\documentclass[twoside]{article} \usepackage{amsfonts, amsmath} % used for R in Real numbers \pagestyle{myheadings} \markboth{Singularities and quasilinear equations on manifolds} {Laurent V\'eron} \begin{document} \setcounter{page}{133} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent 2001-Luminy conference on Quasilinear Elliptic and Parabolic Equations and Systems,\newline Electronic Journal of Differential Equations, Conference 08, 2002, pp 133--154. \newline http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Singular $p$-harmonic functions and related quasilinear equations on manifolds % \thanks{ {\em Mathematics Subject Classifications:} 35J50, 35J60. \hfil\break\indent {\em Key words:} $p$-harmonic, singularity, degenerate equations. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Published October 21, 2002.} } \date{} \author{Laurent V\'eron} \maketitle \begin{abstract} We give here an overview of some recent developments in the study of the description of singular solutions of $$ -\nabla.(|\nabla u|^{p-2}\nabla u) +\varepsilon |u|^{q-1}u=0 %\label{NLE} $$ in $\mathbb{R}^N\setminus \{0\}$, where $p>1$, $\varepsilon \in \{0,1,-1\}$ and $q\geq p-1$. \end{abstract} \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newcommand{\abs}[1]{|#1|} \newcommand{\norm}[1]{\|#1\|} \section {Introduction} Let $\Omega$ be a domain in $\mathbb{R}^N$ containing $0$, $N\geq 2$, and let $$ A: \Omega\times \mathbb{R}\times \mathbb{R}^N\mapsto\mathbb{R}^N, \quad \mbox{and}\quad B:\Omega\times \mathbb{R}\times \mathbb{R}^N\mapsto\mathbb{R}, $$ be two Caratheodory functions. Then a classical problem is the study of the behaviour near $0$ of a solution $u$ of \begin{equation}\label{main equ} -\nabla.A(x,u,\nabla u)+B(x,u,\nabla u)=0 \end{equation} in $\Omega^*=\Omega\setminus\{0\}$. Besides the well known linear case, the first striking results in the nonlinear case were obtained by Serrin in 1964 in a series of celebrated articles \cite {Se1,Se2}. Under the assumptions \begin{eqnarray}\label{p-growth} &(i)& A(x,r,Q).Q \geq c_{1}{\abs Q}^{p}\nonumber\\ &(ii)& |A(x,r,Q)| \leq c_{2}{\abs Q}^{p-1}+c_{3}\\ &(iii)& \abs {B(x,r,Q)}\leq c_{4}{\abs Q}^{p-1}+c_{5}{\abs r}^{p-1}+c_{6} \nonumber \end{eqnarray} for any $(x,r,Q)\in \Omega\times \mathbb{R}\times \mathbb{R}^N\mapsto\mathbb{R}^N$, where the $c_{i}$ are positive constants and $N\geq p>1$. Serrin's results assert that any nonnegative weak solution $u$ of (\ref{main equ}) in $\Omega^*$ belonging to $W^{1,p}_{\rm loc}(\Omega^*)$ is either extendable by continuity as a $C(\Omega)\cap W^{1,p}_{\rm loc}(\Omega)$-solution of the same equation in whole $\Omega$, or satisfies \begin{equation}\label{singular} \theta \leq \frac {u(x)}{\mu_{p}(x)}\leq \theta^{-1}, \end{equation} near $0$, for some positive $\theta$, in which formula the functions $\mu_{p}$ are defined in $\mathbb{R}^N\setminus\{0\}$ by \begin{equation}\label{p-growth 2} \mu_{p}(x)=\begin{cases} {\abs x}^{(p-N)/(p-1)} &\mbox{if }1p-1$; Guedda and V\'eron \cite{GV}, Bidaut-V\'eron \cite{BV}, Serrin and Zou \cite{SZ} in the case $B(x,r,Q)=-{\abs r}^{q-1}r$, always in assuming $q>p-1$. We shall present below an overview or the results of these different authors, writing the equation (\ref{main equ}) in the form \begin{equation}\label{main equ epsilon} -\nabla. ({\abs{\nabla u}^{p-2}}\nabla u)+\varepsilon{\abs u}^{q-1}u=0, \end{equation} with $\varepsilon=1,-1$ or $0$. We put emphasis on separable solutions that are solutions of the form $$u(r,\sigma)=r^{-\beta}\omega (\sigma),\quad (r,\sigma)\in (0,\infty)\times S^{N-1}. $$ Thus $\beta=\beta_{q}=p/(q+1-p)$ and the relation \begin{eqnarray*} \lefteqn{-\nabla_{\sigma}.\left((\omega^{2}+\abs{\nabla_{\sigma} \omega}^{2})^{p/2-1}\nabla_{\sigma}\omega\right)+ \varepsilon {\abs \omega}^{q-1}\omega }\\ &=&\beta_{q}((\beta_{q}+1)(p-1)+1-N) (\omega^{2}+\abs{\nabla_{\sigma} \omega}^{2})^{p/2-1}\omega, \end{eqnarray*} holds on $S^{N-1}$. This equation is not the usual Euler equation of a functional, which makes it more difficult study. However, we give a few results of existence and uniqueness of solutions. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section {Singular $p$-harmonic functions} By looking for radial solutions of the $p$-Laplace equation \begin{equation}\label{singular $p$-Laplace} -\nabla. ({\abs{\nabla u}^{p-2}}\nabla u)=0, \end{equation} in $\mathbb{R}^N\setminus\{(0)\}$, we find that the only solutions are the functions $$u=C_{1}\mu_{p}+C_{2} $$ where the $C_{i}$ are arbitrary constants.The first result obtained by Kichenassamy and V\'eron in \cite{KV} pointed out that any nonnegative singular p-hamonic functions is asymptotically radial near its singularities. They proved the following result. \begin{theorem}\label{singular p-hamonic} Assume $10$ and $R>0$ (this follows from Serrin's result), the scaling transformation $$T_{r}(u)(\xi)=u(r\xi)/\mu (r) $$ and a version of the strong maximum principle which was first noticed by Tolksdorff \cite {To}. Actually, the positivity assumption can be relaxed and replaced by \begin{equation}\label{u/mu}u/\mu_{p}\in L^{\infty}(B_{R}), \end{equation} since Serrin's result asserts that any nonnegative singular $p$-harmonic function does satisfy this estimate. As a consequence, existence and uniqueness of a solution to the singular Dirichlet problem \begin{equation}\label{singular Dirichlet} \begin{gathered} -\nabla. ({\abs{\nabla u}^{p-2}}\nabla u) =c_{N,p}{\abs\gamma^{p-2}}\gamma\delta_{0},\quad\mbox{ in }{\cal D}'(\Omega),\\ u=g,\quad\mbox{ on }\partial\Omega,\end{gathered} \end{equation} can be proved. \begin{corollary} \label{cor1} Assume $11$, then for each positive integer $k$ there exist a $\beta_{k}$ and $\omega_{k}:\mathbb{R}\mapsto\mathbb{R}$ with least period $2\pi/k$, of class $C^\infty$ such that \begin{equation}\label{anisotropic singular 2} u(x)={\abs x}^{-\beta_{k}}\omega_{k} (x/{\abs x}), \end{equation} is $p$-harmonic in $\mathbb{R}^2\setminus\{0\}$; $\beta_{k}$ is the positive root of \begin{equation}\label{algebraic 2} (\beta+1)^2=(1+1/k)^2\left(\beta^2+\beta(p-2)/(p-1)\right). \end{equation} The couple $(\beta_{k},\omega_{k})$ is unique, up to translation and homothety over $\omega_{k}$. \end{theorem} In the case of regular $p$-harmonic functions in the plane, which means that the exponent $\beta=-\tilde\beta$ in (\ref{anisotropic singular}) is negative, the stationary equation becomes \begin{equation}\label{anisotropic equation 3} \big((\tilde\beta^2\tilde\omega^2+\tilde\omega_{\varphi}^2)^{(p-2)/2} \tilde\omega_{\varphi}\big)_{\varphi} +((\tilde\beta-1)(p-1)-1) \tilde\beta(\tilde\beta^2\tilde\omega^2+\tilde\omega_{\varphi}^2)^{(p-2)/2} \tilde\omega=0. \end{equation} Kroll and Mazja \cite {KM} obtained the complete set of solutions of (\ref{anisotropic equation 3}): \begin{theorem}\label{splitted 2} For each positive integer $k$ there exists a couple $(\tilde\beta_{k},\tilde\omega_{k})$, unique up to translation and homothety over $\tilde\omega_{k}$ such that \begin{equation}\label{anisotropic singular 3} x\mapsto u(x)={\abs x}^{\tilde\beta_{k}}\tilde\omega_{k} (x/{\abs x}), \end{equation} is $p$-harmonic in $\mathbb{R}^2$. The exponent $\tilde\beta_{k}$ is the root larger than $1$ of the algebraic equation \begin{equation}\label{algebraic 3} (\tilde\beta-1)^2=(1-1/k)^2\left(\tilde\beta^2-\tilde\beta(p-2)/(p-1)\right). \end{equation} \end{theorem} The derivation of regular or singular $p$-harmonic functions follows in higher dimension under a splitted form. For example, if $N=3$ with $(x_{1},x_{2},x_{3})$ the canonical coordinates in $\mathbb{R}^3$, we put $$ x_{1}=r\cos\varphi\sin\theta,\quad x_{2}=r\sin\varphi\sin\theta,\quad x_{3}=r\cos\theta, $$ where $r>0$, $\varphi\in [0,2\pi]$, $\theta\in [0,\pi]$. Equation (\ref{anisotropic equation}) takes the form \begin{equation}\label{anisotropic equation N=3}\begin{aligned} -\frac{\partial}{\partial\theta} &\left(\sin\theta\left(\beta^2\omega^2+\omega^2_{\theta}+ \sin^{-2}\theta\,\omega^2_{\varphi}\right)^{(p-2)/2}\omega_{\theta}\right)\\ -\frac{\partial}{\partial\varphi} &\left(\sin^{-1}\theta\left(\beta^2\omega^2+\omega^2_{\theta}+ \sin^{-2}\theta\,\omega^2_{\varphi}\right)^{(p-2)/2}\omega_{\varphi}\right)\\ =&\beta(\beta(p-1)+p-3)\sin\theta\left(\beta^2\omega^2+\omega^2_{\theta}+ \sin^{-2}\theta\,\omega^2_{\varphi}\right)^{(p-2)/2}\omega. \end{aligned}\end{equation} We set $$\omega (\varphi,\theta)=\sin^{-\beta}\theta \;v(\varphi)=\sin^{\tilde\beta}\theta \;v(\varphi), $$ then $v$ satisfies (\ref{anisotropic equation 3}). Thanks to Theorem \ref{splitted 2} the set of singular (resp. regular) $p$-harmonic functions under the form $$u(r,\varphi,\theta)=r^{-\beta}\sin^{-\beta}\theta \;v(\varphi), $$ resp. $$u(r,\varphi,\theta)=r^{\tilde\beta}\sin^{\tilde\beta}\theta \;v(\varphi), $$ is explicitly known. Another way for constructing non-isotropic singular $p$-harmonic functions is to use Tolksdorf's shooting method \cite {To}. \begin{theorem} \label{N=3}Let $S\subset S^{N-1}$ be a connected and open, with a $C^2$ relative boundary $\partial S$. Then there exist a unique couple $(\beta,\omega)$, with $\beta>0$, $\omega\in C^1(S)$, $\omega>0$ in $S$, vanishing on $\partial S$, with maximal value $1$ such that the function $u$ defined by (\ref{anisotropic singular}) is $p$-harmonic in $\mathbb{R}^N\setminus \{0\}$. \end{theorem} \paragraph{Proof} Put $K_{S}(R,R')=\{(r,\sigma):\;\sigma\in S,\;R0$ and any $R\in (1,2)$. Thus there exists $k>0$ such that $C(R)\leq kC(2R)$ for any $R\geq 3$. Then $$\abs{\nabla u(x)}\leq C(\abs x){\abs x}^{-1},\quad\mbox{and}\quad \abs{\nabla u(x)-\nabla u(x') }\leq C(\abs x){\abs x}^{-1-\alpha}{\abs {x-x'}}^{\alpha}, $$ for some $C>0$ and $1\leq \abs x\leq \abs {x'}$. Putting $$u_{R}(x)=u(Rx)/C(R), $$ it follows that for any compact subset $K$ of $\overline{K_{S}(0,\infty)}\setminus\{0\}$ there exists $C(K)>0$ such that $$\norm {u_{R}}_{C^{1,\alpha}(K)}\leq C(K). $$ Thus there exist a sequence $R_{n}\to\infty$ and a $p$-harmonic function $u^*$ in $K_{S}(0,\infty)$ such that $u_{R_{n}}\to u^*$ in the $C^1_{\rm loc}$ topology of $\overline {K_{S}(0,\infty)}\setminus \{0\}$. Moreover $u^*>0$, and $\nabla {u^*}\neq 0$ because of (\ref{equiv 1}). In order to prove that there exists $\beta>0$ such that \begin{equation}\label{equiv 2} u^*(r,\sigma)={r}^{-\beta}u^*(1,\sigma), \end{equation} we define $$\Sigma_{R}=\sup\left\{C>0: Cu^*(x)\leq u^*(Rx),\;\forall x\in\overline {K_{S}(0,\infty)}\setminus \{0\}\right\}. $$ Note that $\Sigma_{R}$ exists because of (\ref{equiv 1}). If we assume now that the equality \begin{equation}\label{equiv 3} \Sigma_{R}u^*(x)=u^*(Rx), \end{equation} does not hold in $\overline {K_{S}(0,\infty)}$, then \begin{equation}\label{equiv 4} \Sigma_{R}u^*(x)0$. Clearly $R\mapsto \Sigma_{R}$ is $C^1$ (as $u ^*$) and decreases. For $k\in \mathbb{N}_{*}$ there holds $$\Sigma_{R^k}u^*(x)=u^*(R^k x)=(\Sigma_{R})^ku^*(x). $$ Then $\Sigma_{R^k}=(\Sigma_{R})^k$. Consequently, for any $m\in\mathbb{N}_{*}$, $\Sigma_{R^{k/m}}=(\Sigma_{R})^{k/m}$, and finally $$\Sigma_{R^\alpha}=(\Sigma_{R})^\alpha, $$ for any positive $\alpha$. A straightforward consequence is that (\ref{equiv 6}) holds for some $\beta>0$. If we set \begin{equation}\label{equiv 7}\omega (\sigma)=u^*(1,\sigma), \end{equation} then $\omega$ satisfies (\ref{anisotropic equation}) in $S$, where it is positive, and vanishes on $\partial S$. Uniqueness of the couple $(\beta,\omega)$ with $\sup_{S} \omega=1$ follows from the equivalence principle. \paragraph{Remark} %\rem {\bf 1 } Although the extension is far from being obvious, the regularity requirement on the domain $S$ can be relaxed. It is possible to replace it by the assumption that $\partial S$ is piecewise smooth. In dimension 3, Hopf lemma at a corner is replaced by an expansion in terms of conical functions as in Theorem \ref{N=3}. In higher dimension the proof goes by induction. However, uniqueness of the couple $(\beta,\omega)$ is not clear. From this observation, we can construct $p$-harmonic functions in $\mathbb{R}^N\setminus \{0\}$ under the form (\ref{anisotropic singular}) with a finite symmetry group $G$ generated by reflections through hyperplanes. Taking $S$ to be a fundamental simplicial domain of $G$, we construct $(\beta,\omega)$ in $S$ and then extend $\omega$ to the whole sphere by reflections through the edges.\smallskip It is natural to imbed this problem in a more general setting, by replacing $(S^{N-1},g_{0})$ by a compact and complete $d$-dimensional Riemannian manifold $(M,g)$. Let $\nabla_{g}.$ and $\nabla_{g}$ be respectively the divergence operator acting on vector fields on $M$ and the gradient operator. For $\beta\in \mathbb{R}$ consider the equation \begin{multline} \label{quasi spectrum} -\nabla_{g}.\left((\beta^{2}\psi^{2} +\abs{\nabla_{g}{ \psi}}^{2})^{(p-2)/2}\nabla_{g}{ \psi}\right)\\ =\beta((\beta+1)(p-1)-d) (\beta^{2}\psi^{2}+\abs{\nabla_{g}{\psi}}^{2}) ^{(p-2)/2}\psi. \end{multline} \paragraph{Definition} We denote by $\mathfrak S_{p}(M)$ the set of couples $(\beta,\psi)\in \mathbb{R}\times C^{1}(M)$ satisfying (\ref{quasi spectrum}) and call it the {\it p-quasi-spectrum} of $M$. \begin{theorem} \label{structure} If $(\beta,\psi)\in \mathfrak S_{p}(M)$, then either $\beta((\beta+1)(p-1)-d)=0$ and $\psi$ is any constant, or $\beta((\beta+1)(p-1)-d)>0$ and \begin{equation}\label{mean=0} \int_{M} (\beta^{2}\psi^{2}+\abs{\nabla_{g}{\psi}}^{2})^{(p-2)/2}\psi dv_{g}=0. \end{equation} \end{theorem} \paragraph{Proof} From (\ref{quasi spectrum}), \begin{equation}\label{mean =1} \beta((\beta+1)(p-1)-d) \int_{M} (\beta^{2}\psi^{2}+\abs{\nabla_{g}{\psi}}^{2})^{(p-2)/2}\psi dv_{g}=0. \end{equation} Thus if the integral term is not zero $\beta((\beta+1)(p-1)-d)=0$. Clearly if $\beta=0$, $\psi$ is a constant. If $\beta\neq 0$, $(\beta+1)(p-1)=d$ and from (\ref{quasi spectrum}) there holds $$ -\nabla_{g}.\left((\beta^{2}\psi^{2} +\abs{\nabla_{g}{ \psi}}^{2})^{(p-2)/2}\nabla_{g}{ \psi}\right)=0, $$ which implies $$\int_{M}\left(\beta^{2}\psi^{2}+\abs{\nabla_{g}{\psi}}^{2}\right)^{(p-2)/2} \abs{\nabla_{g}{\psi}}^{2}dv_{g}=0. $$ Thus $\psi$ is constant. Moreover if $\beta((\beta+1)(p-1)-d)=0$ any constant satisfies (\ref{quasi spectrum}). Assume now that $\beta((\beta+1)(p-1)-d)\neq 0$. Then (\ref{mean=0}) holds. Moreover \begin{multline}\label{consistant} \int_{M}\left(\beta^{2}\psi^{2}+\abs{\nabla_{g}{\psi}}^{2}\right)^{(p-2)/2} \abs{\nabla_{g}{\psi}}^{2}dv_{g}\\ =\beta((\beta+1)(p-1)-d)\int_{M}(\beta^{2}\psi^{2} +\abs{\nabla_{g}{\psi}}^{2})^{(p-2)/2}\psi^{2}dv_{g}, \end{multline} and the inequality $\beta((\beta+1)(p-1)-d)>0$ follows. \paragraph{Remark} %2 It should be interesting to study the links between $\mathfrak S_{p}(M)$ and the geometry of $M$, in particular the infimum of the $\beta((\beta +1)(p-1-d)$. Since we conjectured that the set of such $\beta$ is unbounded, as on the sphere, their asymptotic distribution could be of interest. In the particular case where $p=d+1$, the $(d+1)$-quasi-spectrum of $M$ is the set of couples $(\beta,\psi)$ such that $\psi$ is a solution of \begin{equation}\label{(d+1)-quasi spectrum} -\nabla_{g}.