\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 44, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/44\hfil Existence and multiplicity of solutions] {Existence and multiplicity of solutions for a differential inclusion problem involving the $p(x)$-Laplacian} \author[G. Dai\hfil EJDE-2010/44\hfilneg] {Guowei Dai} \address{Guowei Dai \newline Department of Mathematics, Northwest Normal University, Lanzhou, 730070, China} \email{daigw06@lzu.cn} \thanks{Submitted December 31, 2009. Published March 26, 2010.} \thanks{Supported by grants NNSFC 10971087 and NWNU-LKQN-09-1.} \subjclass[2000]{35J20, 35J70, 35R70} \keywords{$p(x)$-Laplacian; nonsmooth mountain pass theorem; \hfill\break\indent differential inclusion} \begin{abstract} In this article we consider the differential inclusion \begin{gather*} -\mathop{\rm div}(|\nabla u|^{p(x)-2}\nabla u)\in \partial F(x,u) \quad\text{in }\Omega,\\ u=0 \quad \text{on }\partial \Omega \end{gather*} which involves the $p(x)$-Laplacian. By applying the nonsmooth Mountain Pass Theorem, we obtain at least one nontrivial solution; and by applying the symmetric Mountain Pass Theorem, we obtain $k$-pairs of nontrivial solutions in $W_{0}^{1,p(x)}(\Omega)$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{proposition}[theorem]{Proposition} \section{Introduction} Let $\Omega$ be bounded open subset of $\mathbb{R}^{N}$ with a $C^1$-boundary $\partial\Omega$. We consider the differential inclusion problem $$\begin{gathered} -\mathop{\rm div}(|\nabla u|^{p(x)-2}\nabla u)\in \partial F(x,u) \quad \text{in }\Omega , \\ u=0\quad \text{on }\partial \Omega, \end{gathered} \label{e1.1}$$ where $p\in C( \overline{\Omega} )$ with $10:\int_{\Omega }|\frac{ u(x)}{\lambda }|^{p(x)}\mathrm{d}x\leq 1\big\} , \] and let $W^{1,p(x) }(\Omega ) =\{ u\in L^{p(x) }(\Omega ) :|\nabla u|\in L^{p(x) }(\Omega ) \}$ with the norm $\| u\| _{W^{1,p(x)}(\Omega )} = |u|_{L^{p(x)}(\Omega )}+|\nabla u|_{L^{p(x)}(\Omega )}.$ Denote by$W_{0}^{1,p(x) }(\Omega ) $the closure of$C_{0}^{\infty }(\Omega ) $in$W^{1,p(x) }(\Omega )$. \begin{proposition}[\cite{f6}] \label{prop2.1} The spaces$L^{p(x)}(\Omega) $,$W^{1,p(x)}(\Omega )$and$W_{0}^{1,p(x) }(\Omega ) $are separable and reflexive Banach spaces. \end{proposition} \begin{proposition}[\cite{f6}] \label{prop2.2} Let$\rho (u)=\int_{\Omega }|u(x)|^{p(x)}\mathrm{d}x$, for$u\in L^{p(x)}(\Omega )$. Then: \begin{itemize} \item[(1)] For$u\neq 0$,$|u|_{p(x)}=\lambda$implies$ \rho (\frac{u}{\lambda })=1$\item[(2)]$|u|_{p(x)}<1(=1;>1) \Leftrightarrow \rho (u)<1(=1;>1)$\item[(3)] If$|u|_{p(x)}>1$, then$|u|_{p(x)}^{p^{-}}\leq \rho (u) \leq |u|_{p(x)}^{p^{+}}$\item[(4)] If$|u|_{p(x)}<1$, then$|u|_{p(x)}^{p^{+}}\leq \rho (u) \leq |u|_{p(x)}^{p^{-}}$\item[(5)]$\lim_{k\to +\infty }|u_{k}| _{p(x)}=0$if and only if$\lim_{k\to +\infty }\rho (u_{k})=0$\item[(6)]$\lim_{k\to +\infty }|u_{k}|_{p(x)}= +\infty$if and only if$\lim_{k\to +\infty }\rho (u_{k})= +\infty$. \end{itemize} \end{proposition} \begin{proposition}[\cite{f6}] \label{prop2.3} In$W_{0}^{1,p(x) }(\Omega ) $the Poincar\'e inequality holds; that is, there exists a positive constant$C_{0}$such that $|u|_{L^{p(x)}(\Omega )}\leq C_{0}| \nabla u|_{L^{p(x)}(\Omega )}, \quad \forall u\in W_{0}^{1,p(x) }(\Omega ).