\documentclass[twoside]{article} \usepackage{amsfonts, amsmath} % used for R in Real numbers \usepackage{graphicx} \pagestyle{myheadings} \markboth{A note on the moving hyperplane method} { C\'eline Azizieh \& Luc Lemaire } \begin{document} \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 1--6. \newline http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % A note on the moving hyperplane method % \thanks{ {\em Mathematics Subject Classifications:} 35J60, 35B99. \hfil\break\indent {\em Key words:} p-Laplacian, symmetry of solutions, nonlinear elliptic equations. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Published October 21, 2002. \hfil\break\indent The second author is supported by an Action de Recherche Concert\'ee de la \hfil\break\indent Communaut\'e Fran\c caise de Belgique }} \date{} \author{C\'eline Azizieh \& Luc Lemaire} \maketitle \begin{abstract} We make precise the domain regularity needed for having the monotonicity and symmetry results recently proved by Damascelli and Pacella on p-Laplace equations. For this purpose, we study the continuity and semicontinuity of some parameters linked with the moving hyperplane method. \textbf{R\'esum\'e.} Dans [1], Ph. Cl\'ement et le premier auteur ont \'etabli par des m\'ethodes de continuation des r\'esultats d'existence pour des probl\`emes du type $-\Delta_pu=f(u)$ dans $\Omega$, $u=0$ sur $\partial\Omega$, $u>0$ sur $\Omega$, o\`u $1< p\le 2$, $\Omega\subset\mathbb{R}^N$ est un domaine born\'e convexe et $f:\mathbb{R}\to[0+\infty)$ est continue. La preuve de ces th\'eor\`emes fait appel aux r\'ecents r\'esultats de monotonie et de sym\'etrie \'etablis par Damascelli et Pacella dans [3], r\'esultats dont la d\'emonstration n\'ecessitait la continuit\'e ou semi-continuit\'e de certains param\`etres g\'eom\'etriques li\'es \`a la m\'ethode des moving hyperplanes. Notre but est ici de pr\'eciser les hypoth`eses de r\'egularit\'e et de convexit\'e du domaine $\Omega$ qui sont n\'ecessaires pour satisfaire les diff\'erentes conditions de continuit\'e des param\`etres en question. \end{abstract} \newtheorem{thm}{Theorem} \newtheorem{proposition}[thm]{Proposition} \numberwithin{equation}{section} \section{Results} Let us consider the problem \begin{equation}\label{problem1} \begin{gathered} -\Delta_pu=f(u) \quad\textrm{in }\Omega,\\ u=0\quad \textrm{on }\partial\Omega,\\ u\in C^1(\Omega), u>0\quad \textrm{in }\Omega \end{gathered} \end{equation} where $10\textrm{ such that } C_0|u|^q\le f(u)\le C_1|u|^q\quad\forall u\in\mathbb{R}^+ $$ where $q>p-1$. In \cite{Az}, Ph. Cl\'ement and the first author proved the existence of a nontrivial positive solution to (\ref{problem1}) by using continuation methods and establishing a~priori estimates for the solutions of some nonlinear eigenvalue problem associated with (\ref{problem1}). The desired a~priori estimates use a blow up argument as well as some monotonicity and symmetry results proved by Damascelli and Pacella in \cite{Da} and generalizing to the p-laplacian operator with $1a(\nu)\,|\,\forall\lambda\in(a(\nu),\mu), \text{ we have (\ref{situation1}) and (\ref{ii})}\big\},\\ &\lambda_1(\nu):=\sup\Lambda_1(\nu) \end{align*} where (\ref{situation1}), (\ref{ii}) are the following conditions: \begin{eqnarray} \label{situation1} &(\Omega_\lambda^\nu)' \textrm{ is not internally tangent to }\partial\Omega \textrm{ at some point }p\notin T_\lambda^\nu&{}\\ \label{ii} &\textrm{for all }x\in\partial\Omega\cap T^\nu_\lambda,\, \nu(x).