\left((\beta^{2}\psi^{2} +\abs{\nabla_{g}{ \psi}}^{2})^{(d-1)/2}\nabla_{g}{ \psi}\right) =d\beta^{2}(\beta^{2}\psi^{2}+\abs{\nabla_{g}{\psi}}^{2})^{(d-1)/2}\psi. \end{equation} As in the case $p=2$, it should be interesting to study the invariance properties of $\mathfrak S_{d+1}(M)$ with respect to the conformal transformations of $M$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section {Equations with strong absorption} In this section we assume $N\geq p>1$ and $q>p-1$. If we look for solutions $u$ of (\ref{main equ epsilon}) with $\varepsilon=1$ under the form (\ref{anisotropic singular}) then $\beta=p/(q+1-p)=\beta_{q}$ and $\omega$ solves \begin{equation}\label{anisotropic equation q} -\nabla_{\sigma}.\left( (\beta_{q}^2\omega^2+{\abs {\nabla_{\sigma}\omega}}^2)^{(p-2)/2}\nabla_{\sigma}\omega\right) +{\abs \omega}^{q-1}\omega=\lambda_{q}(\beta_{q}^2\omega^2 +{\abs {\nabla_{\sigma}\omega}}^2)^{(p-2)/2}\omega, \end{equation} in $S^{N-1}$, where \begin{equation}\label{constant q} \lambda_{q}=\beta_{q}((\beta_{q}+1)(p-1)+1-N)= \Big(\frac{p}{q+1-p}\Big)\Big(\frac{pq}{q+1-p}-N\Big). \end{equation} Since $$\int_{S^{N-1}}\left((\beta_{q}^2\omega^2+{\abs {\nabla_{\sigma}\omega}}^2)^{(p-2)/2} \left({\abs {\nabla_{\sigma}\omega}}^2-\lambda_{q}\omega^2\right)+ {\abs\omega}^{q+1}\right)d\sigma=0, $$ there is no solution if $\lambda_{q}\leq 0$ or equivalently if $q\geq N(p-1)/(N-p)$. This fact corresponds to a removability result which was proved by Vazquez and V\'eron \cite {VV}. \begin{theorem} \label{remov th} Let $\Omega$ be an open subset of $\mathbb{R}^N$containing $0$, $\Omega^*=\Omega\setminus\{0\}$, $N> p>1$, $q\geq N(p-1)/(N-p)=p^\#$ and $g$ a continuous real valued function satisfying \begin{equation}\label{removability} \liminf_{r\to\infty}r^{-p^\#}g(r)>0, \quad \mbox{and}\quad \limsup_{r\to-\infty}{\abs r}^{-p^\#}g(r)<0. \end{equation} If $u\in C(\Omega^*)\cap W_{\rm loc}^{1,p}(\Omega^*)$ is a weak solution of \begin{equation} -\nabla.\left({\abs {\nabla u}}^{p-2}\nabla u\right)+g(u)=0, \quad \mbox{in }\Omega^*, \end{equation} it can be extended to $\Omega$ as a continuous solution of the same equation in whole $\Omega$. \end{theorem} On the contrary, if $p-11$, and $p-10$ such that $ \displaystyle {\lim_{x\to 0}}u(x)/\mu_{p}(x)=\gamma$, and $u$ satisfies \begin{equation}\label{main equ dirac} -\nabla. ({\abs{\nabla u}^{p-2}}\nabla u)+{\abs u}^{q-1}u= c_{N,p}{\abs\gamma^{p-2}}\gamma\delta_{0},\quad\mbox{ in }{\cal D}'(\Omega). \end{equation} (iii) Or $u$ can be extended to whole $\Omega $ as a $C^1$ solution of (\ref{main equ epsilon=1}) in $\Omega$. \end{theorem} \paragraph{Proof} By scaling we can always assume that $B_{1}\subset \Omega$. The starting point is an a priori estimate of Keller-Osserman type due to Vazquez \cite {Va}: if $u$ is any solution of (\ref{main equ epsilon=1}) in $B_{1}^*=\{x\in\mathbb{R}^N:\,0<\abs x<1\}$, there exists a positive constant $K=K_{N,p,q}$ such that \begin{equation}\label{KO} {\abs {u(x)}}\leq K{\abs x}^{-\beta_{q}}, \end{equation} for any $0<\abs x \leq 1/2$. By writting (\ref{main equ epsilon=1}) under the form $$-\nabla. ({\abs{\nabla u}^{p-2}}\nabla u)+d(x) u^{p-1}=0, $$ with $d(x)= u^{q+1-p}$, and using the Trudinger's estimate \cite {Tr} in Harnack inequality, it follows that there exists some $A=A(N,p,q)>0$ such that $$\max_{\abs x=r}u(x)\leq A\min_{\abs x=r}u(x), $$ for any $00$ there exists some $n_{k}$ such that for $n\geq n_{k}$ the function $u$ is bounded from below in $\bar B_{1}\setminus B_{r_{n}}$ by the solution $v_{n}$ of the Dirichlet problem \begin{equation}\label{two points equ} \begin{gathered} -\nabla. ({\abs{\nabla v_{n}}^{p-2}}\nabla v_{n})+{\abs v_{n}}^{q-1}v_{n}=0,\quad \mbox{in } B_{1}\setminus \bar B_{r_{n}},\\ v_{n}(x)=0\quad \mbox{if }\abs x=1,\\ v_{n}(x)=k\mu_{p}(r_{n})\quad \mbox{if }\abs x=r_{n}. \end{gathered}\end{equation} Note that $v_{n}$ is positive, radial and bounded from above by $k\mu_{p}(x)$. Since $q0$. By using the same scaling methods, estimates on $\nabla u$, and the strict comparison principle as in the proof of Theorem \ref{singular p-hamonic}, it can be proved that there exists a real number $\gamma$ such that \begin{equation}\label{weak estimate for u} \lim_{x\to 0}u(x)/\mu_{p}(x)=\gamma, \end{equation} and \begin{equation}\label{weak estimate for Du} \lim_{x\to 0}(\abs x)^{(N-1)/(p-1)}\nabla \left(u(x)-\gamma\mu_{p}(x)\right) =0. \end{equation} Thus $u$ satisfies (\ref{main equ dirac}). If $\gamma=0$, then $$\abs{u(x)}\leq \max_{\abs y=1}\abs {u(y)},\quad\forall x\in B_{1}, $$ by the maximum principle. Thus $u$ is $C^{1,\alpha}$ by the regularity theory of quasilinear equations. \hfill$\square$ \smallskip The construction of nodal singular solutions of (\ref{main equ epsilon=1}) under the form (\ref{anisotropic singular}) is done by a shooting technique, as for the $p$-Laplace equation. \begin{theorem} \label{nodal} Let $00$ be the exponent defined in Theorem \ref{N=3}. If $\beta_{q}>\beta_{S}$ there exists a positive solution $\omega$ of (\ref{anisotropic equation q}) in $S$ which vanishes on $\partial S$. \end{theorem} \paragraph{Proof:} {\bf Step 1} Construction of an approximate solution. For $\varepsilon >0$ small enough denote by $u=u_{\varepsilon}$ the unique solution of \begin{equation}\label{approx epsilon} \begin{gathered} -\nabla.\left({\abs{\nabla u}}^{p-2}\nabla u\right)+ {\abs{u}}^{q-1}u=0,\quad\mbox{in }K_{S}(1,\infty),\\ u=\varepsilon g^{\beta_{q}},\quad\mbox{on }\partial K_{S}(1,\infty),\\ \limsup_{\abs x\to\infty}{\abs x}^{\beta_{q}}u(x)<\infty. \end{gathered} \end{equation} By the monotone operator theory, $u$ is unique and satisfies $0\leq u0$ small enough, the function \begin{equation}\label{subsol} (r,\sigma)\mapsto w_{\delta}(x)=w_{\delta}(r,\sigma) =r^{-\beta_{q}}\delta \omega^\theta_{S}(\sigma) \end{equation} satisfies \begin{equation}\label{approx delta} \begin{gathered} -\nabla.\left({\abs{\nabla w_{\delta}}}^{p-2}\nabla w_{\delta}\right)+ {\abs{w_{\delta}}}^{q-1}w_{\delta}\leq 0,\quad\mbox{in }K_{S}(1,\infty),\\ w_{\delta}=0,\quad\mbox{on }B_{S}(1,\infty). \end{gathered} \end{equation} Set $$\mathcal L w_{\delta}=-\nabla.\left({\abs{\nabla w_{\delta}}}^{p-2}\nabla w_{\delta}\right)+ {\abs{w_{\delta}}}^{q-1}w_{\delta}. $$ Then $\mathcal L (w_{\delta})=r^{-q\beta_{q}}\mathcal T(\delta \omega^\theta_{S})$, where \[ \mathcal T(\eta)=-\nabla_{\sigma}.\left( (\beta_{q}^2\eta^2+{\abs {\nabla_{\sigma}\eta}}^2)^{(p-2)/2}\nabla_{\sigma}\eta\right) -\lambda_{q}(\beta_{q}^2\eta^2+{\abs {\nabla_{\sigma}\eta}}^2)^{(p-2)/2} \eta+{\abs \eta}^{q-1}\eta. \] Putting $\eta=\delta \omega^\theta_{S}$, \[ (\beta_{q}^2\eta^2+{\abs {\nabla_{\sigma}\eta}}^2)^{(p-2)/2} =\delta^{p-2}\theta^{p-2}\omega_{S}^{(\theta-1)(p-2)} (\beta_{S}^2\omega^2+{\abs {\nabla_{\sigma}\omega}}^2)^{(p-2)/2}, \] and \begin{align*} \nabla_{\sigma}.&\big( (\beta_{q}^2\eta^2+{\abs {\nabla_{\sigma}\eta}}^2)^{(p-2)/2}\nabla_{\sigma}\eta\big)\\ =&\delta^{p-1}\theta^{p-1} \nabla_{\sigma}.\Big( \omega_{S}^{(\theta-1)(p-1)} (\beta_{S}^2\omega_{S}^2+{\abs {\nabla_{\sigma}\omega_{S}}}^2)^{(p-2)/2}\nabla_{\sigma}\omega_{S}\Big)\\ =&\delta^{p-1}\theta^{p-1}\omega_{S}^{(\theta-1)(p-1)} \nabla_{\sigma}.\big((\beta_{S}^2\omega_{S}^2+{\abs {\nabla_{\sigma}\omega_{S}}}^2)^{(p-2)/2}\nabla_{\sigma}\omega_{S}\big)\\ &+(\theta-1)(p-1)\delta^{p-1}\theta^{p-1}\omega_{S}^{(\theta-1)(p-1)-1} (\beta_{S}^2\omega_{S}^2+{\abs {\nabla_{\sigma}\omega_{S}}}^2)^{(p-2)/2}{\abs {\nabla_{\sigma}\omega_{S}}}^2 \end{align*} But \[ -\nabla_{\sigma}.\left((\beta_{S}^2\omega_{S}^2+{\abs {\nabla_{\sigma}\omega_{S}}}^2)^{(p-2)/2}\nabla_{\sigma}\omega_{S}\right) =\lambda_{S}(\beta_{S}^2\omega_{S}^2+{\abs {\nabla_{\sigma}\omega_{S}}}^2)^{(p-2)/2}\omega_{S}, \] with $\lambda_{S}=(\beta_{S}+1)(p-1)+1-N)$. Thus, \begin{eqnarray*}\delta^{1-p}\mathcal T(\eta)&=&\delta^{q+1-p}\omega_{S}^{\theta q}+\omega_{S}^{(\theta-1)(p-1)-1} \theta^{p-2}(\beta_{S}^2\omega_{S}^2+{\abs {\nabla_{\sigma}\omega_{S}}}^2)^{(p-2)/2}\\ &&\times \left((\theta\lambda_{S}-\lambda_{q})\omega_{S}^{2} -\theta(\theta-1)(p-1){\abs {\nabla_{\sigma}\omega_{S}}}^2\right). \end{eqnarray*} Since $\theta\lambda_{S}-\lambda_{q}=\beta_{q}(\beta_{S}-\beta_{q})(p-1) =-\beta^{2}_{S}\theta(\theta-1)(p-1)$, \begin{eqnarray*} \lefteqn{\delta^{1-p}\mathcal T(\eta)}\\ &=&\delta^{q+1-p}\omega_{S}^{\theta q}-(p-1)(\theta-1)\theta^{p-1}\omega_{S}^{(\theta-1)(p-1)-1}(\beta_{S}^2\omega_{S}^2+{\abs {\nabla_{\sigma}\omega_{S}}}^2)^{p/2}\\ &\leq&\delta^{q+1-p}\omega_{S}^{\theta q}-(p-1)(\theta-1)\theta^{p-1}\omega_{S}^{\theta(p-1)}. \end{eqnarray*} by assumption $\theta>1$, therefore there exists $\delta>0$ such that $\mathcal T(\eta)\leq 0$. Moreover it can also be assumed that $\delta\omega_{S}^\theta\leq \varepsilon$. Then $w_{\delta}(x)\leq u(x)$ if $\abs x =1$ and $ w_{\delta}\leq u$ in $K_{S}(1,\infty)$ by the maximum principle. Henceforth \begin{equation}\label{two side estim} \delta\omega_{S}^{\theta}(x/\abs x)\leq {\abs x}^{\beta_{q}}u(x)\leq \gamma_{N,p,q}\quad \mbox{in }\;K_{S}(1,\infty). \end{equation} \noindent {\bf Step 3} For $R>0$, define the function $u_{R}$ by $u_{R}=R^{\beta_{q}}u(Rx)$. The function $u_{R}$ satisfies (\ref{main equ epsilon=1}) in $K_{S}(1/R,\infty)$. By the degenerate elliptic equation