\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 33, pp. 1--11.\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-2010/33\hfil Existence of multiple solutions]
{Existence of multiple solutions for a \\
 $p(x)$-Laplace equation}

\author[Duchao Liu\hfil EJDE-2010/33\hfilneg]
{Duchao Liu}

\address{Duchao Liu \newline
School of Mathematics and Statistics,
Lanzhou University, Lanzhou 730000, China}
\email{liudch06@lzu.cn}

\thanks{Submitted September 26, 2008. Published March 3, 2010.}
\thanks{Supported by grant 10671084 from the National Natural Science
Foundation of China}
\subjclass[2000]{35B38, 35D05, 35J20}
\keywords{Critical points; $p(x)$-Laplacian; integral
functionals; \hfill\break\indent generalized Lebesgue-Sobolev spaces}

\begin{abstract}
 This article shows the existence of at least three
 nontrivial solutions to the quasilinear elliptic equation
 \[
 -\Delta_{p(x)}u+|u|^{p(x)-2}u=f(x,u)
 \]
 in a smooth bounded domain $\Omega\subset\mathbb{R}^{n}$,
 with the nonlinear boundary condition
 $|\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu}=g(x,u)$
 or the Dirichlet boundary condition $u=0$ on $\partial\Omega$.
 In addition, this paper  proves that one solution is positive,
 one is negative, and the last one is a sign-changing solution.
 The method used here is based on Nehari results, on three
 sub-manifolds of the space $W^{1,p(x)}(\Omega)$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}

\section{Introduction}

 The study of variational problems with nonstandard growth
conditions is an interesting topic. $p(x)$-growth conditions can
be regarded as an important case of nonstandard $(p,q)$-growth
conditions. Many results have been obtained on this kind of
problems; see for example \cite{a1,a2,f5,f6,m1}.
We refer the reader to the overview papers \cite{h1,s2}
for advances and references in this area.

In this article, we consider the non-homogeneous nonlinear Neumann
boundary\-value problem
\begin{equation} \label{eP}
\begin{gathered}
-\Delta_{p(x)}u+|u|^{p(x)-2}u=f(x,u),\quad x\in\Omega,\\
|\nabla u|^{p(x)-2}\frac{\partial
u}{\partial\nu}=g(x,u),\quad x\in\partial\Omega;
\end{gathered}
\end{equation}
and the problem
\begin{equation}\label{ePprime}
\begin{gathered}
-\Delta_{p(x)}u=f(x,u), \quad x\in\Omega,\\
u=0,\quad x\in\partial\Omega,
\end{gathered}
\end{equation}
where $\Omega\subset\mathbb{R}^{n}$ is a bounded domain with
Lipschitz boundary $\partial\Omega$, $\frac{\partial}{\partial\nu}$
is the outer unit normal derivative,
$ p(x)\in C(\overline{\Omega})$, $\inf_{x\in\Omega}p(x)>1$.

The operator $-\Delta_{p(x)}u:=-div(|\nabla u|^{p(x)-2}\nabla u)$ is
called $p(x)$-Laplacian, which becomes $p$-Laplacian when
$p(x) \equiv p$ (a constant). It possesses more complicated nonlinearities
than the $p$-Laplacian. For related results involving the Laplace
operator, see \cite{a1,h1,s2}.

In this article,  we construct three sub-manifolds of the space
$W^{1,p(x)}(\Omega)$ based on Naheri ideas. And under some
assumptions, there exist three different, nontrivial solutions of
\eqref{eP} or \
eqref{ePprime} on sub-manifold respectively. Moreover these
solutions are, one positive, one negative and the other one has
non-constant sign. This result extends the conclusions of \cite{s1}, and
in \cite{b1}, the author also do some research with the similar method.

Throughout this paper, by (weak) solutions of \eqref{eP} or \eqref{ePprime} we
understand critical points of the associated energy functional
$\Phi$ or $\Psi$ acting on the Sobolev space $W^{1,p(x)}(\Omega)$:
\begin{gather*}
\Phi(v)=\int_{\Omega}\frac{1}{p(x)}(|\nabla
v|^{p(x)}+|v|^{p(x)})\,dx-\int_\Omega F(x,v)\,dx-\int_{\partial\Omega}
G(x,v)dS;\\
\Psi(v)=\int_{\Omega}\frac{1}{p(x)}|\nabla v|^{p(x)}\,dx-\int_\Omega
F(x,v)\,dx,
\end{gather*}
where $F(x,u)=\int^{u}_{0}f(x,z)dz$,
$G(x,u)=\int^{u}_{0}g(x,z)dz$
and $dS$ is the surface measure. We also denote
$\mathcal {F}(v)=\int_{\Omega}F(x,v)\,dx$ and
$\mathcal {G}(v)=\int_{\partial\Omega}G(x,v)dS$.

For a reflexive manifold $M$, $T_{u}M$ denotes the tangential space
at a point $u\in M$, $T_{u}^{*}M$ denotes the co-tangential space.

In this paper span$\{v_{1},\dots,v_{k}\}$ denotes the vector space
generated by the vectors $v_{1},\dots,v_{k}$.



\section{The space $W^{1,p(x)}(\Omega)$}

 To discuss problem \eqref{eP}
or \eqref{ePprime}, we need some results on the space $W^{1,p(x)}(\Omega)$
which we call variable exponent Sobolev space. Firstly, we state some
basic properties of the space $W^{1,p(x)}(\Omega)$ which will be
used later (for more details, see \cite{f2,f3,f4}).