$ \end{proposition} So$|\nabla u|_{L^{p(x)}(\Omega )}$is an equivalent norm in$W_{0}^{1,p(x) }(\Omega)$. We will use the equivalent norm in the following discussion and write$\| u\|=|\nabla u|_{L^{p(x)}(\Omega )}$for simplicity. \begin{proposition}[\cite{f6}] \label{prop2.4} (1) Assume that the boundary of$\Omega$possesses the cone property and$p\in C(\overline{\Omega })$. If$q\in C(\overline{\Omega })$and$1\leq q(x)\leq p^{\ast }(x)$for$x\in \overline{\Omega }$, then there is a continuous embedding$W^{1,p(x)}(\Omega)\hookrightarrow L^{q(x)}(\Omega )$. When$1\leq q(x)< p^{\ast}(x)$, the embedding is compact, where$p^*(x)=\frac{Np(x)}{N-p(x)}$if$p(x)0$depending on$\Omega_{u}$. The generalized directional derivative of$f$at the point$u\in Y$in the direction$v\in X$is $f^{0}(u,v)=\limsup_{w\to u,t\to 0}\frac{1}{t}(f(w+tv)-f(w)).$ The generalized gradient of$f$at$u\in Y$is $\partial f(u)=\big\{u^*\in X^*:\langle u^*,\varphi\rangle \leq f^{0}(u;\varphi) \text{ for all } \varphi\in Y\big\},$ which is a non-empty, convex and$w^*$-compact subset of$Y^*$, where$\langle\cdot,\cdot\rangle$is the duality pairing between$Y^*$and$Y$. We say that$u\in Y$is a critical point of$f$if$0\in \partial f(u)$. For further details, we refer the reader to Chang \cite{c1} or Clarke \cite{c3}. \section{Main results} In this section we give two existence theorems for problem \eqref{e1.1}. For simplicity we write$X=W_{0}^{1,p(x)}(\Omega )$, denote by$c$,$c_i$,$l$and$M$the general positive constant (the exact value may change from line to line). The precise hypotheses are the followings: \begin{itemize} \item[(HF)]$F:\Omega\times \mathbb{R}\to \mathbb{R}$is a Borel measurable locally Lipschitz function with$F(x,0)=0$for a.e.$x\in \Omega$such that \begin{itemize} \item[(i)] there exists a constant$c>0$such that for a.e.$x\in\Omega$, all$u\in \mathbb{R}$and all$\xi(u)\in\partial F(x,u)$$|\xi(u)|\leq c(1+|u|^{\alpha(x)-1}),$ where$\alpha\in C(\overline{\Omega})$and$p^{+}<\alpha^{-}\leq\alpha(x)0$,$\theta>p^+$such that $$0<\theta F(x,u)\leq\langle \xi,u\rangle,\quad \text{a.e. } x\in \Omega, \text{ all } u\in X,\; |u|\geq M,\,\; \xi\in \partial F(x,u);$$ \item[(iii)]$F(x,t)=o(|t|^{p^{+}}), t\to 0$, uniformly for a.e.$x\in\Omega$. \end{itemize} \end{itemize} Because$X$be a reflexive and separable Banach space, there exist$e_i\in X$and$e_j^*\in X^*$such that \begin{gather*} X=\overline{\mathop{\rm span}\{e_i:i=1,2, \ldots\}},\quad X^*=\overline{\mathop{\rm span}\{e_j^*:j=1,2,\ldots\}},\\ \langle e_i,e_j^*\rangle = \begin{cases}1,& i=j,\\ 0, &i\neq j. \end{cases} \end{gather*} For convenience, we write$X_i = \mathop{\rm span}\{e_i\}$,$Y_k =\oplus_{i=1}^kX_i$,$Z_k = \overline{\oplus_{i=k}^\infty X_i}$. In the following we need the nonsmooth version of \emph{Palais-Smale} condition. \begin{definition} \label{def3.1} \rm We say that$I$satisfies the nonsmooth$(\mathrm{PS})_{c}$condition if any sequence$\{u_{n}\}\subset X$such that$I(u_{n})\to c$and$m(u_{n})\to 0$, as$n\to +\infty$, has a strongly convergent subsequence, where$m(u_{n})=\inf\{\| u^*\|_{X^*}:u^*\in \partial I(u_{n})\}$. \end{definition} In what follows we write the$(\mathrm{PS})_{c}$-condition as simply the$\mathrm{PS}$-condition if it holds for every level$c\in \mathbb{R}$for the \emph{Palais-Smale} condition at level$c$. Let $J(u)=\int_{\Omega }\frac{1}{p(x)}|\nabla u|^{p(x)}\mathrm{d}x,\quad \Psi (u)=\int_{\Omega}F(x,u)\mathrm{d}x.