.

If $p=N-1$ we have for $1\leq m<p-1$ and $m^{*}=\frac{m(N-1)}{N-1-m}$

[TABLE]

and $c_{6}=c_{6}(N,p)$, hence

[TABLE]

with $t=\frac{m(N-1)}{p(N-1-m)}=\frac{m}{N-1-m}$. The proof follows again by Moser’s iterative scheme.          $\square$

Proposition 2.2**.**

Let $S_{1}$ and $S_{2}$ be two subdomains of $S^{N-1}$ such that $S_{1}\subset\overline{S}_{1}\subset S_{2}$ and $S_{2}$ not polar. Let $\beta_{j}>0$, j=1,2, such that there exist positive solutions $\omega_{j}\in W^{1,p}_{0}(S_{j})$ solutions of

[TABLE]

Then $\beta_{1}\geq\beta_{2}$.

Proof.  Set $u_{j}(r,\sigma)=r^{-\beta_{j}}\omega_{j}(\sigma)$, $C_{S_{j}}=(0,\infty)\times S_{J}$ and assume $\beta_{1}<\beta_{2}$. By Harnack inequality $\omega_{2}\geq c>0$ on $S_{1}$, thus

[TABLE]

For $\epsilon>0$ there exist $r_{\epsilon}>0$ such that

[TABLE]

Let $\delta>0$, there exists $R_{\delta}>0$ such that

[TABLE]

Hence $\zeta=(u_{1}-\epsilon u_{2}-\delta)_{+}\in W^{1,p}_{0}(Q^{r_{\epsilon},R_{\delta}}_{S_{1}})$, where $Q^{r_{\epsilon},R_{\delta}}_{S_{1}}=\{x\in C_{S_{1}}:r_{\epsilon}<\left|x\right|<R_{\delta}\}$. This implies

[TABLE]

Therefore $\nabla(u_{1}-\epsilon u_{2}-\delta)_{+}=0$ a.e. in $Q^{r_{\epsilon},R\delta}_{S_{1}}$, which leads to $u_{1}-\epsilon u_{2}\leq\delta$ in the same set. Letting $\delta\to 0$ yields $R_{\delta}\to\infty$, thus we obtain $u_{1}\leq\epsilon u_{2}$ in $C_{S_{1}}\cap\overline{B}^{c}_{r_{\epsilon}}$ hence $u_{1}\leq 0$ in $C_{S_{1}}$, contradiction.          $\square$

2.2 Approximations from inside

Proof of Theorem B. Let $\{S_{k}\}$ be an increasing sequence of smooth domains such that $S_{k}\subset\overline{S}_{k}\subset S_{k+1}$. We denote by $\{(\beta_{S_{k}},\omega_{k})\}$ the corresponding sequence of solutions of $(\ref{I-1})$ with $\beta=\beta_{S_{k}}$ and $\omega=\omega_{k}$. The sequence $\{\beta_{S_{k}}\}$ is uniquely determined by [15], it admits a limit $\beta:=\beta_{S}$, and the $\omega_{k}$ are the unique positive solutions such that

[TABLE]

If $p\geq 2$, we have

[TABLE]

Since $\beta_{S_{k}}\leq\beta_{1}$, we derive

[TABLE]

from the normalization assumption with $c_{8}=2c_{7}(N,p,\beta_{1})$.

If $1<p<2$, we have

[TABLE]

and we obtain $(\ref{III-1})$ with $c_{8}=\beta^{p-1}_{1}(p\beta_{1}+(p-1)\beta_{0})$.

Next we extend $\omega_{k}$ by [math] in $S_{k}^{c}$. Then there exists $\omega\in W^{1,p}_{0}(S)$ such that $\omega_{k}\rightharpoonup\omega$ weakly in $W^{1,p}_{0}(S)$, up to subsequence that we still denote $\{\omega_{k}\}$, and $\omega_{k}\to\omega$ in $L^{p}(S)$.

Step 1: We claim that $\nabla^{\prime}\omega_{k}$ converges to $\nabla^{\prime}\omega$ locally in $L^{p}(S)$.

