\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 $$\label{P} \begin{gathered} -\Delta_{p} u = f(x,u) +h \quad \text{in }\Omega,\\ u=0 \quad \text{on }\partial\Omega, \end{gathered}$$ 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 $$\Delta_{p}u \equiv \mathop{\rm div} (|\nabla u|^{p-2}\nabla u), \quad 10 and a coercivity condition of the type \eqref{(F)}. We prove that any minimum u of \Phi, which is not of class \mathcal{C}^1 on W^{1,p}_{0}(\Omega) and may take infinite values too, is a weak solution of \eqref{P} in the sense$$ \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}