$ By a solution of \eqref{e1.1}, we mean a function$u\in X$to which there corresponds a mapping$\Omega\ni x\to g(x)$with$g(x)\in \partial F(x,u)$for a.e.$x\in \Omega$having the property that for every$\varphi\in X$, the function$x\to g(x)\varphi(x)\in L^1(\Omega)$and $\int_{\Omega}|\nabla u|^{p(x)-2}\nabla u\nabla \varphi \mathrm{d}x=\int_{\Omega}g(x)\varphi(x)\mathrm{d}x.$ By standard argument, we show that$u\in X$is a solution of \eqref{e1.1} if and only if$0\in I(u)$, where$I(u)=J(u)-\Psi(u)$. Below we give a proposition that will be used later. \begin{proposition}[\cite{f5}] \label{prop3.1} The functional$J:X\to \mathbb{R}$is convex. The mapping$J':X\to X^{\ast }$is a strictly monotone, bounded homeomorphism, and is of$(S_{+})$type; namely$u_{n}\rightharpoonup u$and$\overline{\lim}_{n\to \infty }(J'(u_{n},u_{n}-u)\leq 0$implies$u_{n}\to u$. \end{proposition} \begin{theorem} \label{thm3.1} If {\rm (HF)} holds, then \eqref{e1.1} has at least one nontrivial solution. \end{theorem} \begin{theorem} \label{thm3.2} If {\rm (HF)} holds and$F(x,-u)=F(x,u)$for a.e.$x\in \Omega$and all$u\in \mathbb{R}$, then \eqref{e1.1} has at least$k$-pairs of nontrivial solutions. \end{theorem} To prove Theorems \ref{thm3.1} and \ref{thm3.2} we need the following generalizations of the classical Mountain pass Theorem (see \cite{c1,g1,k1,k2}) and of the symmetric Mountain pass Theorem \cite{g1,g2}. \begin{lemma} \label{lem3.1} If$X$is a reflexive Banach space,$I:X\to \mathbb{R}$is a locally Lipschitz function which satisfies the nonsmooth {\rm (PS)c}-condition, and for some$r>0$and$e_1\in X $with$\| e_1\|>r$,$\max\{I(0),I(e_1)\}\leq\inf\{ I(u):\| u\|=r\}$. Then$I$has a nontrivial critical$u\in X$such that the critical value$c=I(u)$is characterized by the following minimax principle $c=\inf_{\gamma\in\Gamma}\max_{t\in [0,1]}I(\gamma(t),$ where$\Gamma=\{\gamma\in C([0,1],X): \gamma(0)=0,\gamma(1)=e_1\}$. \end{lemma} \begin{lemma} \label{lem3.2} If$X$is a reflexive Banach space and$I:X\to \mathbb{R}$is even locally Lipschitz functional satisfying the nonsmooth {\rm (PS)c}-condition and \begin{itemize} \item[(i)]$I(0) = 0$; \item[(ii)] there exists a subspace$Y\subseteq X$of finite codimension and number$\beta, \gamma > 0$, such that$\inf \{I(u) : u \in Y \cap \partial B_\gamma (0)\}\geq\beta$, where$B_\gamma = \{u \in X : \| u\| < \gamma\}$and$\partial B_\gamma=\{u \in X : \| u\| = \gamma\}$; \item[(iii)] there is a finite dimensional subspace$V$of$X$with$\mathrm{dim}V>\mathrm{codim}Y$, such that$I(v) \to -\infty$as$\| v\|\to +\infty$for any$v\in V$. \end{itemize} Then$I$has at least$\mathrm{dim} V-\mathrm{codim} Y$pairs of nontrivial critical points. \end{lemma} \section{Proof main results} Let$\widehat{\Psi}$denote its extension to$L^{\alpha(x)}(\Omega)$. We know that$\widehat{\Psi}$is locally Lipschitz on$L^{\alpha(x)}(\Omega)$. In fact, by Proposition \ref{prop2.5}, for$u$,$v\in L^{\alpha(x)}(\Omega)$, we have $$|\widehat{\Psi}(u)-\widehat{\Psi}(v)| \leq\Big(C_1|1|_{\alpha'(x)}+C_2\max_{w\in U}|w^{\alpha(x)-1}|_{\alpha'(x)}\Big)|u-v|_{\alpha(x)},$$ where$U$is an open neighborhood involving$u$and$v$,$w$in the open segment joining$u$and$v$. However, since$\rho(1)=|\Omega|, by Proposition \ref{prop2.2}, we have $$|1|_{\alpha'(x)}<\infty.