\documentclass[english,11pt]{article}
\usepackage[latin1]{inputenc}
\usepackage{amsmath,amsthm,amssymb,babel}
\renewcommand{\arraystretch}{1.7}
\textwidth172mm
\textheight22cm
\hoffset-24mm
\voffset-20mm
\newtheorem{teo}{Theorem}
\newtheorem{defin}{Definition}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem{lemma}{Lemma}
\newtheorem{rem}{Remark}
\renewcommand{\arraystretch}{1.7}
\textwidth172mm
\textheight22cm
\hoffset-24mm
\voffset-20mm
\font\sans=cmss10
\def\proof{{\it Proof.}\ }
\def\endproof{\hfill $\Box$\par\vskip3mm}
\def\pt#1{{\it Proof of Theorem \ref{#1}.}}
\def\eq#1{(\ref{#1})}
\def\th#1{Theorem \ref{#1}}
\def\neweq#1{\begin{equation}\label{#1}}
\def\endeq{\end{equation}}
\def\weak{\rightharpoonup}
\def\ep{\varepsilon}
\def\half{{1\over2}}
\def\phi{\varphi}
\def\RR{{\mathbb R} }
\def\NN{{\mathbb N} }
\def\ZZ{{\mathbb Z} }
\def\supp{{\rm supp}}
\def\di{\displaystyle}
\def\ri{\rightarrow}
\def\intom{\int_\Omega}
\date{}

\title{\sc A multiplicity result for a nonlinear degenerate problem
arising in the theory of electrorheological fluids\thanks{
Correspondence address: Vicen\c{t}iu R\u{a}dulescu, Department of
Mathematics, University of Craiova,  200585 Craiova, Romania. E-mail: 
{\tt
vicentiu.radulescu@u-picardie.fr}}}
\author{\sc Mihai Mih\u ailescu and Vicen\c{t}iu R\u{a}dulescu\\
\small
Department of Mathematics, University of Craiova,  200585 Craiova, 
Romania\\
\small
E-mail addresses: {\tt mmihailes@yahoo.com}\qquad {\tt 
vicentiu.radulescu@u-picardie.fr}}

\begin{document}
\baselineskip16pt
\maketitle
\noindent{\small{\sc Abstract}.
We study the boundary value problem $-{\rm div}(a(x,\nabla
u))=\lambda(u^{\gamma-1}-
u^{\beta-1})$ in $\Omega$,
$u=0$ on $\partial\Omega$, where $\Omega$ is a smooth bounded domain in
$\RR^N$ and
${\rm div}(a(x,\nabla u))$ is a $p(x)$-Laplace type operator, with
$1<\beta<\gamma<\inf_{x\in\Omega}p(x)$. We prove that if $\lambda$ is
large enough
then there exist at least two nonnegative weak solutions. Our approach
relies on the variable exponent theory of generalized Lebesgue-Sobolev
spaces, combined with adequate variational methods and a variant of
Mountain Pass Lemma.  \\
\small{\bf 2000 Mathematics
Subject Classification:}  35D05, 35J60, 35J70, 58E05, 68T40, 76A02. \\
\small{\bf Key words:}  $p(x)$-Laplace operator, 
generalized Lebesgue-Sobolev space, 
critical point, weak solution, electrorheological fluids.}

\section{Introduction and preliminary results}
Most materials can be modeled with sufficient accuracy using classical
Lebesgue and Sobolev spaces, $L^p$ and $W^{1,p}$, where $p$ is a fixed
constant. For some materials with inhomogeneities, for instance
electrorheological fluids (sometimes referred to as ``smart fluids"), 
this
is not adequate, but rather the exponent $p$ should be able to vary. 
This
leads us to the study of variable exponent Lebesgue and Sobolev spaces,
$L^{p(x)}$ and $W^{1,p(x)}$, where $p$ is a real--valued function.