Let $a\in S$ and $r>0$ such that $B_{4r}(a)\subset S$. Then for $k\geq k_{0}$, $\overline{B}_{2r}(a)\subset S_{k}$. Let $\zeta\in C_{0}^{\infty}(B_{2r}(a))$ such that $0\leq\zeta\leq 1$, $\zeta=1$ in $B_{r}(a)$. For test function we choose $\eta_{k}=\zeta(\omega-\omega_{k})$, then

[TABLE]

By the above inequality, we have

[TABLE]

Using the weak convergence of the gradient, we have

[TABLE]

Since $\omega_{k}$ is uniformly bounded in $W^{1,p}_{0}(S)$ and $\omega_{k}\to\omega$ in $L^{p}(S)$, we have

[TABLE]

and

[TABLE]

Combining the above relations we infer

[TABLE]

Next we write

[TABLE]

If $p\geq 2$, we have from $(\ref{II-3})$,

[TABLE]

Since $\omega_{k}\to\omega$ in $L^{p}(S)$, $\beta_{S_{k}}\to\beta$ and $\omega_{k},\omega$ are uniformly bounded in $W^{1,p}_{0}(S)$ , we derive

[TABLE]

as $k\to\infty$. Jointly with $(\ref{III-2})$ we infer that

[TABLE]

If $1<p<2$, then

[TABLE]

Up to extracting a subsequence, we have that $\omega_{k}\to\omega$ a.e. in $S$ and that there exists $\Phi\in L^{1}(S)$ such that

[TABLE]

Since

[TABLE]

and

[TABLE]

we derive that

[TABLE]

which implies that

[TABLE]

where $p^{\prime}$ is the conjugate of $p$, and finally

[TABLE]

For the last term on the right-hand side of $(\ref{III-6})$, we have, for $\gamma\in\mathbb{R}_{+}$ and ${\bf A},{\bf B}\in\mathbb{R}^{N}$,

[TABLE]

This implies

[TABLE]

We observe that

[TABLE]

and since $1<p<2$, we finally obtain

[TABLE]

We plug this estimate into $(\ref{III-6})$ with $\gamma=\beta^{2}_{k}\omega^{2}_{k}$, ${\bf A}=\nabla^{\prime}\omega$ and ${\bf B}=\nabla^{\prime}\omega_{k}$, then

[TABLE]

Set $\phi(.)=\beta^{2}_{k}\omega^{2}_{k}+1+\left|\nabla^{\prime}\omega_{k}\right|^{2}+\left|\nabla^{\prime}\omega\right|^{2}$, then

[TABLE]

Jointly with $(\ref{III-2})$ and $(\ref{III-10})$ we conclude that $(\ref{III-5})$. Step 1 follows by a standard covering argument.

Step 2: We claim that $\omega_{k}$ converges to $\omega$ in $W^{1,p}_{0}(S)$.

Up to a subsequence that we denote again by $\{k\}$, we can assume that $\omega_{k}\to\omega$ and $\nabla^{\prime}\omega_{k}\to\nabla^{\prime}\omega$ a.e. in $S$. Let $\zeta\in C^{\infty}_{0}(S)$, then there exists $k_{\epsilon}\in\mathbb{N}$ such that the support $K$ of $\zeta$ is a compact subset of $S_{k}$ for all $k\geq k_{\epsilon}$. If $1<p<2$,

[TABLE]

which bounded in $L^{p^{\prime}}(K)$, then uniformly integrable in $K$ and by Vitali’s convergence theorem

[TABLE]

in $L^{1}_{loc}(S)$. Similarly

[TABLE]

in $L^{1}_{loc}(S)$. If $p\geq 2$

[TABLE]

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

[TABLE]

we conclude that $\omega$ is a weak solution of $(\ref{I-1})$ with $\beta=\beta_{S}$.          $\square$