Let $\Omega$ be a bounded domain of $\mathbb{R}^{n}$, denote:
\begin{gather*}
C_{+}(\overline{\Omega})=\{h\in
C(\overline{\Omega});h(x)>1\,\forall x\in\overline{\Omega}\},\\
h^{+}=\max_{x\in\overline{\Omega}}h(x),\quad
h^{-}=\text{min}_{x\in\overline{\Omega}}h(x),\quad
 h\in C(\overline{\Omega}),\\
L^{p(x)}(\Omega)=\{u: u\text{ is a measurable real-valued
function, }\int_{\Omega}|u|^{p(x)}\,dx<\infty\}.
\end{gather*}

We  introduce a norm on $L^{p(x)}(\Omega)$:
\[
|u|_{p(x)}=\inf \big\{\lambda>0: \int_{\Omega}
\big|\frac{u(x)}{\lambda}\big|^{p(x)}\,dx\leq 1\big\}.
\]
Then $(L^{p(x)}(\Omega),|\cdot|_{p(x)})$ becomes a Banach space, we
call it a generalized Lebesgue space.

\begin{proposition}[\cite{f5,f6}] \label{prop2.1} \quad
\begin{itemize}
\item[(1)] The space $(L^{p(x)}(\Omega)$, $|\cdot|_{p(x)})$ is  a separable,
uniformly convex Banach space, and has conjugate space
$L^{q(x)}(\Omega)$, where $1/q(x)+1/p(x)=1$.
For $u\in L^{p(x)}(\Omega)$ and $v\in L^{q(x)}(\Omega)$, we have
\[
\big|\int_{\Omega}uv\,dx\big|\leq
(\frac{1}{p^{-}}+\frac{1}{q^{-}})|u|_{p(x)}|v|_{q(x)};
\]

\item[(2)] If  $p_{1}, p_{2}\in C_{+}(\overline{\Omega})$,
$p_{1}(x)\leq  p_{2}(x)$, for any $x\in\Omega$, then
$L^{p_{2}(x)}(\Omega)\hookrightarrow L^{p_{1}(x)}(\Omega)$, and the
imbedding is continuous.
\end{itemize}
\end{proposition}

 \begin{proposition}[\cite{f5,z1}] \label{prop2.2}
If $f:\Omega\times \mathbb{R}\to \mathbb{R}$
is a Caratheodory function and satisfies
\[
|f(x,s)|\leq  a(x)+b|s|^{\frac{p_{1}(x)}{p_{2}(x)}},\quad
\text{for any }x\in\Omega,s\in\mathbb{R},
\]
where $p_{1}(x)$, $p_{2}(x)\in C_{+}(\overline{\Omega})$,
$a(x)\in L^{p_{2}(x)}(\Omega)$, $a(x)\geq 0$, and
$b\geq 0$ is a constant, then the Nemytsky operator
from $L^{p_{1}(x)}(\Omega)$ to
$L^{p_{2}(x)}(\Omega)$ defined by $(N_{f}(u))(x)=f(x,u(x))$ is a
continuous and bounded operator.
\end{proposition}

\begin{proposition}[\cite{f5,z2}] \label{prop2.3}
Denote
\begin{align*}
\rho(u)=\int_{\Omega}|u|^{p(x)}\,dx,\quad \forall u\in
L^{p(x)}(\Omega).
\end{align*}
Then \begin{itemize}
\item[(1)] $|u(x)|_{p(x)}<1(=1;>1)$ if and only if
$\rho(u)<1(=1;>1)$;

\item[(2)] $|u(x)|_{p(x)}>1$ implies
$|u|_{p(x)}^{p^{-}}\leq \rho(u)\leq |u|_{p(x)}^{p^{+}}$;
$|u(x)|_{p(x)}<1$ implies
$|u|_{p(x)}^{p^{-}}\geq \rho(u)\geq |u|_{p(x)}^{p^{+}}$;

\item[(3)] $|u(x)|_{p(x)}\to 0$ if and only if
$\rho(u)\to 0$;
$|u(x)|_{p(x)}\to \infty$ if and only if
$\rho(u)\to \infty$.
\end{itemize}
\end{proposition}

\begin{proposition}[\cite{f5,z2}] \label{prop2.4}
 If $u, u_{n}\in L^{p(x)}(\Omega)$, $n=1,2,\dots$, then
the following statements are equivalent:
\begin{itemize}
\item[(1)] $\lim_{k\to \infty}|u_{k}-u|_{p(x)}=0$;
\item[(2)] $\lim_{k\to \infty}\rho(u_{k}-u)=0$;
\item[(3)] $u_{k}\to  u$ in measure in $\Omega$ and
$\lim_{k\to \infty}\rho(u_{k})=\rho(u)$.
\end{itemize}
\end{proposition}

Let us define the space
\[
W^{1,p(x)}(\Omega)=\{u\in L^{p(x)}(\Omega): |\nabla u|\in
L^{p(x)}(\Omega)\},
\]
 equipped with the norm
\begin{align*}
\|u\|=|u|_{p(x)}+|\nabla u|_{p(x)},\quad \forall u\in
W^{1,p(x)}(\Omega).
\end{align*}
Let $W^{1,p(x)}_{0}(\Omega)$ be the closure of
$C_{0}^{\infty}(\Omega)$ in
 $W^{1,p(x)}(\Omega)$ and $p^{*}(x)=np(x)/(n-p(x))$,
$p^{*}_{\partial}(x)=(n-1)p(x)/(n-p(x))$, when $p(x)<n$.


\begin{proposition}[\cite{f1,f5}]  \label{prop2.5}
\begin{itemize}
\item[(1)] $W^{1,p(x)}_{(0)}(\Omega)$ is separable reflexive Banach space;

\item[(2)] If $q\in C_{+}(\overline{\Omega})$ and $q(x)<p^{*}(x)$ for any
$x\in\overline{\Omega}$, then the imbedding from
$W^{1,p(x)}(\Omega)$
to $L^{q(x)}(\Omega)$ is compact and continuous;

\item[(3)] If $q\in C_{+}(\overline{\Omega})$ and
$q(x)<p_{\partial}^{*}(x)$ for any $x\in\overline{\Omega}$, then the
trace imbedding from $W^{1,p(x)}(\Omega)$ to
$L^{q(x)}(\partial\Omega)$ is compact and
continuous;

\item[(4)](Poincar\'e) There is a constant $C>0$, such that
\[
|u|_{p(x)}\leq  C|\nabla u|_{p(x)}\quad \forall u\in
W^{1,p(x)}_{0}(\Omega).
\]
\end{itemize}
\end{proposition}


\section{Assumptions and statement of main result}

 The  assumptions on the source terms $f$ and $g$ are
as follows:
\begin{itemize}
\item[(F1)] $f:\Omega\times\mathbb{R}\to \mathbb{R}$ is a measurable
function with respect to the first argument and continuous
differentiable with respect to the second argument for a.e.
$x\in\Omega$. Moreover, $f(x,0)=0$ for every $x\in\overline{\Omega}$.