regularity theory, the set of functions $\{u_{R}\}$ remains bounded in the $C_{\rm loc}^{1,\alpha}$-topology of $\overline{K_{S}(0,\infty)}\setminus \{0\}$. Let $0\beta_{S}$ is sharp. \begin{theorem} \label{no nodal} Let $00$, it follows from the proof of Theorem \ref{nodal}-Step 2 that, for any $\delta>0$, \begin{eqnarray*} \delta^{1-p}\mathcal T(\eta)&=&\delta^{q+1-p}\omega_{S}^{\theta q}\\ &&+(p-1)(1-\theta)\theta^{p-1}\omega_{S}^{(\theta-1)(p-1)-1}(\beta_{S}^2\omega_{S}^2+{\abs {\nabla_{\sigma}\omega_{S}}}^2)^{p/2}>0. \end{eqnarray*} We take $\delta=\delta_{0}$ as the smallest parameter such that $\eta=\eta_{\delta}\geq \omega$. Notice that such a choice is always possible since $\omega\in C^{1}(\bar S)$, the normal derivative of $\omega_{S}$ on the relative boundary $\partial S$ is negative from the Hopf boundary lemma and therefore $\omega_{S}^\theta(\sigma)\geq c (dist(\sigma,\partial S)^\theta$ for some $c>0$. We shall distinguish according there exists $\sigma_{0}\in S$ such that \begin{equation}\label{tangential} \eta(\sigma)\geq \omega (\sigma),\;\forall \sigma\in \bar S,\quad\mbox{and}\quad \eta(\sigma_{0})= \omega (\sigma_{0}), \end{equation} or not. If (\ref{tangential}) holds true, which is always the case if $\beta_{S}>\beta_{q}$, the function $\psi=\eta-\omega$ is nonnegative in $\bar S$, not identically $0$ and achieves its minimal value $0$ in an interior point $\sigma_{0}$. Let $g=(g_{ij})$ be the metric tensor on $S^{N-1}$. We write in local coordinates $\sigma_{j}$ around $\sigma_{0}$, \begin{gather*} {\abs {\nabla\varphi }^{2}}= \sum_{j,k}g^{jk}\frac {\partial \varphi}{\partial \sigma_{j}}\frac {\partial \varphi}{\partial \sigma_{k}}, \\ \nabla.X=\frac {1}{\sqrt {\abs g}} \sum_{\ell} \frac {\partial}{\partial \sigma_{\ell}}\left(\sqrt {\abs g}X^\ell\right) =\frac {1}{\sqrt {\abs g}} \sum_{\ell,i}\frac {\partial}{\partial \sigma_{\ell}}\left(\sqrt {\abs g}g^{\ell i}X_{i}\right), \end{gather*} if we lower the indices by setting $\displaystyle {X^\ell=\sum_{i}g^{\ell i}X_{i}}$. From the Mean Value Theorem, we obtain \begin{multline*} (\beta^{2}_{q}\eta^2+\abs {\nabla_{\sigma}\eta}^2)^{(p-2)/2}\frac {\partial \eta}{\partial \sigma_{i}}- (\beta^{2}_{q}\omega^2+\abs {\nabla_{\sigma}\omega}^2)^{(p-2)/2}\frac {\partial \omega}{\partial \sigma_{i}}\\ =\sum_{j}\alpha^i_{j}\frac {\partial (\eta-\omega)}{\partial \sigma_{j}}+b^i(\eta-\omega), \end{multline*} where \begin{eqnarray*}b^i&=&(p-2)\left(\beta_{q}^2(\omega+t(\eta-\omega))^2+{\abs {\nabla_{\sigma}(\omega+t(\eta-\omega))}}^2\right)^{(p-4)/2}\\ &&\times (\omega+t(\eta-\omega))\frac {\partial (\omega+t(\eta-\omega))}{\partial\sigma_{i}}, \end{eqnarray*} and \begin{eqnarray*} \alpha^i_{j}&=&(p-2)\left(\beta_{q}^2(\omega+t(\eta-\omega))^2+{\abs {\nabla_{\sigma}(\omega+t(\eta-\omega))}}^2\right)^{(p-4)/2}\\ &&\times\frac {\partial (\omega+t(\eta-\omega))}{\partial\sigma_{i}}\sum_{k}g^{jk}\frac {\partial (\omega+t(\eta-\omega))}{\partial\sigma_{k}}\\ &&+\delta_{i}^j\left(\beta_{q}^2(\omega+t(\eta-\omega))^2+{\abs {\nabla_{\sigma}(\omega+t(\eta-\omega))}}^2\right)^{(p-2)/2}. \end{eqnarray*} Since the graph of $\eta$ and $\omega $ are tangent at $\sigma_{0}$, $$\eta(\sigma_{0})=\omega(\sigma_{0})=P_{0}>0\quad \mbox{and } \nabla{\eta(\sigma_{0})}=\nabla{\omega(\sigma_{0})=Q}.$$ Thus $$b^i(\sigma_{0})= (p-2)\left(\beta_{q}^2P_{0}^2+{\abs {Q}}^2\right)^{(p-4)/2}P_{0}Q_{i}, $$ and \begin{eqnarray*} \alpha^i_{j}(\sigma_{0})=\big(\beta_{q}^2P_{0}^2+{\abs Q}^2\big)^{(p-4)/2} \Big(\delta_{i}^j(\beta_{q}^2P_{0}^2+{\abs Q}^2)+(p-2) Q_{i} \sum_{k}g^{jk}Q_{k}\Big). \end{eqnarray*} Now \begin{align*} \mathcal T&(\eta)-\mathcal T(\omega)\\ =& \frac {-1}{\sqrt {\abs g}}\sum_{\ell,i}\frac {\partial}{\partial \sigma_{\ell}} \Big[\sqrt {\abs g}g^{\ell i} \Big((\beta^{2}_{q}\eta^2+\abs {\nabla_{\sigma}\eta}^2)^{\frac {p}{2}-1}\frac {\partial \eta}{\partial \sigma_{i}}-(\beta^{2}_{q}\omega^2+\abs {\nabla_{\sigma}\omega}^2)^{\frac {p}{2}-1}\frac {\partial \omega}{\partial \sigma_{i}}\Big)\Big]\\ &-\lambda_{q}\left((\beta^{2}_{q}\eta^2+\abs {\nabla_{\sigma}\eta}^2)^{\frac {p}{2}-1}\eta-(\beta^{2}_{q}\omega^2+\abs {\nabla_{\sigma}\omega}^2)^{\frac {p}{2}-1}\omega\right)+\eta^q-{\abs\omega}^{q-1}\omega),\\ =&-\frac {1}{\sqrt {\abs g}}\sum_{\ell,i}\frac {\partial}{\partial \sigma_{\ell}}\Big[\sqrt {\abs g}g^{\ell i} \Big(\sum_{j}\alpha^i_{j}\frac {\partial (\eta-\omega)}{\partial \sigma_{j}} +b^i(\eta-\omega)\Big)\Big] \\ &+\sum_{i}C_{i}\frac {\partial (\eta-\omega)}{\partial \sigma_{i}}+C(\eta-\omega)\\ =&-\frac {1}{\sqrt {\abs g}}\sum_{\ell,j}\frac {\partial}{\partial \sigma_{\ell}}\left[a^\ell_{j}\frac {\partial (\eta-\omega)}{\partial \sigma_{j}}\right] +\sum_{i}C_{i}\frac {\partial (\eta-\omega)}{\partial \sigma_{i}}+C(\eta-\omega), \end{align*} where the $C_{i}$ and $C$ are continuous functions and $$a^\ell_{j}=\sqrt{\abs g}\sum_{i}g^{\ell i}\alpha^i_{j}. $$ The