2.3 Approximations from outside

Proof of Theorem C. Since $\overline{S}^{c}$ has a non-empty interior, the existence of a sequence $\{\omega^{\prime}_{k}\}$ corresponding to solutions of $(\ref{I-1})$ in $S^{\prime}_{k}$ with $\beta=\beta_{S^{\prime}_{k}}$ is the consequence of [13]. The fact that $\{\beta_{S^{\prime}_{k}}\}$ is increasing follows from Proposition 2.2. We denote by $\hat{\beta}:=\hat{\beta}_{S}$ its limit, and it is smaller or equal to $\beta_{S}$. Estimates $(\ref{II-3})$ are valid with $S^{\prime}_{k}$, $\omega^{\prime}_{k}$ and $\beta_{S^{\prime}_{k}}$ instead of $S$, $\omega$ and $\beta$. If we extend $\omega^{\prime}_{k}$ by [math] in $S^{\prime c}_{k}$ these estimates are valid with $S^{N-1}$ instead of $S^{\prime}_{k}$. Then up to a subsequence the exists $\omega\in W^{1,p}(S^{N-1})$ and a subsequence stil denoted by $\{k\}$ such that $\omega^{\prime}_{k}\rightharpoonup\omega$ weakly in $W^{1,p}(S^{N-1})$, strongly in $L^{p}(S^{N-1})$ and a.e. in $S^{N-1}$. Furthermore, as in the proof of Theorem A, for any compact set $K\subset S$, $\nabla^{\prime}\omega^{\prime}_{k}\to\nabla^{\prime}\omega^{\prime}$ in $L^{p}(K)$. This is sufficient to assert that $\omega$ is a weak solution of

[TABLE]

Moreover $\omega^{\prime}\lfloor_{S^{\prime}_{k}}$ belongs to $W_{0}^{1,p}(S^{\prime}_{k})$ for all $k$. Since $\omega^{\prime}_{k}=0$ in $S_{k}^{c}$ and converges a.e. to $\omega$, this last function vanishes a.e. in $\cup_{k}S_{k}^{c}=(\cap_{k}S_{k})^{c}=\overline{S}^{c}$. Therefore $\omega$ vanishes a.e. in $\overline{S}^{c}$ and since it is quasi continuous, it vanishes, $(1-p)$- quasi everywhere in $\overline{S}^{c}$. From Netrusov’s theorem (see [1, Th 10.1.1]-(iii)) there exists a sequence $\{\eta_{n}\}\subset C^{\infty}_{0}(S)$ which converges to $\omega$ in $W^{1,p}(S)$, thus $\omega\in W_{0}^{1,p}(S)$.          $\square$

3 Uniqueness

3.1 Uniqueness of exponent $\beta$

Proof of Theorem D. If $S$ is Lipschitz, $C_{S}$ is also Lipschitz. We fix $z\in S\approx S^{N-1}\cap\partial C_{S}$ and we apply [10, Th 2] in $G_{z}=C_{S}\cap B_{\frac{1}{2}}(z)$ to two separable $p$-harmonic functions $u(r,\sigma)=r^{-\beta}\omega(\sigma)$ and $u^{\prime}(r,\sigma)=r^{-\beta^{\prime}}\omega^{\prime}(\sigma)$. There exist $\gamma\in(0,\frac{1}{2})$, $c_{10}>0$ and $\alpha\in(0,1)$ such that

[TABLE]

Assume $|y_{1}|=|y_{2}|=1$, then

[TABLE]

We denote by $\ell(x,y)$ the geodesic distance on $S^{N-1}$ and by $\ell(x,K)$ the geodesic distance from a point $x\in S^{N-1}$ to a subset $K$. Since the set $S_{\gamma}=\{\sigma\in S:\ell(\sigma,\partial S)\leq\tfrac{\gamma}{2}\}$ can be covered by a finite number of balls with center on $\partial S$, we infer that

[TABLE]

In $S\setminus\overline{S}_{\tfrac{\gamma}{2}}$ we can use Harnack inequality to obtain

[TABLE]

Hence there exists a constant $c_{13}>0$ such that $(\ref{IV-4})$ holds for any $y_{1},y_{2}\in S\setminus\overline{S}_{\tfrac{\gamma}{2}}$, with $c_{12}$ replaced by $c_{13}$. Furthermore $\omega^{\prime}$ satisfies the same inequality in $S\setminus\overline{S}_{\tfrac{\gamma}{2}}$. Combining the two inequalities we obtain