\item[(F2)] There exist %functions
 $q(x),s(x),t(x)$, such that
$p(x)\leq  p^{+}< q^{-}\leq  q(x)<p^{*}(x)$,
$s(x)>{p^{*}(x)}/{(p^{*}(x)-q(x))}$,
\[
t(x)=s(x)q(x)/(2+(q(x)-2)s(x))>{p^{*}(x)}/{(p^{*}(x)-2)}
\]
and there exist functions $a(x)\in L^{s(x)}(\Omega)$,
$b(x)\in L^{t(x)}(\Omega)$, such that for
$x\in\Omega$, $u\in\mathbb{R}$,
\[
|f_{u}(x,u)|\leq  a(x)|u|^{q(x)-2}+b(x).
\]

\item[(F3)] There exist constants $c_{1}\in(0,1/(p^{+}-1))$,
$c_{2}>p^{+}$, $0<c_{3}<c_{4}$, such that for any
 $u\in L^{q(x)}(\Omega)$,
\begin{align*}
c_{3}\int_{\Omega}|u|^{q(x)}\,dx
&\leq  c_{2}\int_{\Omega}F(x,u)\,dx\leq \int_{\Omega}f(x,u)u\,dx\\
&\leq  c_{1}\int_{\Omega}f_{u}(x,u)u^{2}\,dx\leq
c_{4}\int_{\Omega}|u|^{q(x)}\,dx.
\end{align*}

\item[(G1)] $g:\partial\Omega\times\mathbb{R}\to \mathbb{R}$ is a
measurable function with respect to the first argument and
continuous differentiable with respect to the second argument for
a.e. $x\in\partial\Omega$. Moreover, $g(x,0)=0$ for every
$x\in\partial\Omega$.

\item[(G2)] There exist functions $r(x),\sigma(x),\tau(x)$, such that
$p(x)\leq  p^{+}<r^{-}\leq  r(x)< p^{*}_{\partial}(x)$,
$\sigma(x)>{p^{*}_{\partial}(x)}/{(p^{*}_{\partial}(x)-r(x))}$,
$\tau(x)=\sigma(x)r(x)/(2+(r(x)-2)\sigma(x))>
{p^{*}_{\partial}(x)}/{(p^{*}_{\partial}(x)-2)}$ and there exist
functions $\alpha(x)\in L^{\sigma(x)}(\partial\Omega)$, $\beta(x)\in
L^{\tau(x)}(\partial\Omega)$, such that for $x\in\partial\Omega$,
$u\in\mathbb{R}$,
\[
|g_{u}(x,u)|\leq  \alpha(x)|u|^{r(x)-2}+\beta(x).
\]

\item[(G3)] There exist constants $k_{1}\in(0,1/(p^{+}-1))$,
$k_{2}>p^{+}$, $0<k_{3}<k_{4}$, such that for any
$u\in L^{r(x)}(\partial\Omega)$
satisfies
\begin{align*}
k_{3}\int_{\Omega}|u|^{r(x)}\,dx
&\leq  k_{2}\int_{\partial\Omega}G(x,u)dS\leq \int_{\partial\Omega}g(x,u)udS\\
&\leq  k_{1}\int_{\partial\Omega}g_{u}(x,u)u^{2}dS\leq
k_{4}\int_{\Omega}|u|^{r(x)}\,dx.
\end{align*}
\end{itemize}

We remark that by assumptions (F1), (F2) there exists
$\pi(x)<p^{*}(x)$ such that the operators $u\to  f(u)$ is
bounded and Lipschitz continuous from $L^{\pi(x)}(\Omega)\to
(W^{1,p(x)}(\Omega))^{-1}$. Moreover, by the compactness of the
embedding $W^{1,p(x)}(\Omega)\hookrightarrow L^{\pi(x)}(\Omega)$ the
operator is compact if viewed as acting on $W^{1,p(x)}(\Omega)$.
Hence the expression
\[
\psi(u,\cdot)=\langle\nabla\Phi(u),\cdot\rangle
\]
is well defined and Lipschitz continuously differentiable in
$W^{1,p(x)}(\Omega)$, where $\langle\cdot,\cdot\rangle$ is the
duality pairing between $(W^{1,p(x)}(\Omega))^{-1}$ and
$W^{1,p(x)}(\Omega)$.

The main result of the paper reads as follows:

\begin{theorem} \label{thm3.1}
 Under assumptions {\rm (F1)-(F3), (G1)-(G3)},
there exist three different, non trivial solutions of \eqref{eP} and
\eqref{ePprime}. Moreover these solutions are, one positive, one negative and
the other one has non-constant sign.
\end{theorem}

\section{The proof for the Neumann problem}

 The proof uses the similar approach as in \cite{s1}. That is, we
will construct three disjoint sets $K_{i}\neq\emptyset$ not
containing 0 such that $\Phi$ has a critical point in $K_{i}$. These
sets will be subsets of smooth manifolds
$N_{i}\in W^{1,p(x)}(\Omega)$ which we will obtain by imposing a sign
condition and a normalization condition on admissible functions.
Let
\begin{gather*}
N_{1}=\{u\in W^{1,p(x)}(\Omega):\int_{\Omega}
u_{+}\,dx>0,\langle\nabla\Phi(u),u^{+}\rangle=0\},\\
N_{2}=\{u\in W^{1,p(x)}(\Omega):\int_{\Omega}
u_{-}\,dx>0,\langle\nabla\Phi(u),u^{-}\rangle=0\},\\
N_{3}=N_{1}\cap N_{2},
\end{gather*}
where $u_{+}=\max\{u,0\}$, $u_{-}=\max\{-u,0\}$ are the positive and
negative parts of $u$.