matrix $\left(\alpha^i_{j}(\sigma_{0})\right)$ is symmetric, definite and positive since it is the Hessian of the strictly convex function $$ X=(X_{1},\ldots,X_{n-1})\mapsto \frac {1}{p}\left(P_{0}^{2}+{\abs X}^2\right)^{p/2} =\frac {1}{p}\Big(P_{0}^{2}+\sum_{j,k}g^{jk}X_{j}X_{k}\Big)^{p/2}. $$ Therefore, $\left(\alpha^i_{j}\right)$ has the same property in some neighborhood of $\sigma_{0}$, and the same holds true with $\left(a^\ell_{j}\right)$. Finally the function $\psi=\eta-\omega$ is nonnegative, vanishes at $\sigma_{0}$ and satisfies \begin{equation}\label{max princ} -\frac {1}{\sqrt {\abs g}}\sum_{\ell,j}\frac {\partial}{\partial \sigma_{\ell}}\big[a^\ell_{j}\frac {\partial \psi}{\partial \sigma_{j}}\big] +\sum_{i}C_{i}\frac {\partial \psi}{\partial \sigma_{i}}+C_{+}\psi\geq 0. \end{equation} Then $\psi=0$ in a neighborhood of $S$. Since $S$ is connected, $\psi$ is identically $0$, which a contradiction. If (\ref{tangential}) does not hold, then $\theta=1$ and that the graphs of $\eta$ and $\omega$ are tangent at some point $\sigma_{0}$ of the relative boundary $\partial S$. Proceeding as above and using the fact that $\partial \eta/\partial\nu$ exists and never vanishes on the boundary, we see that $\psi=\eta-\omega$ satisfies (\ref{max princ}) with a strongly elliptic operator in a neighborhood $\mathcal N$ of $\sigma_{0}$. Moreover $\psi >0$ in $\mathcal N$, $\psi (\sigma_{0})=0$ and $\partial \psi/\partial\nu(\sigma_{0})=0$. This is a contradiction, which ends the proof.\medskip \paragraph{Remark} %rem {\bf 3} The existence result of Theorem \ref{nodal} is valid if $S$ is no longer a $C^{2}$ domain but a domain with a piecewise regular boundary since only the existence of $(\beta_{S},\omega_{S})$ is needed. We conjecture that the condition $\beta_{q}>\beta_{S}$ is still necessary. As is section 2, we can construct nodal solutions of (\ref{anisotropic equation q}) with a finite symmetry group $G$ generated by reflections through hyperplanes. Taking $S$ to be a fundamental simplicial domain of $G$, we construct $(\beta,\omega)$ in $S$ and then extend $\omega$ to the whole sphere by reflections through the edges. It follows that there exists nodal singular solutions of (\ref{main equ epsilon=1}) in $\mathbb{R}^N\setminus \{0\}$. \smallskip \paragraph{Remark} %\rem {\bf 4} Under the assumptions of Theorem \ref{nodal}, we conjecture that uniqueness of the positive solution $\omega$ of (\ref{anisotropic equation q}) which vanishes on $\partial S$ holds. If $S=S^{N-1}$ and $p-1p-1$). In the range $1N(p-1)/(N-p)$ the constant function $$\omega_{0}=(\beta^{p-1}_{q}(N-q\beta_{q})^{1/(q+1-p)} $$ is a solution of (\ref{anisotropic equation q-2}), and a natural question is to look for nonconstant solutions. As in Section 2, we imbed this problem in the more general setting of a compact $d$-dimensional Riemannian manifold $(M,g)$ without boundary. For $\beta$ and $\lambda \in \mathbb{R}$ consider the equation \begin{equation} \label {mainM} -\nabla_{g}.\left((\beta^{2}\omega^{2} +\abs{\nabla_{g}{ \omega}}^{2})^{(p-2)/2}\nabla_{g}{ \omega}\right) +\lambda (\beta^{2}\omega^{2}+\abs{\nabla_{g}{ \omega}}^{2})^{(p-2)/2} \omega=\abs \omega^{q-1}\omega . \end{equation} We shall assume $\lambda >0$ in order for the constant solution $$\omega_{\ast}=(\beta^{p-2}\lambda)^{1/(q+1-p)} $$ to exist. We assume also that the starting equation is super-quasilinear in the sense that $\beta>0$ and $q>q+1-p$. We can linearize (\ref{mainM}) in a neighborhood of $\omega_{*}$, and we obtain \begin{gather*} \begin{split} \frac {d}{dt} \nabla_{g}.\left((\beta^{2}(\omega_{*}+t\varphi)^{2} +\abs{\nabla_{g}{ (\omega_{*}+t\varphi)}}^{2})^{(p-2)/2} \nabla_{g}{ (\omega_{*}+t\varphi)}\right)&\Big|_{t=0}\\ &=\beta^{p-2}\omega_{*}^{p-2}\Delta_{g}\varphi. \end{split} \\ \frac {d}{dt}\left((\beta^{2}(\omega_{*}+t\varphi)^{2} +\abs{\nabla_{g}{ (\omega_{*}+t\varphi)}}^{2})^{(p-2)/2} (\omega_{*}+t\varphi) \right)\Big|_{t=0} =(p-1)\beta^{p-2}\omega_{*}^{p-2}\varphi. \\ \frac {d}{dt}\left(\omega_{*}+t\varphi\right)^{q} \Big|_{t=0}=q\omega_{*}^{q-1}\varphi. \end{gather*} Since $\omega_{*}=(\beta^{p-2}\lambda)^{1/(q+1-p)}$, the linearized equation is \begin{equation}-\Delta_{g}\varphi=(q+1-p)\lambda\varphi. \end{equation} where $\Delta_{g}=\nabla_{i}\nabla^i$ is the laplacian on $M$. \begin{theorem} \label{bifur} Let $\mu_{1}$ be the first nonzero eigenvalue of $\Delta_{g}$, and assume it is simple. Then for any $\lambda>\mu_{1}/(q+1-p)$ equation (\ref{mainM}) admits a nonconstant positive solution $\omega_{\lambda}$. \end{theorem} \paragraph{Proof} The existence of a global and unbounded branch of bifurcation $\mathcal B=\{(\lambda,\omega_{\lambda})\}\subset \mathbb R\times C^{1}(M)$ issued from $(\mu_{1}/(q+1-p),\omega_{\ast})$ follows from the application in the space $C^{1}(M)$ of the classical bifurcation theorem from a simple