This paper is motivated by phenomena which are described by nonlinear
boundary value problems of the type
\begin{equation}\label{1}
\left\{\begin{array}{lll}
-{\rm div}(a(x,\nabla u))=f(x,u), &\mbox{for}&
x\in\Omega\\
u=0, &\mbox{for}& x\in\partial\Omega
\end{array}\right.
\end{equation}
where $\Omega\subset\RR^N$ ($N\geq 3$) is a bounded domain with
smooth boundary, $1<p(x)$ and $p(x)\in C(\overline\Omega)$.
The interest in studying such problems consists in the presence
of the $p(x)$--Laplace type operator ${\rm div}(a(x,\nabla u))$.
We remember that the $p(x)$-Laplace operator is defined by
$\Delta_{p(x)}u={\rm div}(|\nabla u|^{p(x)-2}\nabla u)$.
The study of differential equations
and variational problems involving $p(x)$-growth conditions is
a consequence of their applications.
Materials requiring such more advanced theory have been studied
experimentally since the middle of last century. The first major 
discovery
on electrorheological fluids is due to Willis Winslow in  1949. These
fluids have the
interesting property that their viscosity depends on the electric field 
in
the fluid. He noticed that in such fluids (for instance lithium
polymetachrylate) viscosity in an electrical field is inversely
proportional to the strength of the field. The field induces 
string-like formations in the fluid, which are parallel to the field. They can 
raise the viscosity by as much as five orders of magnitude. This phenomenon 
is known as the Winslow effect. For a general account of the underlying physics 
consult Halsey (1992) and for some technical applications Pfeiffer {\it et al.} (1999).
Electrorheological fluids have been used in robotics and space technology.
The experimental research has been done mainly in the USA, for instance 
in NASA laboratories. For more information on properties, modelling and 
the application of variable exponent spaces to these fluids we refer to 
Acerbi {\it et al.} (2001), Chabrowski {\it et al.} (2005), Diening (2002), 
Fan {\it et al.} (2005), Halsey (1992), Ruzicka (2002). 

We point out a recent mathematical model developed by Rajagopal and Ruzicka (2001). 
The model takes into account the delicate interaction between
the electromagnetic fields and the moving fluids. Particularly, in the
context of continuum mechanics, these fluids are seen as non-Newtonian
fluids. The system modeling the phenomenon arising from this study is 
\begin{equation}\label{raj}
\left\{\begin{array}{lll}
{\rm div}\;{\bf E}=0 \;\;\;\;\;\;\;\;\;\; {\rm curl}\;{\bf E}=0\\
\di\frac{\partial v}{\partial t}-{\rm div}S(x,{\bf E},{\cal E}(v))+[\nabla v]v
+\nabla\pi=g(x,{\rm E})\\
{\rm div}\;v=0
\end{array}\right.
\end{equation}
where ${\bf E}(x)$ is the electromagnetic field, $v:\Omega(\subset\RR^3)
\rightarrow\RR^3$ is the velocity of the field, ${\cal E}(v)$ is the symmetric
part of the gradient, $S$ is the extra stress tensor and $\pi$ is the pressure
(according to notations in Rajagopal and Ruzicka (2001)).

The constitutive relation for the extra stress tensor $S$ is
$$S(x,{\bf E},z)=\nu({\bf E})(1+|z|^2)^{(p-2)/2}z+\;{\rm terms}\;{\rm of}
\;{\rm the}\;{\rm same}\;{\rm growth}$$
for all symmetric $3\times 3$ matrices $z$ and where $p=p(|{\bf E}|^2)$. 
The structure of the system allows to determine ${\bf E}$ so that it depends
on $x$ and thus $p=p(x)$.

The extra stress tensor $S$ is chosen such that it is a monotone vector
field satisfying the ellipticity condition
$$D_zS(x,{\bf E},z)\lambda\otimes\lambda\geq\nu({\bf E})(1+|z|^2)^{(p-2)/2}
|\lambda|^2$$
where $\nu(\bf E)\geq\overline\nu>0$, for any $3\times 3$ symmetric matrices
$z$, $\lambda$ with null trace.