In fact, following Nehari \cite{n1}, conditions in $N_{i}$, $(i=1,2,3)$,
are \emph{norming conditions} for the positive and negative parts of
$u$.

Also we define
\[
K_{1}=\{u\in N_{1}|u\geq 0\},\quad
K_{2}=\{u\in N_{2}|u\leq 0\},K_{3}=N_{3}.
\]
To prove that the sets $N_{i},K_{i}$ possess the properties stated at
the beginning of the paragraph we establish the following
estimate:

\begin{lemma} \label{lem4.1}
There exist constants $c_{j}>0, j=1,2$,
such that for any $u\in K_{i}$,
\begin{align*}
c_{1}\min\{\|u\|^{p^{-}}, \|u\|^{p^{+}}\}
&\leq \int_{\Omega}(|\nabla u|^{p(x)}+|u|^{p(x)})\,dx\\
&\leq \int_{\Omega}f(x, u)u\,dx+\int_{\partial\Omega}g(x, u)udS\\
&\leq  c_{2}\Phi(u).
\end{align*}
\end{lemma}

\begin{proof} Since $u\in K_{i}$, we have
\[
\int_{\Omega}(|\nabla u|^{p(x)}+|u|^{p(x)})\,dx=\int_{\Omega}f(x,
u)u\,dx+\int_{\partial\Omega} g(x, u)udS.
\]
This proves the second inequality. Now, by (F3) and (G3)
\begin{gather*}
\int_{\Omega}F(x, u)\,dx \leq \frac{1}{c_{2}}\int_{\Omega}f(x, u)u\,dx, \\
\int_{\partial\Omega}G(x,
u)dS \leq \frac{1}{k_{2}}\int_{\partial\Omega}g(x, u)udS.
\end{gather*}
So for $c=\max\{1/c_{2}, 1/k_{2}\}<1/p^{+}$,
\[
\Phi(u)\geq (\frac{1}{p^{+}}-c)\int_{\Omega}|\nabla
u|^{p(x)}+|u|^{p(x)}\,dx.
\]
This proves the third inequality. The first inequality is easily
obtained by proposition \ref{prop2.3}.
\end{proof}

\begin{lemma} \label{lem4.2}
There exists $c>0$ such that
\begin{gather*}
\int_{\Omega}(|\nabla u|^{p(x)}+|u|^{p(x)})\,dx\geq  c\quad
\emph{for }u\in K_{1}, \\
\int_{\Omega}(|\nabla u|^{p(x)}+|u|^{p(x)})\,dx\geq  c\quad
\emph{for }u\in K_{2}, \\
\int_{\Omega}(|\nabla u_{\pm}|^{p(x)}+|u_{\pm}|^{p(x)})\,dx\geq c\quad
\emph{for }u\in K_{3}.
\end{gather*}
\end{lemma}

\begin{proof}
By the definitions of $K_{i}$, conditions (F3), (G3)
and proposition \ref{prop2.3},
\begin{align*}
&\int_{\Omega}(|\nabla u_{\pm}|^{p(x)}+|u_{\pm}|^{p(x)})\,dx \\
&=\int_{\Omega}f(x, u_{\pm})u_{\pm}\,dx+\int_{\partial\Omega}g(x, u_{\pm})u_{\pm}dS\\
&\leq  c(\int_{\Omega}|u_{\pm}|^{q(x)}\,dx+\int_{\Omega}|u_{\pm}|^{r(x)}\,dx)\\
&\leq  c(\|u_{\pm}\|^{q^{\pm}}+\|u_{\pm}\|^{r^{\pm}})\\
&\leq  c(|\nabla
u_{\pm}|_{p(x)}^{q^{\pm}}+|u_{\pm}|_{p(x)}^{q^{\pm}}+|\nabla u_{\pm}|_{p(x)}^{r^{\pm}}+|u_{\pm}|_{p(x)}^{r^{\pm}})\\
&\leq  c[(\int_{\Omega}|\nabla
u_{\pm}|^{p(x)}+|u_{\pm}|^{p(x)}\,dx)^{\frac{q^{\pm}}{p\pm}}+(\int_{\Omega}|\nabla
u_{\pm}|^{p(x)}+|u_{\pm}|^{p(x)}\,dx)^{\frac{r^{\pm}}{p\pm}}].
\end{align*}
Here by $U^{p^{\pm}}$ we mean $\max\{U^{p^{+}},U^{p^{-}}\}$.
Then obviously the conclusion holds since $p^{+}<q^{-},r^{-}$.
\end{proof}

\begin{lemma} \label{lem4.3}
There exists $c>0$ such that
$\Phi(u)\geq  c\int_{\Omega}(|\nabla u|^{p(x)}+|u|^{p(x)})\,dx$
for every $u\in W^{1, p(x)}(\Omega)\emph{ s.t. }\|u\|\leq  c$.
\end{lemma}

\begin{proof}
 By (F3), (G3) and the Sobolev inequalities in
proposition \ref{prop2.5} we have
\begin{align*}
\Phi(u)
&=\int_{\Omega}\frac{1}{p(x)}(|\nabla u|^{p(x)}+|u|^{p(x)})\,dx-\mathcal {F}(u)-\mathcal {G}(u)\\
&\geq  \int_{\Omega}\frac{1}{p(x)}(|\nabla
u|^{p(x)}+|u|^{p(x)})\,dx-c(\int_{\Omega}|u|^{q(x)}\,dx+\int_{\Omega}|u|^{r(x)}\,dx)\\
&\geq  \frac{1}{p^{+}}\int_{\Omega}(|\nabla
u|^{p(x)}+|u|^{p(x)})\,dx-c(\int_{\Omega}|u|^{q(x)}\,dx+\int_{\Omega}|u|^{r(x)}\,dx)\\
&\geq  c\int_{\Omega}(|\nabla u|^{p(x)}+|u|^{p(x)})\,dx,
\end{align*}
if $\|u\|$ is sufficiently small, since
$p^{+}< q^{-},r^{-}$ ,for all $x\in \Omega$, and the embedding
from $W^{1,p(x)}(\Omega)$ to $L^{q(x)}(\Omega)$.
\end{proof}