[TABLE]

Combining this estimate with $(\ref{IV-2})$ we derive that it holds for all $y_{1},y_{2}\in S$. This implies

[TABLE]

Assume now that there exist two exponents $\beta>\beta^{\prime}>0$ such that $r^{-\beta}\omega(.)$ and $r^{-\beta^{\prime}}\omega^{\prime}(.)$ are $p$-harmonic and positive in the cone $C_{S}$ and vanishes on $\partial C_{S}$. Put $\theta=\frac{\beta}{\beta^{\prime}}$, $\eta=\omega^{\prime\theta}$ and

[TABLE]

then

[TABLE]

Up to multiplying $\omega^{\prime}$ by $\lambda$, we can assume that $\eta\leq\omega$ and that the graphs of $\eta$ and $\omega$ are tangent in $\overline{S}$. Since $\omega^{\prime}\leq c\omega$, $\eta=o(\omega)$ near $\partial S$. Hence there exists $\sigma_{0}\in S$ such that $\omega(\sigma_{0})=\eta(\sigma_{0})$ and the coincidence set of $\eta$ and $\omega$ is a compact subset of $S$. We put $w=\omega-\eta$, since $\nabla\omega(\sigma_{0})=\nabla\eta(\sigma_{0})$ we proceed as in [14, Th 4.1] (see also [4] in the flat case) and derive that $w$ satisfies, in a system of local coordinates $(\sigma_{1},...,\sigma_{N-1})$ near $\sigma_{0}$,

[TABLE]

where the matrix $(A_{j,\ell})$ is smooth, symmetric and positive near $\sigma_{0}$ and the $C_{j}$ and $C$ are bounded. Hence $w$ is locally zero. By a standard argument of connectedness, this implies that the zero set of $w$ must be empty, contradiction. Hence $\beta=\beta^{\prime}$.          $\square$

3.2 Uniqueness of eigenfunction

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

3.2.1 The convex case

Theorem 3.1**.**

Assume $S$ is a convex spherical subdomain. Then two positive solutions of $(\ref{I-1})$ are proportional.

Proof.  We recall that a domain $S$ is (geodesically) convex if a minimal geodesic joining two points of $S$ is contained in $S$. If $S\subset S^{N-1}$ is convex, the cone $C_{S}$ is convex too. Since $S$ is convex, it is Lipschitz and by Theorem D, $\beta_{S}=\hat{\beta}_{S}:=\beta$. Let $\omega$ and $\omega^{\prime}$ be two positive solutions of $(\ref{I-1})$ satisfying $\sup_{S}\omega=\sup_{S}\omega^{\prime}=1$. We denote by $u_{\omega}(x)=|x|^{-\beta}\omega(.)$ and $u_{\omega^{\prime}}(x)=|x|^{-\beta}\omega^{\prime}(.)$ the corresponding separable $p$-harmonic functions defined in $C_{S}$. If $0<a<b$, we set $C_{S}^{a,b}=C_{S}\cap(B_{b}\setminus\overline{B}_{a})$. Then for $0<\epsilon<1$ we denote by $u_{\epsilon}$ the unique function which satisfies

[TABLE]

Then

[TABLE]

Furthermore $\epsilon\mapsto u_{\epsilon}$ is increasing. When $\epsilon\downarrow 0$, $u_{\epsilon}\uparrow u_{0}$ where $u_{0}$ is positive and $p$-harmonic in $C_{S}^{1,0}$, vanishes on $\partial C_{S}^{1,0}\setminus\{0\}$ and satisfies $(\ref{V-1c})$ with $\epsilon=0$. In particular

[TABLE]

We construct the same approximation $u^{\prime}_{\epsilon}$ in $C_{S}^{\epsilon,1}$ with $\omega^{\prime}$ instead of $\omega$. Mutadis mutandis $(\ref{V-2c})$ holds and $u^{\prime}_{\epsilon}\uparrow u^{\prime}_{0}$ which is positive and $p$-harmonic in $C_{S}^{1}$, satisfies

[TABLE]

and thus

[TABLE]

However, by [10, Th 4] $u_{0}$ and $u^{\prime}_{0}$ are proportional. Combined with $(\ref{V-3c})$, $(\ref{V-4c})$ 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 next nondegeneracy property of positive $p$-harmonic functions is proved in [10, Lemma 4.28].