$$ Meanwhile, since \begin{align*} \rho(w^{\alpha(x)-1}) &= \int_\Omega |w^{\alpha(x)-1}|^{\alpha'(x)}\mathrm{d}x\\ &\leq \int_\Omega |w|^{\alpha(x)}\mathrm{d}x \\ &\leq 2^{\alpha^+}(\int_\Omega | u|^{\alpha(x)}\mathrm{d}x+\int_\Omega | u|^{\alpha(x)}\mathrm{d}x) < \infty, \end{align*} by Proposition \ref{prop2.2}, we also have| w^{\alpha(x)-1}|_{\alpha'(x)}<\infty$. Then, using Proposition \ref{prop2.4} and \cite[Theorem 2.2]{c1}, we have that${\Psi}=\widehat{\Psi}|_{X}$is also locally Lipschitz, and$\partial{\Psi}(u)\subseteq\int_{\Omega}\partial F(x,u)\mathrm{d}x$(see \cite{k3}), where$\widehat{\Psi}|_{X}$stands for the restriction of$\widehat{\Psi}$to$X$. The interpretation of$\partial{\Psi}(u)\subseteq\int_{\Omega}\partial F(x,u)\mathrm{d}x$is as follows: For every$\xi\in\partial{\Psi}(u)$there corresponds a mapping$\xi(x)\in\partial F(x,u)$for a.e.$x\in \Omega$having the property that for every$\varphi\in X$the function$\xi(x)\varphi(x)\in L^1(\Omega)$and$\langle g, \varphi\rangle=\int_{\Omega}\xi(x)\varphi(x)\mathrm{d}x$(see \cite{k3}). Therefore,$I$is a locally Lipschitz functional and we can use the nonsmooth critical point theory. \begin{lemma} \label{lem4.1} If hypotheses {\rm (i)} and {\rm (ii)} hold, then$I$satisfies the nonsmooth {\rm (PS)}-condition. \end{lemma} \begin{proof} Let$\{u_{n}\}_{n\geq1}\subseteq X$be a sequence such that$|I(u_{n})|\leq c$for all$n\geq1$and$m(u_{n})\to 0$as$n\to \infty. Then, from (ii), we have \begin{align*} c&\geq I(u_n)=\int_\Omega\frac{|\nabla u_n|^{p(x)}}{p(x)}\mathrm{d}x-\int_\Omega F(x,u)\mathrm{d}x\\ &\geq \frac{\| u_n\|^{p^-}}{p^+}-\int_\Omega \frac{1}{\theta}\langle \xi(u_n),u_n\rangle\mathrm{d}x-c_1\\ &\geq \big(\frac{1}{p^+}-\frac{1}{\theta}\big)\| u_n\|^{p^-}+\int_\Omega \frac{1}{\theta}(\| u_n\|^{p^-}-\langle \xi(u_n),u_n\rangle)\mathrm{d}x-c_1\\ &\geq \big(\frac{1}{p^+}-\frac{1}{\theta}\big)\| u_n\|^{p^-}- \frac{1}{\theta}\| \xi\|_{X^*}\| u_n\|-c_1. \end{align*} Hence\{u_{n}\}_{n\geq1}\subseteq X$is bounded. Thus by passing to a subsequence if necessary, we may assume that$u_{n}\rightharpoonup u$in$X$as$n\to \infty$. We have $\langle J'(u_{n}),u_{n}-u\rangle -\int_{\Omega}\xi_{n}(x)(u_{n}-u)\mathrm{d}x \leq\varepsilon_{n}\| u_{n}-u\|$ with$\varepsilon_{n}\downarrow0$, where$\xi_n\in \partial\Psi(u_n)$. From Chang \cite{c1} we know that$\xi_n\in L^{\alpha'(x)}(\Omega)$($\alpha'(x)=\frac{\alpha(x)}{\alpha(x)-1}$). Since$X$is embedded compactly in$L^{\alpha(x)}(\Omega)$, we have that$u_{n}\to u$as$n\to \infty$in$L^{\alpha(x)}(\Omega)$. So using Proposition \ref{prop2.5}, we have $$\int_{\Omega}\xi_{n}(x)(u_{n}-u)\,dx\to 0 \quad \text{as } n\to \infty.