\begin{lemma} \label{lem4.4}
Put $P^{+}(u)=u_{+}$, $P^{-}(u)=u_{-}$,
then the mapping $P^{+}$ and $P^{-}$: $W^{1,p(x)}(\Omega)\to
W^{1,p(x)}(\Omega)$ is continuous.
\end{lemma}

\begin{proof}
Here we prove only that the projection
$P^{+}:W^{1,p(x)}(\Omega)\to  W^{1,p(x)}(\Omega)$ is
continuous. Let $u_{n}\to  u$ in $W^{1,p(x)}(\Omega)$, we
prove $(u_{n})_{+}\to  u_{+}$ in $W^{1,p(x)}(\Omega)$.
It is obvious the following inequality holds:
\[
|(u_{n})_{+}(x)-u_{+}(x)|\leq |u_{n}(x)-u(x)|,\quad \text{for a.e.
}x\in\Omega,
\]
which implies
\[
|(u_{n})_{+}(x)-u_{+}(x)|^{p(x)}\leq |u_{n}(x)-u(x)|^{p(x)},\quad
\text{for a.e. }x\in\Omega.
\]
So $\int_{\Omega}|u_{n}(x)-u(x)|^{p(x)}\,dx\to  0$ implies
$\int_{\Omega}|(u_{n})_{+}(x)-u_{+}(x)|^{p(x)}\,dx\to 0 $. Then
$(u_{n})_{+}\to  u_{+}$ in $L^{p(x)}(\Omega)$.

Next we  prove $|\nabla (u_{n})_{+}-\nabla
u_{+}|_{p(x)}\to 0$.
Define $g:\mathbb{R}\to \mathbb{R}$ by
\[
g(t)=\begin{cases}
 1,& t>0,\\
 0,& t\leq 0,
\end{cases}
\]
then
\begin{align*}
&|\nabla (u_{n})_{+}(x)-\nabla u_{+}(x)|\\
&=|g(u_{n}(x))\nabla
u_{n}(x)-g(u(x))\nabla u(x)|\\
&\leq  |g(u_{n}(x))(\nabla u_{n}(x)-\nabla
u(x))|+|(g(u_{n}(x))-g(u(x)))\nabla u(x)|\\
&\leq  |\nabla u_{n}(x)-\nabla
u(x)|+|(g(u_{n}(x))-g(u(x)))\nabla u(x)|,
\end{align*}
so
\[
|\nabla (u_{n})_{+}(x)-\nabla u_{+}(x)|_{p(x)}\leq |\nabla
u_{n}-\nabla u|_{p(x)}+|(g(u_{n})-g(u))\nabla u|_{p(x)}.
\]
We already know that $|\nabla u_{n}-\nabla u|_{p(x)}\to 0$. And by
$|(g(u_{n})-g(u))\nabla u|\to 0$ and the Lebesgue
Dominated Convergence Theorem, we conclude that
$\int_{\Omega}|(g(u_{n})-g(u))\nabla u|^{p(x)}\,dx\to 0.$ That
is $|(g(u_{n})-g(u))\nabla u|_{p(x)}\to 0$. So $(|\nabla
u_{n})_{+}-\nabla
u_{+}|_{p(x)}\to 0$, which ends the proof.
\end{proof}

\noindent \textbf{Remark.}
 By the above Lemma, for any  $c>0$,
the set $\{u\in W^{1,p(x)}(\Omega): \|u_{\pm}\|<c\}$ is  open
in $W^{1,p(x)}(\Omega)$, and the set $\{u\in
W^{1,p(x)}(\Omega):\|u_{\pm}\|\leq  c\}$ is  closed  in
$W^{1, p(x)}(\Omega)$.

The regularity properties of the sets $N_{i}$ are stated in the
following Lemma.

\begin{lemma} \label{lem4.5}
\begin{itemize}
\item[(1)]  $N_{i}$ is a $C^{1, 1}$ sub-manifold of
$W^{1, p(x)}(\Omega)$ of co-dimension $1(i=1, 2), 2(i=3)$
respectively;
\item[(2)] The sets $K_{i}$ are complete;
\item[(3)] For any $u\in N_{i}$, we have the direct decomposition
\begin{gather*}
T_{u}W^{1, p(x)}(\Omega)=T_{u}N_{1}\oplus \emph{span}\{u_{+}\}, \\
T_{u}W^{1, p(x)}(\Omega)=T_{u}N_{2}\oplus \emph{span}\{u_{-}\}, \\
T_{u}W^{1, p(x)}(\Omega)=T_{u}N_{3}\oplus \emph{span}\{u_{+},
u_{-}\},
\end{gather*}
where $T_{u}N_{i}$ is the tangent space at $u$ of the Banach
manifold $N_{i}$. Finally, the projection onto the first component
in this decomposition is uniformly continuous on bounded sets of
$N_{i}$.
\end{itemize}
\end{lemma}