Proposition 3.2**.**

Let $\Omega\subset\mathbb{R}^{N}$ be a bounded Lipschitz domain and $1<p<\infty$. Then there exist constants $\rho>0$, $c_{14},c_{15}>0$ depending respectively on $\Omega$ (for $\rho$), and $p$, $N$ and the Lipschitz norm $M$ of $\partial\Omega$ (for $c_{14}$ and $c_{15}$) with the property that for any $w\in\partial\Omega$ and any positive $p$-harmonic function $u$ in $\Omega$, continuous in $\overline{\Omega}\cap\overline{B}_{2\rho}(w)$ and vanishing on $\partial\Omega\cap B_{\rho}(w)$, one can find $\xi\in S^{N-1}$, independent of $u$, such that

[TABLE]

for all $y\in C_{S}\cap\overline{B}_{\frac{\rho\left|w\right|}{c_{15}}}(w)$.

If $\Omega$ is replaced by a cone $C_{S}$, the nondegeneracy property still holds uniformly on $\partial C_{S}\setminus\{0\}$.

Corollary 3.3**.**

Let $1<p<\infty$, $S\subset S^{N-1}$ is a Lipschitz domain and $C_{S}$ the cone generated by $S$.

(i) Then there exist constants $\rho<\frac{1}{2}$, $c_{14},c_{15}>0$ depending respectively on $S$ (for $\rho$), and $p$, $N$ and the Lipschitz norm $M$ of $\partial S$ and $diam(S)$ (for $c_{14}$ and $c_{15}$) with the property that for any $w\in\partial C_{S}$ and any positive $p$-harmonic function $u$ in $C_{S}$, continuous in $\overline{C}_{S}\cap\overline{B}_{2\rho\left|w\right|}(w)$ and vanishing on $\partial C_{S}\cap\overline{B}_{\rho\left|w\right|}(w)$ continuous, one can find $\xi\in S^{N-1}$, independent of $u$, such that

[TABLE]

for all $y\in B_{\frac{\rho}{c_{15}}}(w)\cap C_{S}$.

(ii) Then there exist positive constants $\kappa$ and $c_{16},c_{17}$ depending on $S$ (for $\kappa$), and $p$, $N$ and the Lipschitz norm $M$ of $\partial S$ and $diam(S)$ (for $c_{16},c_{17}$ such that for any $a>0$ and any positive $p$-harmonic function $u$ in $C^{a}_{S}$ vanishing on $\partial C_{S}\cap B_{a}^{c}$, there holds

[TABLE]

Let $\omega,\omega^{\prime}\in W^{1,p}_{0}(S)\cap C(\overline{S})$ be positive solutions $(\ref{I-1})$. Since $\frac{\omega}{\omega^{\prime}}$ is bounded from above and from below in $S$ by positive constants, we can assume, as in the proof of Theorem D, that $\omega\geq\omega^{\prime}$ in $S$ and that the graphs of $\omega$ and $\omega^{\prime}$ are tangent. hence, if $\omega\neq\omega^{\prime}$, then $\omega>\omega^{\prime}$ in $S$ and there exists a sequence $\{\sigma_{n}\}$ converging to $\sigma_{0}\in\partial S$ as $n\to\infty$ such that

[TABLE]

We define $\delta_{1}=\sup\{\delta>0:\delta\omega<\omega^{\prime}\}$. For $t\in(\delta_{1},1)$, we set

[TABLE]

We also set

[TABLE]

Lemma 3.4**.**

The functions $\phi_{t}$ and $\psi_{t}$ are respectively a subsolution and a supersolution of $(\ref{I-1})$ in $W^{1,p}_{0}(S)$, $v_{\phi_{t}}$ and $v_{\psi_{t}}$ are respectively a subsolution and a supersolution of $-\Delta_{p}$ in $C_{S}$, and there exists $\eta\in W^{1,p}_{0}(S)$ solution of $(\ref{I-1})$ such that

