\documentclass[twoside]{article} \usepackage{amsfonts, amsmath} % used for R in Real numbers \pagestyle{myheadings} \markboth{Strongly nonlinear elliptic problem without growth condition } { Aomar Anane \& Omar Chakrone } \begin{document} \setcounter{page}{41} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent 2002-Fez conference on Partial Differential Equations,\newline Electronic Journal of Differential Equations, Conference 09, 2002, pp 41--47. \newline http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Strongly nonlinear elliptic problem without growth condition % \thanks{ {\em Mathematics Subject Classifications:} 49R50, 74G65, 35D05. \hfil\break\indent {\em Key words:} $p$-Laplacian, growth condition. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Published December 28, 2002. } } \date{} \author{Aomar Anane \& Omar Chakrone} \maketitle \begin{abstract} We study a boundary-value problem for the $p$-Laplacian with a nonlinear term. We assume only coercivity conditions on the potential and do not assume growth condition on the nonlinearity. The coercivity is obtained by using similar non-resonance conditions as those in \cite{An-Go}. \end{abstract} \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{prop}[theorem]{Proposition} \newtheorem{coro}[theorem]{Corollary} \section{Introduction} Consider the boundary-value problem \begin{equation} \label{P} \begin{gathered} -\Delta_{p} u = f(x,u) +h \quad \text{in }\Omega,\\ u=0 \quad \text{on }\partial\Omega, \end{gathered} \end{equation} where $\Omega$ is a bounded domain of $\mathbb{R}^N$, $-\Delta_{p}\colon W^{1,p}_{0}(\Omega)\to W^{-1,p'}(\Omega)$ is the $p$-Laplacian operator defined by $$ \Delta_{p}u \equiv \mathop{\rm div} (|\nabla u|^{p-2}\nabla u), \quad 10 \end{equation} and a coercivity condition of the type \eqref{(F)}. We prove that any minimum $u$ of $\Phi$, which is not of class $\mathcal{C}^1$ on $W^{1,p}_{0}(\Omega)$ and may take infinite values too, is a weak solution of \eqref{P} in the sense $$ \int_\Omega |\nabla u|^{p-2}\nabla u \nabla v\,dx =\int_\Omega f(x,u)v\,dx +\langle h,v\rangle, $$ for $v$ in a dense subspace of $W^{1,p}_0(\Omega)$. This result is proved by Degiovanni-Zani~\cite{Degio-Zani1} in the case $p=2$. In the autonomous case $f(x,s)=f(s)$, De~Figueiredo and Gossez~\cite{Defi-Go} have proved the existence of solutions for any $h\in L^{\infty}(\Omega)$ by a topological method. They supposed only a coercivity condition and established that $$ \int_{\Omega}|\nabla u|^{p-2}\nabla u\nabla v\,dx=\int_{\Omega}f(x,u) v\,dx +\langle h,v\rangle $$ for all $v\in W^{1,p}_{0}(\Omega)\cap L^{\infty}(\Omega) \cup \{u\}$ but the solution obtained may not minimize $\Phi$. Indeed, an example is given in~\cite{Defi-Go} in the case $p=2$ and an other one is given in~\cite{Ch} where $p$ may be different from~2. Note that in our case, the condition \eqref{(f_0)} implies no growth condition on $f$ as it may be seen in the following example. \paragraph{Example} Consider the function $$ f(x,s)= \begin{cases} d(x)\Big( \sin (\frac{\pi s}{2})-\frac{\mathop{\rm sign}(s)}{2}\Big) \exp \Big( \frac{2\cos\left( \frac{\pi s}{2}\right)}{\pi} +\frac{|s|-1}{2} \Big) & \text{if } |s|\geq 1\\ d(x)\frac{s}{2}(10 s^{2}-9) &\text{if } |s|\leq 1\,, \end{cases} $$ where $d(x)\in L^{1}_{\rm loc}(\Omega)$ and $d(x)\geq 0$ almost everywhere in $\Omega$, so that $$ F(x,s)= \begin{cases} -d(x)\exp\Big(\frac{2\cos \left( \frac{\pi s}{2}\right)}{\pi} \Big) \exp \Big( \frac{|s|-1}{2} \Big) & \text{if } |s|\geq 1\\ -d(x)\frac{s^{2}}{4}(-5 s^{2}+9) &\text{if } |s|\leq 1\,. \end{cases} $$ Then $F(x,s)\leq 0$ for all $s\in\mathbb{R}$ almost everywhere in $\Omega$. So, $\Phi$ is coercive. Nevertheless, as we can check easily, $f$ satisfies no growth condition. \section{Theoretical approach} We will show that when \eqref{(f_0)} is fulfilled, any minimum $u$ of $\phi$ is a weak solution of \eqref{P} in an acceptable sense. \paragraph{Definition} The space $L^{\infty}_0(\Omega)$ is defined by $$ L^{\infty}_0(\Omega)=\big\{v\in L^{\infty}(\Omega);\;v(x)=0\text{ a.e. outside a compact subset of }\Omega \big\}. $$ For $u\in W^{1,p}_0(\Omega)$, we set $$ V_{u}=\big\{v\in W^{1,p}_0(\Omega)\cap L^{\infty}_0(\Omega);\;u\in L^\infty(\{x\in \Omega;\;v(x)\not=0\}) \big\}. $$ \begin{prop}[Brezis-Browder {\cite{B-B}}]\label{bre-bro-prop} If $u\in W^{1,p}_0(\Omega)$, there exists a sequence $(u_n)_n\subset W^{1,p}_0(\Omega)$ such that: \begin{itemize} \item[(i)] $(u_n)_n\subset W^{1,p}_{0}(\Omega)\cap L^{\infty}_{0}(\Omega)$. \item[(ii)] $|u_n(x)|\leq |u(x)|$ and $u_n(x).u(x) \geq 0$ a.e. in $\Omega$. \item[(iii)] $u_n\to u$ in $W^{1,p}_0(\Omega)$, as $n\to\infty$. \end{itemize} \end{prop} The linear space $V_{u}$ enjoys some nice properties. \begin{prop} \label{prop2} The space $V_u$ is dense in $W^{1,p}_0(\Omega)$. And if we assume that \eqref{(f_0)} holds, then $$ A_u=\big\{\varphi\in W^{1,p}_0(\Omega);\;f(x,u)\varphi\in L^1(\Omega) \big\} $$ is a dense subspace of $W^{1,p}_0(\Omega)$ as $V_u\subset A_u$. More precisely, Brezis-Browder's result holds true if we replace $W^{1,p}_{0}(\Omega)\cap L^{\infty}_{0}(\Omega)$ by $V_{u}$. \end{prop} \paragraph{Proof} It suffices to show that $V_{u}$ is dense in $W^{1,p}_{0}(\Omega)$ and that $V_{u}\subset A_{u}$ when \eqref{(f_0)} holds.\\ \textbf{The density of $V_{u}$ in $W^{1,p}_{0}(\Omega)$:} We have to show that for any $\varphi\in W^{1,p}_{0}(\Omega)$, there exists a sequence $(\varphi_{n})_{n}\subset V_{u}$ satisfying (ii) and (iii). This is done in two steps. First, we show it is true for all $\varphi\in W^{1,p}_{0}(\Omega)\cap L_{0}^{\infty}(\Omega)$. Then, using Proposition~\ref{bre-bro-prop}, we show it is true in $W^{1,p}_{0}(\Omega)$. \noindent \textbf{First Step:} Suppose $\varphi\in W^{1,p}_{0}(\Omega)\cap L_{0}^{\infty}(\Omega)$ and consider a sequence $(\Theta_{n})_{n}\subset \mathcal{C}^{\infty}_{0}(\mathbb{R})$ such that:\\ (1) $\mathop{\rm supp} \Theta_{n}\subset [-n,n]$,\\ (2) $\Theta_n\equiv 1$ on $[-n+1,n-1]$,\\ (3) $0\leq \Theta_{n}\leq 1$ on $\mathbb{R}$ and\\ (4) $|\Theta_{n}'(s)|\leq 2$. The sequence we are looking for is obtained by setting $$ \varphi_{n}(x)=(\Theta_n\circ u)(x)\varphi(x) \quad\text{for a.e. }x \text{ in }\Omega. $$ Indeed, let's check the following three statements\\ (a) $\varphi_{n}\in V_{u}$,\\ (b) $|\varphi_{n}(x)|\leq |\varphi(x)|$ and $\varphi_{n}(x)\varphi(x)\geq 0$ a.e. in $\Omega$ and\\ (c) $\varphi_{n}\to \varphi$ in $W^{1,p}_{0}(\Omega)$.