\begin{proof} (1) Denote
\begin{gather*}
N'_{1}=\{u\in W^{1, p(x)}(\Omega)|\int_{\Omega}u_{+}\,dx>0\}, \\
N'_{2}=\{u\in W^{1, p(x)}(\Omega)|\int_{\Omega}u_{-}\,dx>0\}, \\
N'_{3}=N'_{1}\cap N'_{1}.\quad\quad\quad\quad\quad\quad
\end{gather*}
By the continuous embedding of $W^{1, p(x)}(\Omega)\to
L^{1}(\Omega)$, the set $N'_{i}$ is open in $W^{1, p(x)}(\Omega)$.
Therefore it suffices to show that $N_{i}$ is a smooth sub-manifold
of $N'_{i}$. By the implicit function theorem, part (1) of the lemma
will be followed from a representation of $N_{i}$ as the inverse
image of a regular value of a $C^{1, 1}$-function
$\psi_{i}:N'_{i}\to  \mathbb{R}^{m}$ with $m=1(i=1, 2),
m=2(i=3)$ respectively.
In fact we define: for $u\in N'_{1}$
\[
\psi_{1}(u)=\int_{\Omega}(|\nabla
u_{+}|^{p(x)}+|u_{+}|^{p(x)})\,dx-\langle\mathcal{F}'(u),
u_{+}\rangle-\langle\mathcal{G}'(u), u_{+}\rangle;
\]
for $u\in N'_{2}$
\[
\psi_{2}(u)=\int_{\Omega}(|\nabla
u_{-}|^{p(x)}+|u_{-}|^{p(x)})\,dx-\langle\mathcal{F}'(u),
u_{-}\rangle-\langle\mathcal{G}'(u), u_{-}\rangle;
\]
For $u\in N'_{3}$
\[
\psi_{3}(u)=(\psi_{1}(u), \psi_{2}(u)).
\]
Then $N_{i}=\psi^{-1}_{i}(0)$ and that $0$ is a regular value of
$\psi_{i}$ may be seen from the estimates:
\begin{align*}
\langle\nabla\psi_{1}(u), u_{+}\rangle
&=\int_{\Omega}p(x)(|u_{+}|^{p(x)} +|\nabla
u_{+}|^{p(x)})\,dx-\int_{\Omega}f_{u}(x, u)u_{+}^{2}\\
&\quad +f(x, u)u_{+}\,dx-
\int_{\partial\Omega}g_{u}(x, u)u_{+}^{2}+g(x, u)u_{+}dS\\
&\leq (p^{+}-1)\int_{\Omega}f(x, u)u_{+}\,dx-\int_{\Omega}
f_{u}(x, u)u_{+}^{2}\,dx\\
&\quad +(p^{+}-1)\int_{\partial\Omega}g(x, u)u_{+}dS-\int_{\partial\Omega}
g_{u}(x, u)u_{+}^{2}dS.
\end{align*}
By (F3), (G3) the last term is bounded by
\[
(p^{+}-1-c_{1}^{-1})\int_{\Omega}f(x,
u)u_{+}\,dx+(p^{+}-1-k_{1}^{-1})\int_{\partial\Omega}g(x, u)u_{+}dS.
\]
Recall that $c_{1}, k_{1}<1/(p^{+}-1)$, and by Lemma \ref{lem4.1} the above
formula is bounded by
\[
-c\int_{\Omega}(|\nabla u_{+}|^{p(x)}+|u_{+}|^{p(x)})\,dx,
\]
which is strictly negative by Lemma \ref{lem4.2}. Therefore, $N_{1}$ is a
smooth sub-manifold of $W^{1, p(x)}$. The exact same argument
applies to $N_{2}$.
Since, trivially,
\[
\langle\nabla\psi_{1}(u), u_{-}\rangle=\langle\nabla\psi_{2}(u),
u_{+}\rangle=0
\]
for $u\in N_{3}$, the same argument holds for $N_{3}$.

(2) Let $\{u_{j}\}$ be a Cauchy sequence in $K_{i}$. Then
$u_{j}\to  u\in W^{1, p(x)}(\Omega)$ and also
$u_{j\pm}\to  u_{\pm}\in W^{1, p(x)}(\Omega)$. By Lemma \ref{lem4.2},
$u\in\{u\in W^{1, p(x)}(\Omega)|u\geq 0, u\neq0\}$. Since
continuity of $\psi_{i}$ that $u\in K_{i}$.

(3) By (1), we have the direct decomposition $T_{u}W^{1,
p(x)}(\Omega)=T_{u}N_{1}\oplus \text{span}\{u_{+}\}$,
$T_{u}W^{1,p(x)}(\Omega)=T_{u}N_{2}\oplus \text{span}\{u_{-}\}$,
$T_{u}W^{1, p(x)}(\Omega)=T_{u}N_{3}\oplus \text{span}\{u_{+},
u_{-}\}$. Let $v\in T_{u}W^{1, p(x)}(\Omega)$ be a unit tangential
vector. Then $v=v'+v''$ where $v', v''$ are given by
\[
v''=(\langle\nabla \psi_{1}(u)\big|_{\text{span}\{u_{+}\}},
\cdot\rangle)^{-1}\langle\nabla\psi_{1}(u), v\rangle\in
\text{span}\{u_{+}\},\quad  v'=v-v''\in T_{u}N_{1}.
\]
Obviously the mapping $\nabla\psi_{1}$ is uniformly bounded on
bounded subsets of $K_{1}$ and the uniform boundedness of
$(\langle\nabla \psi_{1}(u)\big|_{\text{span}\{u_{+}\}},
\cdot\rangle)^{-1}$ on such sets is a consequence of the estimate
proved in part (1) of
this proof. So we have the conclusion of the lemma.
The similar results hold for $i=2, 3$.
\end{proof}

\begin{lemma} \label{lem4.6}
The function $\Phi\big|_{N_{i}}$ satisfies the
Palais-Smale condition.
\end{lemma}