For the system described above, Rajagopal and Ruzicka established an existence 
theory which is particularly satisfying in the steady case
$$-{\rm div}S(x,{\cal E}(v))+[\nabla v]v+\nabla\pi=g(x).$$

Our paper can be regarded as a starting point for investigations of
models like those described above, since we treat the existence and
multiplicity of solutions for problems with $p(x)$ growth as in \eq{1}. 
We point out that even if our results will be formulated in a variational
context, our methods and techniques can be applied to systems as well 
(see e.g. the work of El Hamidi (2004) for a nice generalization of
such results to the study of elliptic systems of gradient type with
$p(x)$ growth).

A complete description regarding the development of variable exponent
spaces, based on a rich bibliography can be found
in the paper of L. Diening {\it et al.} (2004). We resume in what follows some 
basic facts from the above quoted study. According to that paper, variable
exponent Lebesgue spaces appeared in the literature for the 
first time already in a 1931 article by W.~Orlicz (1931). In
the 1950's this study was carried on by Nakano who
made the first systematic study of spaces with variable exponent
(called {\it modular spaces}). Nakano mentioned explicitly variable 
exponent Lebesgue spaces as an example of more general spaces he
considered, see Nakano (1950) [p.~284]. 
Later, the Polish mathematicians investigated the modular function 
spaces (see e.g. Musielak (1983)). Variable exponent Lebesgue 
spaces on the real line have been independently developed by Russian
researchers. In that context we refer to the work of Tsenov   
(1961), Sharapudinov (1978) and Zhikov (1987).

We recall in what follows some definitions and basic properties
of the generalized Lebesgue--Sobolev spaces $L^{p(x)}(\Omega)$
and $W_0^{1,p(x)}(\Omega)$, where $\Omega$ is
a bounded domain in $\RR^N$.

Throughout this paper we assume that $p(x)>1$, $p(x)\in C^{0,\alpha}
(\overline\Omega)$ with $\alpha\in(0,1)$.

Set
$$C_+(\overline\Omega)=\{h;\;h\in C(\overline\Omega),\;h(x)>1\;{\rm 
for}\;
{\rm all}\;x\in\overline\Omega\}.$$
For any $h\in C_+(\overline\Omega)$ we define
$$h^+=\sup_{x\in\Omega}h(x)\qquad\mbox{and}\qquad h^-=
\inf_{x\in\Omega}h(x).$$
For any $p(x)\in C_+(\overline\Omega)$, we define the variable exponent
Lebesgue space
$$L^{p(x)}(\Omega)=\{u;\ u\ \mbox{is a
 measurable real-valued function such that }
\int_\Omega|u(x)|^{p(x)}\;dx<\infty\}.$$
We define a norm, the so-called {\it Luxemburg norm}, on this space by 
the
formula
$$|u|_{p(x)}=\inf\left\{\mu>0;\;\int_\Omega\left|
\frac{u(x)}{\mu}\right|^{p(x)}\;dx\leq 1\right\}.$$
Variable exponent Lebesgue spaces resemble classical Lebesgue spaces
in many respects: they are Banach spaces (Kov\'a\v cik and R\'akosn\'{\i}k (1991) 
[Theorem 2.5]), the H\"older inequality holds
(Kov\'a\v cik and R\'akosn\'{\i}k (1991) [Theorem 2.1]), 
they are reflexive if and only if $1 < p^-\leq p^+<\infty$
(Kov\'a\v cik and R\'akosn\'{\i}k (1991) [Corollary 2.7]) and continuous functions 
are dense if $p^+<\infty$ (Kov\'a\v cik and R\'akosn\'{\i}k (1991) [Theorem 2.11]). 
The inclusion between Lebesgue spaces also generalizes naturally 
(Kov\'a\v cik and R\'akosn\'{\i}k (1991) [Theorem 2.8]): if 
$0 <|\Omega|<\infty$ and $p_1$, $p_2$ are variable exponent so that $p_1(x) \leq p_2(x)$ 
almost everywhere in $\Omega$ then there exists the continuous embedding
$L^{p_2(x)}(\Omega)\hookrightarrow L^{p_1(x)}(\Omega)$, whose norm does
not exceed $|\Omega|+1$.