[TABLE]

If $S_{t}$ is the subset of $\eta\in W^{1,p}_{0}(S)$ solutions of $(\ref{I-1})$ and satisfying ($\ref{V-2-3})$, then $\omega_{t}=\sup\{\eta:\eta\in S_{t}\}$ belongs to $S_{t}$. It is increasing with respect to $t$ with uniform limits $\omega^{\prime}$ when $t\downarrow\delta_{1}$ and $\omega$ when $t\uparrow 1$. Finally, if $\theta_{t}=\frac{t-\delta_{1}}{1-\delta_{1}}$, there holds

[TABLE]

Proof.  Clearly $\phi_{t}$ and $\psi_{t}$ are respectively a subsolution and a supersolution of the operator ${\mathcal{T}}$, they belong to $W^{1,p}_{0}(S)\cap L^{\infty}(S)$ and they satisfy $\omega^{\prime}\leq\phi_{t}\leq\psi_{t}\leq\omega$. Furthermore, by Dini convergence theorem

[TABLE]

uniformly in $\overline{S}$. Moreover, in spherical coordinates,

[TABLE]

Hence, if $u(r,\sigma)=r^{-\beta}\eta(\sigma)$,

[TABLE]

Thus $v_{\phi_{t}}$ is a subsolution $-\Delta_{p}$ in $C_{S}$ and $v_{\psi_{t}}$ is a supersolution. Since the operator ${\mathcal{T}}$ is a Leray-Lions operator, it follows by [3] that there exists $\eta\in W^{1,p}_{0}(S)\cap L^{\infty}(S)$ satisfying ${\mathcal{T}}(\eta)=0$ and $\phi_{t}\leq\eta\leq\psi_{t}$ in $S$. We denote by $S_{t}$ the set of $\eta\in W^{1,p}_{0}(S)\cap L^{\infty}(S)$ satisfying ${\mathcal{T}}(\eta)=0$ and $\phi_{t}\leq\eta\leq\psi_{t}$ in $S$. Then there exists a sequence $\{\eta_{n}\}\subset S_{t}$ and $\omega_{t}\in W^{1,p}_{0}(S)\cap L^{\infty}(S)$ such that $\eta_{n}(\sigma)\uparrow\omega_{t}(\sigma)$ for all $\sigma\in\Sigma$, where $\Sigma$ is a countable dense subset of $S$. By Lemma 2.1 $\{\eta_{n}\}$ is bounded in $L^{p}(S)$, hence in $C^{\gamma}(S)$ for some $\gamma\in(0,1)$. By the estimates of the proof of Theorem B-Step 2, $\{\eta_{n}\}$ is bounded in $W^{1,p}_{0}(S)$. By standard regularity theory, we can also assume that $\eta_{n}\to\omega_{t}$ in the $C^{1}_{loc}(S)$-topology. Hence $\omega_{t}$ is a weak solution of $(\ref{I-1})$, it belongs to $W^{1,p}_{0}(S)\cap L^{\infty}(S)$ and satisfies $\phi_{t}\leq\omega_{t}\leq\psi_{t}$. Therefore it is the maximal element of $S_{t}$. The monotonity of $\omega_{t}$ is a consequence of the monotonicity of $\phi_{t}$ and $\psi_{t}$ and the last statement $(\ref{V-2-4})$ is a straightforward computation.           $\square$

Next we recall the deformation of $p$-harmonic functions already used in [10]. If $\tau\in(0,1)$ and $0<a<b$, we denote by $v_{\tau,a,b}$ the $p$-harmonic function defined in $C_{S}^{a,b}$ satisfying

[TABLE]

Lemma 3.5**.**

The mapping $(\tau,b)\mapsto v_{\tau,a,b}$ is continuous and increasing. If $v_{\tau,a}=\displaystyle\lim_{b\to\infty}v_{\tau,a,b}$, then it is a positive p-harmonic function in $C_{S}^{a,\infty}$ vanishing on $\partial S\cap B_{a}^{c}$, and there holds