\nu\ne0,& \end{eqnarray} where $\nu(x)$ denotes the inward unit normal to $\partial\Omega$ at $x$. Notice that $\Lambda_1(\nu)\ne\emptyset\quad\textrm{and}\quad\lambda_1(\nu)<\infty$ since for $\lambda>a(\nu)$ close to $a(\nu)$, (\ref{situation1}) and (\ref{ii}) are satisfied and $\Omega$ is bounded. \begin{figure}[ht] \begin{center} \includegraphics[width=5cm]{fig11.eps} \caption{Illustration of the notations} \end{center} \end{figure} Propositions \ref{luc} and \ref{lem} below give sufficient conditions on $\Omega$ to guarantee the continuity of the functions $a(\nu)$ and $\lambda_1(\nu)$, as well as the lower semicontinuity of $\lambda_1(\nu)$. \begin{proposition}\label{luc} Let $\Omega$ be a bounded domain with $C^1$ boundary. Then the function $a(\nu)$ is continuous with respect to $\nu\in S^{N-1}$. \end{proposition} \begin{proposition}\label{lem} Let $\Omega \subset\mathbb{R}^N$ be a bounded domain with $C^1$ boundary. Then the function $\lambda_1(\nu)$ is lower semicontinuous with respect to $\nu\in S^{N-1}$. If moreover $\Omega$ is strictly convex, then $\lambda_1(\nu)$ is continuous. \end{proposition} As a consequence of these results, we can give more precision on the conditions to impose to $\Omega$ in the monotonicity result of \cite{Da}. This result becomes:\medskip \textbf{Theorem 1.1 in Damascelli-Pacella \cite{Da}} \emph{ Let $\Omega$ be a bounded domain in $\mathbb{R}^N$ with $C^1$ boundary, $N\ge2$ and $g:\mathbb{R}\to\mathbb{R}$ be a locally Lipschitz continuous function. Let $u\in C^1(\bar{\Omega})$ be a weak solution of \begin{gather*} -\Delta_pu=g(u) \quad \mbox{in}\Omega\\ u>0\quad {in }\Omega, \\ u=0\quad \mbox{on }\partial\Omega \end{gather*} where $10$ for all $x\in\Omega_{\lambda_1(\nu)}^\nu\backslash Z$ where $Z=\{x\in\Omega\,|\,\nabla u(x)=0\}$.}\smallskip Below we prove Propositions \ref{luc} and \ref{lem} and we give a counterexample of a $C^\infty$ convex but not strictly convex domain for which $\lambda_1(\nu)$ is not continuous everywhere. \paragraph{Proof of Proposition \ref{luc}:} Let us fix a direction $\nu\in S^{N-1}$. We shall prove that for all sequence $\nu_n\to \nu$ with $|\nu_n|=1$, there exists a subsequence still denoted by $\nu_n$ such that $a(\nu_n)\to a(\nu)$. Since $\Omega$ is bounded, $(a(\nu_n))$ is also bounded, so passing to an adequate subsequence, there exists $\bar a\in\mathbb{R}$ such that $ a(\nu_n)\to \bar a $. We will show that $\bar a= a(\nu)$. Suppose by contradiction that $\bar a\ne a(\nu)$. Then either $\bar a a(\nu)$. \noindent\textsc{Case 1: $\bar a a(\nu)$:} There exists $x\in\partial\Omega$ with $x.\nu=a(\nu)$. For $n$ large, $|x.\nu_n-x.\nu|=|x.\nu_n-a(\nu)|$ is small, and since $a(\nu_n)\to \bar a>a(\nu)$, for $n$ large enough we have $x.