We denote by $L^{q(x)}(\Omega)$ the conjugate space
of $L^{p(x)}(\Omega)$, where $1/p(x)+1/q(x)=1$. For any
$u\in L^{p(x)}(\Omega)$ and $v\in L^{q(x)}(\Omega)$ the H\"older
type inequality
\begin{equation}\label{Hol}
\left|\int_\Omega uv\;dx\right|\leq\left(\frac{1}{p^-}+
\frac{1}{q^-}\right)|u|_{p(x)}|v|_{q(x)}
\end{equation}
holds true.

An important role in manipulating the generalized Lebesgue-Sobolev 
spaces
is played by the {\it modular} of the $L^{p(x)}(\Omega)$ space, which 
is
the mapping
 $\rho_{p(x)}:L^{p(x)}(\Omega)\rightarrow\RR$ defined by
$$\rho_{p(x)}(u)=\int_\Omega|u|^{p(x)}\;dx.$$
If $(u_n)$, $u\in L^{p(x)}(\Omega)$ and $p^+<\infty$ then the following
relations
holds true
\begin{equation}\label{L4}
|u|_{p(x)}>1\;\;\;\Rightarrow\;\;\;|u|_{p(x)}^{p^-}\leq\rho_{p(x)}(u)
\leq|u|_{p(x)}^{p^+}
\end{equation}
\begin{equation}\label{L5}
|u|_{p(x)}<1\;\;\;\Rightarrow\;\;\;|u|_{p(x)}^{p^+}\leq
\rho_{p(x)}(u)\leq|u|_{p(x)}^{p^-}
\end{equation}
\begin{equation}\label{L6}
|u_n-u|_{p(x)}\rightarrow 0\;\;\;\Leftrightarrow\;\;\;\rho_{p(x)}
(u_n-u)\rightarrow 0.
\end{equation}
Spaces with $p^+ =\infty$ have been studied by Edmunds, Lang and 
Nekvinda (1999).

Next, we define $W_0^{1,p(x)}(\Omega)$ as the closure of
$C_0^\infty(\Omega)$ under the norm
$$\|u\|=|\nabla u|_{p(x)}.$$
The space $(W_0^{1,p(x)}(\Omega),\|\cdot\|)$ is a separable and
reflexive Banach space. We note that if $q\in C_+(\overline\Omega)$
and $q(x)<p^\star(x)$ for all $x\in\overline\Omega$ then the
embedding
$W_0^{1,p(x)}(\Omega)\hookrightarrow L^{q(x)}(\Omega)$
is compact and continuous, where $p^\star(x)=\frac{Np(x)}{N-p(x)}$
if $p(x)<N$ or $p^\star(x)=+\infty$ if $p(x)\geq N$. We refer to 
Edmunds {\it et al.} (1992), Edmunds {\it et al.} (2000), Fan {\it et al.} (2001), 
Kov\'a\v cik and R\'akosn\'{\i}k (1991) for further properties of variable exponent
Lebesgue-Sobolev spaces.
\smallskip

 The paper contains two sections. In the next section
we describe the problem  and we state the main
result. Some remarks and connections regarding similar results are also
included in the end of this section. In  Section 3 
we prove the main result of the paper. We also include some generalizations
of standard results involving the generalized Lebesgue-Sobolev spaces in
order to offer clarity and strictness to our paper. These auxiliary results
aim to be a guide which facilitates the reading of the paper. 

 \section{The main result}
Assume that $a(x,\xi):\overline\Omega\times\RR^N\rightarrow\RR^N$
is the continuous derivative with respect to $\xi$ of
the mapping $A:\overline\Omega\times\RR^N\rightarrow\RR$,
$A=A(x,\xi)$, that is, $a(x,\xi)=\nabla_{\xi}A(x,\xi)$. Suppose that
$a$ and $A$ satisfy the following hypotheses:
\smallskip

\noindent (A1) The following equality holds
$$A(x,0)=0,$$
for all $x\in\overline\Omega$.
\smallskip

\noindent (A2) There exists a positive constant $c_1$ such that
$$|a(x,\xi)|\leq c_1(1+|\xi|^{p(x)-1}),$$
for all $x\in\overline\Omega$ and $\xi\in\RR^N$.
\smallskip

\noindent (A3) The following inequality holds
$$0\leq(a(x,\xi)-a(x,\psi))\cdot(\xi-\psi),$$
for all $x\in\overline\Omega$ and $\xi,\psi\in\RR^N$, with equality
if and only if $\xi=\psi$.