\\ For (a), since $\varphi\in L^{\infty}_{0}(\Omega)$, we have that $\varphi_{n}\in L^{\infty}_{0}(\Omega)$ and it's clear by~(4) that $\varphi_{n}\in W^{1,p}_{0}(\Omega)$. Finally, by (1), $u(x)\in[-n,n]$ for a.e. $x$ in $\{ x\in \Omega;\; \varphi_{n}(x)\not= 0\}$. The assumption (b) is a consequence of (3). For (c), by (2), $\varphi_{n}(x)\to\varphi(x)$ a.e. in $\Omega$ and $$ \frac{\partial \varphi_{n}}{\partial x_{i}}(x)=\Theta'_{n}(u(x))\frac{\partial u}{x_{i}}\varphi(x)+\Theta_{n}(u(x))\frac{\partial \varphi}{\partial x_{i}}\ \to\ \frac{\partial \varphi}{\partial x_{i}}\text{ in }\Omega. $$ And by (4), $$ \Big|\frac{\partial \varphi_{n}}{\partial x_{i}}(x)\Big| \leq 2\Big|\frac{\partial u}{\partial x_{i}}(x)\Big| |\varphi(x)| +\Big|\frac{\partial \varphi}{\partial x_{i}}(x) \Big|\in L^{p}(\Omega). $$ Finally, by the dominated convergence theorem we get (c). \noindent \textbf{Second Step:} Suppose that $\varphi\in W^{1,p}_{0}(\Omega)$. By Proposition~\ref{bre-bro-prop}, there is a sequence $(\psi_{n})_{n}\subset W^{1,p}_{0}(\Omega)$ satisfying (i), (ii) and (iii). For $k=1,2,\ldots$, there is $n_{k}\in \mathbb{N}$ such that $||\psi_{n_{k}}-\varphi||_{1,p}\leq {1}/{k}$. Since $\psi_{n_{k}}\in W^{1,p}_{0}(\Omega)\cap L^{\infty}_{0}(\Omega)$, by the first step, there is $\varphi_{k}\in V_{u}$ such that $|\varphi_{k}(x) |\leq |\psi_{n_{k}}(x)|$ and $\varphi_{k}(x)\psi_{n_{k}}(x)\geq 0$ almost everywhere in $\Omega$ and $||\varphi_{k}-\psi_{n_{k}}||_{1,p}\leq {1}/{k}$, so that $(\varphi_{k})_{k}$ is the sequence we are seeking. Indeed, $|\varphi_{k}(x) |\leq |\psi_{n_{k}}(x)|\leq |\varphi(x) |$, $\varphi_{k}(x)\varphi(x)\geq 0$ a.e. in $\Omega$ and $||\varphi_{k}-\varphi(x)||_{1,p}\leq ||\varphi_{k}-\psi_{n_{k}}||_{1,p} + ||\psi_{n_{k}}-\varphi(x)||_{1,p}\leq {2}/{k}$. \noindent\textbf{The inclusion $V_{u}\subset A_{u}$:} Indeed, for $\varphi\in V_{u}$, set $E=\big\{ x\in \Omega;\; \varphi(x)\neq 0\big\}$ so that $$ \begin{array}{rl} |f(x,u)\varphi | & = \left| f(x,u) \chi_{E}\varphi(x)\right|\\[2mm] & \leq \max \big\{ | f(x,s) \varphi(x)|; |s|\leq||u ||_{L^{\infty}(E)}\big\} \end{array} $$ where $\chi_{E}$ is the characteristic function of the set $E$. By \eqref{(f_0)}, the last term lies to $L^{1}(\Omega)$, so that $\varphi\in A_{u}$. \hfill$\square$ \begin{theorem}\label{thmfond1} Assume \eqref{(f_0)}. If $u\in W^{1,p}_{0}(\Omega)$ is a minimum of $\Phi$ such that $F(x,u)\in L^1(\Omega)$, then \begin{itemize} \item[(i)] $\int_\Omega |\nabla u|^{p-2}\nabla u \nabla \phi\,dx = \int_\Omega f(x,u)\phi\,dx + \langle h,\phi\rangle $ for all $\phi\in A_u$. \item[(ii)] $f(x,u)\in W^{-1,p'}(\Omega)$ in the sense that the mapping $T: V_u\to{\mathbb R} : T(\phi)=\int_\Omega f(x,u)\phi\,dx$ is linear, continuous and admits an unique extension $\tilde{T}$ to the whole space $W_0^{1,p}(\Omega)$. \item[(iii)] $\langle f(x,u),\phi\rangle= \int_\Omega f(x,u)\phi\,dx \ \quad \forall \phi\in A_u$. \item[(iv)] $-\Delta_p u=f(x,u)+h$ in $W^{-1,p'}(\Omega)$. \end{itemize} \end{theorem} \paragraph{Remark} % rmk1 There are in In~\cite{An-Go} some conditions that guarantee the existence of a minimum $u$ of $\Phi$ in $W_0^{1,p}(\Omega)$ and consequently $F(x,u)\in L^1(\Omega)$. \paragraph{Proof of Theorem \ref{thmfond1}} We will prove that the assertion (i) holds for all $\phi\in V_{u}$ as a first step, then prove (iii), (iv) and (i). Let $\phi\in V_{u}$ and $s\in{\mathbb R}$ such that $00\}$, \item[(4)] $f(x,u(x))\geq\eta_1(x)$ a.e. in $\{x\in\Omega;\;u(x)<0\}$ and $f(x,u(x))\leq\eta_2(x)$ a.e. in $\{x\in\Omega;\;u(x)>0\}$. \end{itemize} Then $f(x,u)\in L^1_{\rm loc}(\Omega)$ and consequently $L^{\infty}_c(\Omega)\cap W_0^{1,p}(\Omega)\subset A_u$. \end{coro} \paragraph{Proof} Assume (3) (the same argument works for (4)). Let $\phi\in C_c^{\infty}(\Omega)$. We set $\Omega_1=\{x\in \Omega;\;u(x)\leq -1 \mbox{ a.e.