\begin{proof}
Let $\{u_{k}\}\in N_{1}$ be a Palais-Smale sequence,
that is $\Phi(u_{k})$ is uniformly bounded and $\nabla
\Phi\big|_{N_{1}}(u_{k})\to 0$ strongly. We need to
show that there exists a sub-sequence $u_{k_{i}}$,
that converges strongly. \\
In fact, the assumptions imply that $\nabla\Phi(u_{k})\to 0$.
To see this let $v_{j}\in T_{u_{j}}W^{1, p(x)}(\Omega)$ be a unit
tangential vector such that
\[
\langle\nabla\Phi(u_{j}), v_{j}\rangle=\|\nabla\Phi(u_{j})\|_{(W^{1,
p(x)}(\Omega))^{-1}}.
\]
By Lemma \ref{lem4.5} (3), $v_{j}=v_{j}'+v_{j}''\in
T_{u_{j}}N_{1}+\mathop{\rm span}\{(u_{j})_{+}\}$, since
$\Phi((u_{j})_{+})\leq \Phi(u_{j})\leq  c$ and by Lemma \ref{lem4.1},
then the sequence $\{(u_{j})_{+}\}$ is uniformly bounded. Hence
$\|v_{j}\|\geq  \|v_{j}'\|-\|v_{j}''\|$ implies $v_{j}'$ is
uniformly bounded in $W^{1,p(x)}(\Omega)$.
$(\langle\nabla\Phi\big|_{{\rm span}\{(u_{j})_{+}\}}(u_{j})$,
$(u_{j})_{+}\rangle=\psi_{1}(u_{j})=0.)$ Hence
\[
\|\nabla\Phi(u_{j})\|_{(W^{1, p(x)}(\Omega))^{-1}}=\langle\nabla
\Phi(u_{j}), v_{j}\rangle=\langle\nabla\Phi\big|_{N_{1}}(u_{j}),
v_{j}'\rangle\to 0.
\]
As $u_{j}$ is bounded in $W^{1, p(x)}(\Omega)$, there exists $u\in
W^{1, p(x)}(\Omega)$ such that $u_{j}\rightharpoonup u$, weakly in
$W^{1, p(x)}(\Omega)$. By condition (F2) and (G2), it is well known
that the unrestricted functional $\Phi$ satisfies the Palais-Smale
condition, the lemma
then follows. Similaly when $i=2, 3$ the lemma also holds.
\end{proof}

 From the proof of lemma \ref{lem4.6}, we immediately obtain the following
result.

\begin{lemma}[Nehari result] \label{lem4.7}
Let $u\in N_{i}$ be a critical point of the restricted functional
$\Phi\big|_{N_{i}}$.
Then $u$ is also a critical point of the unrestricted functional
$\Phi$.
\end{lemma}

\begin{lemma} \label{lem4.8}
There exists a critical point of the energy
functional $\Phi$ in $K_{i}$.
\end{lemma}

\begin{proof}
 From Lemma \ref{lem4.3} we know that there exists a
sufficient small constant $\tau>0$, such that
\begin{equation} \label{e1}
\Phi(u)\geq  \tau\int_{\Omega}|\nabla
u|^{p(x)}+|u|^{p(x)}\,dx\geq  \tau\|u\|^{p_{+}},
\end{equation}
when $\|u\|\leq \tau$. Let
\[
U=\{u\in N_{1}: \|u^{-}\|<\tau\},
\]
then $U$ is an open set in $N_{1}$ which contains $K_{1}$, and
$\overline{U}$ is complete. As $\Phi$ is bounded from below, we
denote $c=\inf _{u\in \overline{U}} \Phi(u)$. Let $\{u_{n}\}$
be the sequence minimizing $\Phi$ to $c$; i.e.m $u_{n}\in
\overline{U}$, such that $\Phi(u_{n})\to  c$ as $n\to
\infty$. From Lemma \ref{lem4.3} we know that $\Phi((u_{n})_{+})\leq
\Phi(u_{n})$, so $\{(u_{n})_{+}\}$ is also the minimizing sequence
tending $\Phi$ to $c$. (In fact, $(u_{n})_{+}\in K_{1}$.) Now we
have $\Phi((u_{n})_{+})<c+\varepsilon_{n}$, in which
$\tau^{p_{+}+1}>\varepsilon_{n} $ for $\forall n\in N^{+}$. Put
$\delta_{n}=\sqrt{\varepsilon_{n}}$, from Ekeland's
variational principle we know there exists a sequence
$\{v_{n}\}\subset\overline{U}$ such that the following holds,
\begin{equation} \label{e2}
\begin{gathered}
\|(u_{n})_{+}-v_{n}\|\leq \delta_{n},\quad
\Phi(v_{n})\leq \Phi((u_{n})_{+})<c+\varepsilon_{n},\\
\Phi(v_{n})<\Phi(w)+\frac{\varepsilon_{n}}{\delta_{n}}\|v_{n}-w\|,\quad
\forall w\in \overline{U},\; w\neq v_{n}.
\end{gathered}
\end{equation}
We assert that
\begin{equation} \label{e3}
v_{n}\in U,\quad \text{i.e., }\|(v_{n})_{-}\|<\tau.
\end{equation}
In fact, if the above assertion doesn't hold, i.e.
$\|(v_{n})_{-}\|=\tau$, then from \eqref{e1} we have
\[
\Phi((v_{n})_{-})\geq \tau\|(v_{n})_{-}\|^{p_{+}}
=\tau^{p_{+}+1}>\varepsilon_{n}.
\]
Observe that $(v_{n})_{+}\in K_{1}\subset U$, we have
\[
\Phi(v_{n})=\Phi((v_{n})_{+})+\Phi((v_{n})_{-})\geq
c+\varepsilon_{n},
\]
which contradicts $\Phi(v_{n})<c+\varepsilon_{n}$.
So assertion \eqref{e3} holds.