[TABLE]

where $\tau^{*}=(1-\delta_{1})\tau+\delta_{1}$ and as a consequence

[TABLE]

Furthermore

[TABLE]

Proof.  The uniqueness and the (strict) monotonicity of $(\tau,b)\mapsto v_{\tau,a,b}$ follow from the monotonicity of $\tau\mapsto\tau\omega+(1-\tau)\omega^{\prime}$ and the strong maximum principle. The continuity is a consequence of uniqueness and regularity theory for $p$-harmonic functions. It follows from $(\ref{V-2-4})$ with $t=\tau^{*}$ and the fact that $v_{\phi_{\tau^{*}}}$ and $v_{\psi_{\tau^{*}}}$ are respectively a subsolution and a supersolution of $-\Delta_{p}$, that we have

[TABLE]

which yields $(\ref{V-2-7})$. Similarly, we have on $\partial C_{S}^{a,b}$

[TABLE]

equivalently

[TABLE]

By the maximum principle $(\ref{V-2-9})$ holds in $C_{S}^{a,b}$. This implies $(\ref{V-2-8})$.          $\square$

As a consequence of $(\ref{V-2-8})$, $\partial_{\tau}v_{\tau,a}$ exists for almost all $\tau\in(0,1)$ in $W^{1,p}_{0}(C_{S}^{a,b})$ for all $b>a$ and it is a solution of

[TABLE]

where

[TABLE]

$\mathbb{L}$ satisfies the following ellipticity condition

[TABLE]

It is important to notice that $\mathbb{L}v_{\tau,a}=(p-1)\Delta_{p}v_{\tau,a}=0$. The estimate $(\ref{V-4})$ combined with $(\ref{V-2-b})$ and the decay of $v_{\tau,a}$ and $\partial_{\tau}v_{\tau,a}$ implies that they satisfy Harnack inequality and boundary Harnack inequality in $C_{S}^{a}$. There exists a constant $\hat{c}>c_{17}>1$ (see 3.36) such that

[TABLE]

where $x_{a}=(\hat{c}a,\sigma_{0})$ for some $\sigma_{0}\in S$ fixed. We set

[TABLE]

Lemma 3.6**.**

For $t>\hat{c}a$ there holds

[TABLE]

Proof.  There holds

[TABLE]

Estimate $(\ref{V-5})$ is valid for any couple of positive solutions $(h_{1},h_{2})$ of $\mathbb{L}h=0$ in $C_{S}^{a}$ vanishing on $\partial C_{S}^{a}\cap B_{a}^{c}$, in particular for $\left(\partial_{\tau}v_{\tau,a}-m(t)v_{\tau,a},v_{\tau,a}\right)$ and $\left(M(t)v_{\tau,a}-\partial_{\tau}v_{\tau,a},v_{\tau,a}\right)$. Hence

[TABLE]

This implies

[TABLE]

and

[TABLE]

Finally

[TABLE]

Similarly

[TABLE]

Summing the two inequalities we get

[TABLE]

which yields $(\ref{V-7})$.          $\square$

End of the proof. By the differentiability property of $v_{\tau,a}$ with respect to $\tau$, there exists two countable dense sets $\{(r_{\nu}\}\subset[a,\infty)$ and $\{\sigma_{\mu}\}\subset[a,\infty)$ such that $\partial_{\tau}v_{\tau,a}(r_{\nu},\sigma_{\mu})$ exists for almost all $\tau$. We put $x_{\nu,\mu}=(r_{\nu},\sigma_{\mu})$, hence

[TABLE]

Using the continuity of $\frac{\omega}{\omega^{\prime}}$ and the density of $\{\sigma_{m}\}$ we derive

[TABLE]

We can assume that $r_{\nu}\geq\hat{c}^{\nu_{n}}a$ for some sequence $\{\nu_{n}\}$ tending to infinity with $n$, hence

[TABLE]

where $\theta=\frac{\hat{c}^{2}-1}{\hat{c}^{2}+1}<1$. Letting $n\to\infty$ implies the claim.          $\square$