\nu_n0$ and a sequence $(\nu_n)\subset S^{N-1}$ such that $\nu_n\to\nu$ and $|\lambda_1(\nu)-\lambda_1(\nu_n)|>\epsilon$ for all $n\in\mathbb{N}$. Passing to a subsequence still denoted by $(\nu_n)$, we can suppose that $$ \textrm{either}\quad\lambda_1(\nu)>\lambda_1(\nu_n)+\epsilon\quad\forall n\in\mathbb{N}\quad\textrm{or}\quad \lambda_1(\nu)<\lambda_1(\nu_n)-\epsilon\quad\forall n\in\mathbb{N}. $$ \noindent\textsc{Case 1:} $\lambda_1(\nu)>\lambda_1(\nu_n)+\epsilon$ for all $n\in\mathbb{N}$. \quad For any fixed $n\in\mathbb{N}$, we have the following alternative: either there exists $x_n\in T_{\lambda_1(\nu_n)}^{\nu_n}\cap\partial\Omega $ with $\nu(x_n).\nu_n=0$, or there exists $x_n\in(\partial\Omega\cap\overline {\Omega_{\lambda_1(\nu_n)}^{\nu_n}})\setminus T_{\lambda_1(\nu_n)}^{\nu_n}$ with $\left(x_n\right)^{\nu_n}_{\lambda_1(\nu_n)}\in\partial\Omega$. Passing once again to subsequences, we can suppose that we are in one of the two situations above for all $n\in\mathbb{N}$. We treat below each situation and try to reach a contradiction. \\ (1.a) For all $n\in\mathbb{N}$, there exists $x_n\in T_{\lambda_1(\nu_n)}^{\nu_n}\cap\partial\Omega $ with $\nu(x_n).\nu_n=0$. \\ Passing if necessary to a subsequence, there exist $\bar\lambda\le\lambda_1(\nu)-\epsilon$ and $x\in T_{\bar\lambda}^\nu\cap\partial\Omega$ such that $x_n\to x$ and $\nu(x).\nu=0$. This contradicts the definition of $\lambda_1(\nu)$. \\ (1.b) For all $n\in\mathbb{N}$, there exists $x_n\in(\partial\Omega\cap\overline {\Omega_{\lambda_1(\nu_n)}^{\nu_n}})\setminus T_{\lambda_1(\nu_n)}^{\nu_n}$ with $\left(x_n\right)^{\nu_n}_{\lambda_1(\nu_n)}\in\partial\Omega$. \\ Passing if necessary to a subsequence, there exist $\bar\lambda\le\lambda_1(\nu)-\epsilon$ and $x\in\partial\Omega\cap\overline{\Omega_{\bar\lambda}^\nu}$ such that $x_n\to x$ and $x_{\bar\lambda}^\nu\in\partial\Omega$. If $x\not\in T_{\bar\lambda}^\nu$, we reach a contradiction with the definition of $\lambda_1(\nu)$. Suppose now that $x\in T_{\bar\lambda}^\nu$. Let us denote $(x_n)^{\nu_n}_{\lambda_1(\nu_n)}$ by $u_n$. Since $\Omega$ is a $C^1$ domain, it holds that $\nu(u_n).\nu_n\le0$ for all $n$. By definition of $\lambda_1(\nu_n)$, $\nu(x_n).\nu_n\ge0$. If $x\in T_{\bar\lambda}^\nu$, $x=\lim x_n=\lim u_n$ and so $\nu(x).\nu=0$, which contradicts the definition of $\lambda_1(\nu)$.\\ Observe that we do not use the convexity of the domain in Case 1.\smallskip \noindent\textsc{Case 2:} $\lambda_1(\nu)<\lambda_1(\nu_n)-\epsilon$ for all $n\in\mathbb{N}$:\quad As in the first case, either there exists $x\in T_{\lambda_1(\nu)}^\nu\cap\partial\Omega$ with $\nu(x).\nu=0$ or there exists $x\in(\partial\Omega\cap\overline{\Omega_{\lambda_1(\nu)}^\nu})\setminus T_{\lambda_1(\nu)}^\nu$ such that $x_{\lambda_1(\nu)}^\nu\in\partial\Omega$. We treat the first situation in (2.a) and the second one in (2.b). \\ (2.a) For $\epsilon$ small enough, $T_{\lambda_1(\nu)+\frac{\epsilon}{2}}^\nu\cap \partial\Omega\ne\emptyset$. Since $\Omega$ is strictly convex, there exists $x'\in T_{\lambda_1(\nu)+\frac{\epsilon}{2}}^\nu\cap\partial\Omega$ such that \begin{equation}\label{eau} \nu(x').