eigenvalue. \hfill$\square$ \paragraph{Remark} %rem {\bf 5 } The condition on the simplicity of $\mu_{1}$ can be avoided in many cases where symmetries occur. When $(M,g)=(S^{N-1},g_{0})$, we have the parametric representation $$S^{N-1}=\{\sigma=(\cos\varphi,\sin\varphi \sigma')\;:\;\varphi\in [0,\pi],\,\sigma'\in S^{N-2}\}, $$ and $$\Delta_{S^{N-1}}\omega=\sin^{2-N}\varphi \frac {\partial}{\partial \varphi} \big(\sin^{N-2}\varphi\frac {\partial \omega}{\partial \varphi}\big)+\sin^{-2}\varphi\Delta_{S^{N-2}}\omega. $$ If we only consider function depending on $\varphi$ (they are called zonal functions), $\mu_{1}=N-1$ is a simple eigenvalue. Moreover any eigenspace of $S^{N-1}$ contains a 1-dimensional sub-eigenspace of functions depending only on $\varphi$. Therefore all the corresponding eigenvalues are simple. Thus from each of the couples $(\mu_{k}/(q+1-p),\omega_{*})$ is issued a $C^{1}$ curve of positive solutions $(\lambda,\omega_{\lambda})$ with $\lambda>\mu_{k}/(q+1-p)$. \paragraph{Open question} An interesting problem is to find sufficient conditions besides $\lambda\leq \mu_{1}/(q+1-p)$ and probably $q\leq dp/(d-p)-1$, in order the constant $\omega_{*}$ be the only positive solution of (\ref{mainM}). We believe additional conditions linked to the curvature should be found (see \cite {GS}, \cite {BVV}, \cite {LV} in the case $p=2$).\smallskip We define the critical Sobolev exponent $q_{c}$ by \begin{equation}\label{crit exp}q_{c}=\frac {Np}{N-p}-1=\frac {N(p-1)+p}{N-p}. \end{equation} A particular case of equation (\ref{epsilon=-1}) is when $q=q_{c}$. Then $$q_{c}+1-p=\frac {p^{2}}{N-p},\quad \beta_{q_{c}}=\frac{N-p}{p}\quad {\rm and } \quad \lambda_{q_{c}}=-\beta^{2}_{q_{c}}. $$ The critical equation is therefore \begin{equation}\label{maincrit} \nabla_{\sigma}.\Big((\beta_{q_{c}}^{2}\omega^{2} +\abs{\nabla_{\sigma}{ \omega}}^{2})^{p/2-1}\nabla_{\sigma}\omega\Big)+\abs \omega^{q_{c}-1}\omega -\beta^{2}_{q_{c}}(\beta_{q_{c}}^{2}\omega^{2} +\abs{\nabla_{\sigma}{ \omega}}^{2})^{p/2-1}\omega= 0, \end{equation} on $S^{N-1}$. A natural question is to explore the connection between the positive solutions of (\ref{maincrit}) and the positive solutions of \begin{equation} -\nabla.\Big({\abs{\nabla u}}^{p-2}\nabla u\Big) v=v^{q_{c}}\quad \mbox{ in }\;\mathbb{R}^{N}\label{sobolev}. \end{equation} Notice that the radial solutions of this equation, depending of a parameter $a>0$, are known: \begin{equation} v_{a}(x)=\Big(Na\big(\frac{N-p}{p-1}\big)^{p-1}\Big)^{(N-p)/{p^{2}}} \left(a+{\abs x}^{p/(p-1)}\right)^{(p-N)/p}. \label{hill} \end{equation} The solutions of (\ref{maincrit}) are the critical points of the functional \begin{equation}\label{Euler} J_{q_{c}}(\psi)=\int_{S^{N-1}}\Big(\frac {1}{p}(\beta_{q_{c}}^{2}\psi^{2} +\abs{\nabla_{\sigma}{ \psi}}^{2})^{p/2}-\frac {1}{q_{c}+1}{\abs \psi}^{q_{c}+1}\Big)\,d\sigma, \end{equation} where $\psi\in W^{1,p}(S^{N-1})$. \paragraph{Remark} % rem {\bf 6 } Let $00$. If we look for particular solutions of (\ref{plaplacexp}) under the form $$ u(r,\sigma)=\alpha\ln r+ bw(\sigma)+k, $$ where $\alpha$, $b$ and $k$ are constants, one finds $\alpha=-p$ and \[ b\nabla_{\sigma}.\Big(\big[p^{2}+b^{2} \abs{{\nabla_{\sigma}w}}^{2}\big]^{p/2-1} \nabla_{\sigma}w\Big)+\lambda e^ke^{bw} -p(N-p)\big[p^{2}+b^{2}\abs{{\nabla_{\sigma}w}}^{2}\big]^{p/2-1}=0 \] on $S^{N-1}$. A necessary condition for the existence of a solution is \begin{equation} p-N<0. \label{cond}\end{equation} Assuming this condition, we take $b=p$ and get $$\nabla_{\sigma}.\Big(\big[1+\abs{{\nabla_{\sigma}w}}^{2}\big]^{p/2-1} \nabla_{\sigma}w\Big) -(N-p)\big[1+\abs{{\nabla_{\sigma}w}}^{2}\big]^{p/2-1} +\lambda p^{1-p}e^{k}e^{pw}=0. $$ Now choose $k=\ln (p^{p-1} \lambda^{-1})$. Assuming $10$ such that $\displaystyle {\lim_{x\to 0}}u(x)/\mu_{p}(x)=\alpha$, and $u$ satisfies \begin{equation}\label{main equ dirac 2} -\nabla. ({\abs{\nabla u}^{p-2}}\nabla u)- u^q= c_{N,p}\alpha^{p-1}\delta_{0},\quad\mbox{ in }{\cal D}'(\Omega). \end{equation} (ii) Or $u$ can be extended as a $C^{1}$ solution of (\ref{epsilon=-1}) in $\Omega$. \end{theorem} The general proof of this result is based upon the extension obtained in \cite {BV} of the Brezis-Lions lemma \cite {BL} dealing with singular super-harmonic functions. \begin{lemma} Let $10$ such that \begin{equation}\label{inequality with delta p-3} u(x)\leq C\mu_{p}(x), \end{equation} holds in a neighborhood of $0$. With this estimate, a scaling methods similar to the one used in \cite {FV} ends the proof. Actually, in \cite {GV}, a more general convergence result is proved: if $1Np/(N-p)-1$ can be found in \cite {BV}. For a long time, the non-radial case appeared out of reach up to the recent work of Serrin and Zou \cite {SZ}. In this striking paper they proved, among other results, that Gidas and Spruck classical a priori estimate in the case $p=2$, $N/(N-2)\leq q1$ and $p^{\#}\leq p