\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{equation} \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} \end{equation} where $p\in C( \overline{\Omega} )$ with $1
0:\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 $p(x) 0$, $\theta>p^+$ such that
\begin{equation}
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);
\end{equation}
\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
\begin{equation}
|\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)},
\end{equation}
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
\begin{equation}
|1|_{\alpha'(x)}<\infty.
\end{equation}
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
\begin{equation}
\int_{\Omega}\xi_{n}(x)(u_{n}-u)\,dx\to 0 \quad \text{as }
n\to \infty.
\end{equation}
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
\begin{equation} \label{e4.4}
F(x,t) \leq\varepsilon|t|^{p^{+}}+c(\varepsilon)|t|^{\alpha(x)}.
\end{equation}
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
$1