\nu<0. \end{equation} For $\epsilon>0$ small enough, there exists $n_0\in\mathbb{N}$ such that for all $n\ge n_0$, the sets $T_{\lambda_1 (\nu)+\frac{\epsilon}{2}}^{\nu_n}\cap\partial\Omega$ are non empty and since they are compact, we can choose a sequence $(x_n)$ satisfying $$ x_n \in T_{\lambda_1 (\nu)+\frac{\epsilon}{2}}^{\nu_n}\cap\partial\Omega, \qquad |x'-x_n|=\min\left\{|x'-y|\,:\,y\in T_{\lambda_1 (\nu)+\frac{\epsilon}{2}}^{\nu_n}\cap\partial\Omega\right\}. $$ Passing if necessary to a subsequence, $x_n\to y$ for some $y\in T_{\lambda_1 (\nu)+\frac{\epsilon}{2}}^\nu\cap\partial\Omega$ such that $$ |x'-y|=\lim_{n\to\infty}\mathop{\rm dist}( x',T_{\lambda_1 (\nu)+\frac{\epsilon}{2}}^{\nu_n}\cap\partial\Omega), $$ but since this limit is equal to $0$, we infer that $x'=y$. Now, since $\lambda_1(\nu)<\lambda_1(\nu_n)-\epsilon$ for all $n\in\mathbb{N}$, $\nu(x_n).\nu_n>0$ for all $n$ and thus $\nu(x').\nu\ge0$, a contradiction with (\ref{eau}). \\ (2.b) The convexity of $\Omega$ implies that $ x_{\lambda_1(\nu)+\frac{\epsilon}{2}}^\nu\notin\Omega. $ Now, $ x_{\lambda_1(\nu)+\frac{\epsilon}{2}}^{\nu_n}\to x_{\lambda_1(\nu)+\frac{\epsilon}{2}}^\nu, $ so that \begin{equation}\label{mercure} x_{\lambda_1(\nu)+\frac{\epsilon}{2}}^{\nu_n}\notin\Omega \end{equation} for $n$ large enough. But since $x.\nu<\lambda_1(\nu)$ by definition of $x$, we also have $ x.\nu_n<\lambda_1(\nu)<\lambda_1(\nu)+\frac{\epsilon}{2} $ for $n$ sufficiently large, and so $$x\in\big(\partial\Omega\cap \Omega_{\lambda_1(\nu) +\frac{\epsilon}{2}}^{\nu_n}\big)\setminus T_{\lambda_1(\nu)+\frac{\epsilon}{2}}^{\nu_n}$$ for these values of $n$. This fact together with (\ref{mercure}) contradicts the definition of $\lambda_1(\nu_n)$. The proof of the lower semicontinuity follows from Case 1, which uses only the $C^1$ regularity of the domain. \hfill$\square$ \subsection*{A counterexample in $\mathbb{R}^2$} \begin{figure}[ht] \begin{center} \includegraphics[width=5cm,height=2.5cm]{fig10.eps} \raise1.2cm\hbox{$\lambda_1(\nu_n)>\lambda_1(\nu)+\epsilon$} \end{center} \caption{Counterexample of a smooth convex but not strictly convex domain for which $\lambda_1(\nu)$ is not continuous everywhere.}\end{figure} This is an example of a convex but not strictly convex domain in $\mathbb{R}^2$. It contradicts case (2.a) in the proof and indeed, case (2.a) is the only one using the \emph{strict} convexity. The example can be made smooth. In fact all is required is a convex domain in $\mathbb{R}^2$ whose boundary contains a piece of (straight) line, say of length $L$. Then for $\nu$ parallel to the line, there exists a sequence $\nu_n\to\nu$ such that $\lambda_1(\nu_n)\ge\lambda_1(\nu) +\frac{L}{2}$. A variation of this construction will produce similar examples in higher dimensions. \begin{thebibliography}{10} \frenchspacing \bibitem{Az}C. Azizieh and Ph. Cl\'ement, A priori estimates for positive solutions of p-Laplace equations, \emph{J. Differential Equations}, \textbf{179}, No 1, (2002), 213-245. \bibitem{BN} H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, \emph{Bol. Soc. Brasil. Mat. }\textbf{22} (1991), 1-37. \bibitem{brock} F. Brock, Continuous rearrangement and symmetry of solutions of elliptic problems, \emph{Proc. Indian Acad. Sci. Math. Sci.,} \textbf{110}, No 2, (2000), 157-204. \bibitem{Da} L. Damascelli and F. Pacella, Monotonicity and symmetry of solutions of $p-$Laplace equations, $1