$$ Therefore we obtain$\limsup_{n\to \infty}\langle J'(u_{n}),u_{n}-u\rangle\leq0$. But we know that$J'$is a mapping of type ($S_{+}$). Thus we have $u_{n}\to u \quad \text{in } X.$ \end{proof} \begin{lemma} \label{lem4.2} If hypotheses {\rm (i), (iii)} hold, then there exist$r>0$and$\delta>0$such that$I(u)\geq\delta>0$for every$u\in X$and$\| u\|=r$. \end{lemma} \begin{proof} Let$\varepsilon>0$be small enough such that$\varepsilon c_{0}^{p^{+}}\leq\frac{1}{2p^{+}}$, where$c_{0}$is the embedding constant of$X\hookrightarrow L^{p^+}(\Omega)$. From hypothesis (i) and (iii), we have $$\label{e4.4} F(x,t) \leq\varepsilon|t|^{p^{+}}+c(\varepsilon)|t|^{\alpha(x)}.$$ Therefore, for every$u\in X, we have \begin{align*} I(u)&\geq \frac{1}{p^{+}}\| u\|^{p^{+}}-\varepsilon c_{0}^{p^{+}}\| u\|^{p^{+}}-c(\varepsilon)\| u\|^{\alpha^{-}} \\ &\geq \frac{1}{2p^{+}}\| u\|^{p^{+}}-c(\varepsilon)\| u\|^{\alpha^{-}}, \end{align*} when\| u\|\leq 1$. So we can find$r>0$small enough and$\delta>0$such that$I(u)\geq\delta>0$for every$u\in X$and$\| u\|=r$. \end{proof} \begin{lemma} \label{lem4.3} If hypotheses {\rm (ii)} holds, then there exists$u_1\in X$such that$I(u_1)\leq0$. \end{lemma} \begin{proof} From (ii), there exist$M>0$,$c_2>0$such that (see \cite[p. 298]{g1}) $F(x,u)\geq c_2|u|^{\theta}$ for all$|u|> M$and a.e.$x\in \Omega$. Thus for$1M\}}F(x,tu)\mathrm{d}x+ \int_{\{t|u|\leq M\}}F(x,tu)\mathrm{d}x \\ &\geq c_2t^{\theta}\int_{\{t|u|> M\}}| u|^{\theta}\mathrm{d}x-c_3. \end{align*} Therefore, for $t>1$, we have \label{e4.5} \begin{aligned} I(tu)&\leq \frac{1}{p^{-}}t^{p^{+}}\int_{\Omega}|\nabla u|^{p(x)}\mathrm{d}x-c_2t^{\theta}\int_{\{t| u|> M\}}|u|^{\theta}\mathrm{d}x+c_3\\ &= \frac{1}{p^{-}}t^{p^{+}}\int_{\Omega}|\nabla u|^{p(x)}\mathrm{d}x-c_2t^{\theta}\int_{\Omega}| u|^{\theta}\mathrm{d}x+c_2t^{\theta}\int_{\{t|u|\leq M\}}|u|^{\theta}\mathrm{d}x+c_3. \end{aligned} Noting that $c_2t^{\theta}\int_{\{t|u|\leq M_{3}\}}| u|^{\theta}$ is bounded, it follows that $I(tu)\to -\infty\quad \text{as } t\to +\infty.$ \end{proof} \begin{proof}[Proof of Theorem \ref{thm3.1}] Using Lemma \ref{lem3.1} and Lemmas \ref{lem4.1}-\ref{lem4.3}, we can find an $u\in X$ such that $I(u)>0$ (hence $u\neq0$) and $0\in\partial I(u)$. Hence $u\in X$ is a nontrivial solution of \eqref{e1.1}. \end{proof} \begin{proof}[Proof of Theorem \ref{thm3.2}] Firstly, we can easily see that $I$ is even functional on $X$. We claim that $I(u)\to -\infty$ as $\| u\|\to +\infty$, for any $u\in Y_k$. We assume $\| u\|\geq1$. From \eqref{e4.4}, we have $I(u)\leq\frac{1}{p^{-}}\| u\|^{p^{+}}-c_4| u|_{\theta}^{\theta}+c_4\int_{\{|u|\leq M\}}| u|^{\theta}\mathrm{d}x+c_5.$ Since $Y_k$ is finite dimensional, all norms of $Y_k$ are equivalent. For $p^+<\theta$, we get $I(u)\to -\infty$ as $\| u\|\to +\infty$. We can apply Lemma \ref{lem3.2} with $V=Y_k$ and $Y=X$. From Lemma \ref{lem4.1} and Lemma \ref{lem4.2}, we get $k$-pairs of nontrivial critical points, which are solutions of \eqref{e1.1}. \end{proof} We remark that using the same method as in hte proof of Theorems \ref{thm3.1} and \ref{thm3.2}, we can obtain the same results for the corresponding differential inclusion problems with Neumann boundary data. 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