}\}$, $\Omega_2=\{x\in \Omega;\;|u(x)|\leq 1 \mbox{ a.e.}\}$ and $\Omega_3=\{x\in \Omega;\;u(x)\geq 1 \mbox{ a.e.}\}$. It suffices to prove that $f(x,u)|\phi|\chi_{\Omega_i}\in L^1(\Omega)$ for $i=1,2$, $3$. By \eqref{(f_0)} we have $f(x,u)\phi\chi_{\Omega_2}\in L^1(\Omega)$. Let $\theta\in C^{\infty}({\mathbb R})$ : $$ \theta(s)=\begin{cases} 1&\mbox{if } s\geq 1,\\ 0\leq\theta(s)\leq 1 &\mbox{if } 0\leq s\leq 1,\\ 0 &\mbox{if } s\leq 0. \end{cases} $$ It is clear that $(\theta\circ u)|\phi|\in W_0^{1,p}(\Omega)$ and that $$ f(x,u(x))(\theta\circ u(x))|\phi(x)| \geq (\theta\circ u(x))|\phi(x)|\eta_2(x)\in L^1(\Omega). $$ By Proposition \ref{prop4}, we have $f(x,u)(\theta\circ u)|\phi|\in L^1(\Omega) $, then $f(x,u)\phi\chi_{\Omega_3}\in L^1(\Omega)$ (the same argument to prove $f(x,u)\phi\chi_{\Omega_1}\in L^1(\Omega)$). We conclude that $f(x,u)\phi\in L^1(\Omega)$ for all $\phi\in C_c^{\infty}(\Omega)$, which implies $f(x,u)\in L^1_{\rm loc}(\Omega)$. Now assume (1) (the same argument works for (2)). For all $\phi\in C_c^{\infty}(\Omega)$ we have $f(x,u)|\phi|\geq \eta(x)|\phi|\in L^1(\Omega)$, then $f(x,u)|\phi|\in L^1(\Omega)$; therefore, $f(x,u)\phi\in L^1(\Omega)$. Then we conclude that $f(x,u)\in L_{\rm loc}^1(\Omega)$. \hfill$\square$ \begin{thebibliography}{00} \frenchspacing \bibitem{An-Go} A. Anane and J.-P. Gossez, \newblock Strongly non-linear elliptic eigenvalue problems. \newblock \textit{Comm. Partial Diff. Eqns.}, \textbf{15}, 1141--1159, (1990). \bibitem{Degio-Zani1} M. Degiovanni and S. Zani, \newblock Euler equations involving nonlinearities without growth conditions. \newblock \emph{Potential Anal.}, \textbf{5}, 505--512, (1996). \bibitem{Ch} O. Chakrone, \newblock Sur certains probl\`emes non-lin\'eaires \`a la r\'esonance. \newblock D.E.S. Thesis, Faculty of sciences, Oujda, (1995). \bibitem{An} A. Anane, \newblock Simplicit\'e et isolation de la premi\`ere valeur propre du $p$-Laplacien avec poids. \newblock \textit{C. R. Acad. Sci. Paris}, \textbf{305}, 725--728, (1987). \bibitem{jabri-moussaoui2} Y. Jabri and M. Moussaoui, \newblock A saddle point theorem without compactness and applications to semilinear problems. \newblock \emph{Nonlinear Analysis, TMA}, \textbf{32}, No.~3, 363--380, (1997). \bibitem{Defi-Go} D.G. de Figueiredo and J.-P. Gossez, \newblock Un probl\`eme elliptique semi-lin\'eaire sans conditions de croissance. \newblock \emph{C.R. Acad. Sci. Paris}, \textbf{308}, 277--280, (1989). \bibitem{B-B} H. Brezis and F.E. Browder, \newblock A property of Sobolev spaces. \newblock \emph{Comm. Partial Differential equations}, \textbf{4}, 1077--1083, (1979). \end{thebibliography} \noindent\textsc{Aomar Anane} (e-mail: anane@sciences.univ-oujda.ac.ma)\\ \textsc{Omar Chakrone } (e-mail: chakrone@sciences.univ-oujda.ac.ma)\\[2pt] University Mohamed I, Department of Mathematics,\\ Faculty of Sciences, Box 524, 60000 Oujda, Morocco\\ \end{document}