Since $v_{n}\in U$ and \eqref{e2} imply $\nabla
\Phi\big|_{N_{1}}(v_{n})\to 0$ as $n\to \infty$. By
Lemma \ref{lem4.6}, $\{v_{n}\}$ contains a convergence subsequence, which we
also denote as $\{v_{n}\}$, and $v_{n}\to  v_{0}\in
W^{1,p(x)}(\Omega)$ as $n\to \infty$. From
$\|(u_{n})_{+}-v_{n}\|\to 0$, we have $(u_{n})_{+}\to
v_{0}$ in $W^{1,p(x)}(\Omega)$. And from the completeness of $K_{1}$
we get $v_{0}\in K_{1}$. At the same time
$\nabla\Phi\big|_{N_{1}}(v_{0})=0$. Then from Lemma \ref{lem4.7},
$\nabla\Phi(v_{0})=0$. Similarly, we can prove the lemma
when $i=2,3$.
\end{proof}

Finally since the sets $K_{i}$ are disjoint and
$0\notin K_{i}$ the
proof of Theorem \ref{thm3.1} is complete.

\noindent\textbf{Remark}
For the Dirichlet problem \eqref{ePprime},
let
\begin{gather*}
N_{1}=\{u\in W_{0}^{1, p(x)}(\Omega): \int_{\Omega}
u_{+}\,dx>0, \langle\nabla\Psi(u), u^{+}\rangle=0\}, \\
N_{2}=\{u\in W_{0}^{1, p(x)}(\Omega): \int_{\Omega}
u_{-}\,dx>0, \langle\nabla\Psi(u), u^{-}\rangle=0\}, \\
N_{3}=N_{1}\cap N_{2}, \quad
K_{1}=\{u\geq 0: u\in N_{1}\}, \\
K_{2}=\{u\leq 0: u\in N_{2}\}, \quad
K_{3}=N_{3},
\end{gather*}
with a similar approach, we can prove that \eqref{ePprime} has three
nontrivial solutions.

\subsection*{Acknowledgements}
The author would like to thank Professor Xianling Fan
adn the anonymous referee for their
valuable suggestions.

\begin{thebibliography}{00}

\bibitem{a1} E. Acerbi, N. Fusco;
\emph{Partial regularity under anisotropic $(p, q)$-growth
conditions}, J. Differential Equations, \textbf{107} (1994) 46-47.

\bibitem{a2} E. Acerbi, N. Fusco;
\emph{A transmission problem in the calculus of variations},
 Calc. Var. \textbf{2} (1994) 1-16.

\bibitem{b1} J. Bonder;
\emph{Multiple solutions for the p-Laplace equation with nonlinear
boundary conditions}, Electronic Journal of Differential Equations,
\textbf{2006} (2006), No. 37, pp. 1-7.

\bibitem{f1} X. L. Fan;
\emph{Boundary trace embedding theorems for variable exponent
Soblev spaces}, J. Math. Anal. Appl. (2007), doi: 10.
1016/j. jmaa. 2007. 08. 003.

\bibitem{f2} X. L. Fan;
\emph{Solutions for $p(x)-$Laplacian Dirichlet problems with singular
coefficients}, J. Math. Anal. Appl. \textbf{312} (2005) 464-447.

\bibitem{f3} X. L. Fan, Q. H. Zhang;
\emph{Existence of solutions for $p(x)-$Laplacian Direchlet problem},
Nonlinear Anal. \textbf{52} (2003) 464-477.

\bibitem{f4} X. L. Fan, D. Zhao;
\emph{On the spaces $L^{p(x)}(\Omega)$ and $W^{m, p(x)}(\Omega)$},
J. Math. Anal. Appl. \textbf{263} (2001) 424-446.

\bibitem{f5} X. L. Fan, D. Zhao;
\emph{On the generalized Olicz-Sobolev space $W^{1, p(x)}(\Omega)$},
J. Gansu Educ. college, \textbf{12(1)} (1998) 1-6.

\bibitem{f6} X. L. Fan;
\emph{Regularity of minimizers of variational intrgrals with
$p(x)-$growth conditions}, Ann. Math. Sinica, \textbf{17A(5)}
(1996) 557-564.

\bibitem{h1} P. Harjulehto, P. Hasto;
\emph{An overview of variable exponent Lebesgue and Soblev spaces},
in: D. Herron (Ed.), Future Trends in Geometric Function Theory,
RNC Workshop, Jyvaskyla, 2003, pp. 85-93.

\bibitem{l1} N. Levinson;
\emph{Positive eigenfunctions of the equation $\Delta u+\lambda f(u)=0$},
Arch. Rat. Mech. Anal. \textbf{11} (1962) 259-272.

\bibitem{m1} P. Marcellini;
\emph{Regularity of minimizers of intergrals of the calculus of
variations with nonstandard growth conditions},
Arch. Rational Mech. Anal. \textbf{105} (1989) 267-284.

\bibitem{n1} Zeev Nehari;
\emph{Characteristic values associated with a class of nonlinear
second order differential equations}, Acta. math.
 Acta Math.  105  1961 141--175. 

\bibitem{s1} R. S. Palais, S. Smale;
\emph{A generalized Morse theory}, Bull.
AMS \textbf{70} (1964) 165-171. 105(1961) 141-175.

\bibitem{s2} M. Struwe;
\emph{Three nontrivial solutions of anticoercive boundary value
problems for the Pseudo-Laplace operator}, J. Reine Angew. Math.
\textbf{325} (1981), 68-74.

\bibitem{s3} S. Samko;
\emph{On a progress in the theory of Lebesgue spaces with variable
exponent, Maximal and singular operators}, Integral Transforms Spec.
Funct. \textbf{16} (2005) 461-482.

\bibitem{s4} J. T. Schwartz;
\emph{Generalizing the Lusternik-Schnirelman theory of critical points},
Comm. Pure Appl. Math. \textbf{17} (1964) 307-315.

\bibitem{z1} D. Zhao, X. L. Fan, W. J. Qiang;
\emph{On generalized Orlicz spaces $L^{p(x)}(\Omega)$},
J. Gansu Sci. \textbf{9(2)} (1996) 1-7.

\bibitem{z2} D. Zhao, X. L. Fan;
\emph{On the Nemytsky operators from $L^{p_{1}(x)}$ to $L^{p_{2}(x)}$},
J. Lanzhou Univ. \textbf{34} (1) (1998) 1-5.

\end{thebibliography}


\end{document}