\noindent (A4) There exists $k>0$ such that
$$A\left(x,\frac{\xi+\psi}{2}\right)\leq\frac{1}{2}A(x,\xi)+
\frac{1}{2}A(x,\psi)-k|\xi-\psi|^{p(x)}$$
for all $x\in\overline\Omega$ and $\xi,\psi\in\RR^N$.
\smallskip

\noindent (A5) The following inequalities hold true
$$|\xi|^{p(x)}\leq a(x,\xi)\cdot\xi\leq p(x)\;A(x,\xi),$$
for all $x\in\overline\Omega$ and $\xi\in\RR^N$.
\bigskip

\noindent{\bf Examples.}\\
1. Set $A(x,\xi)=\frac{1}{p(x)}|\xi|^{p(x)}$, $a(x,\xi)=|\xi|^{p(x)-2}
\xi$, where $p(x)\geq 2$. Then we get the $p(x)$-Laplace operator
$${\rm div}(|\nabla u|^{p(x)-2}\nabla u).$$
\smallskip

\noindent 2. Set $A(x,\xi)=\frac{1}{p(x)}[(1+|\xi|^2)^{p(x)/2}-1]$,
$a(x,\xi)=(1+|\xi|^2)^{(p(x)-2)/2}\xi$, where
$p(x)\geq 2$. Then we obtain the generalized mean curvature operator
$${\rm div}((1+|\nabla u|^2)^{(p(x)-2)/2}\nabla u).$$
\bigskip

In this paper we study problem \eq{1} in the particular case
$$f(x,t)=\lambda(t^{\gamma-1}-t^{\beta-1})$$
with $1<\beta<\gamma<\inf_{x\in\overline\Omega}p(x)$ and $t\geq 0$.
More precisely, we consider the degenerate boundary value problem
\begin{equation}\label{2}
\left\{\begin{array}{lll}
-{\rm div}(a(x,\nabla u))=\lambda(u^{\gamma-1}-
u^{\beta-1}), &\mbox{for}& x\in\Omega\\
u=0, &\mbox{for}& x\in\partial\Omega\\
u\geq 0,&\mbox{for}& x\in\Omega.
\end{array}\right.
\end{equation}

We say that $u\in W_0^{1,p(x)}(\Omega)$ is a {\it weak
solution} of problem \eq{2} if
$u\geq 0$ a. e. in $\Omega$
and
$$\int_\Omega a(x,\nabla u)\cdot\nabla\phi\;dx-\lambda
\int_\Omega u^{\gamma -1}\phi\;dx+\lambda\int_\Omega
u^{\beta-1}\phi\;dx=0$$
for all $\phi\in W_0^{1,p(x)}(\Omega)$.
\smallskip

Our main result asserts that problem \eq{2} has at least two nontrivial
weak solutions provided that $\lambda>0$ is large enough and
operators $A$ and $a$ satisfy conditions (A1)-(A5). More precisely,
we prove

\begin{teo}\label{t1}
Assume hypotheses (A1)-(A5) are fulfilled. Then there exists
$\lambda^\star>0$ such that for all $\lambda>\lambda^\star$
problem \eq{2} has at least two distinct non-negative,
nontrivial weak solutions, provided that $p^+<\min\{N,
Np^-/(N-p^-)\}$.
\end{teo}

{\bf Remark.} By Theorem 4.3 in Fan {\it et al.} (2003) problem \eq{2} has
at least a weak solution in the particular case
$a(x,\xi)=|\xi|^{p(x)-1}\xi$. However, the proof in Fan {\it et al.} (2003)
does not state the fact that the solution is non-negative and not even
nontrivial in the case when $f(x,0)=0$.
\smallskip

We point out that our result is inspired
by  Theorem 1.2 in Perera (2003), where a related property
 is proved in the case of the $p$-Laplace operators. We
point out that the extension from $p$-Laplace operator to
$p(x)$-Laplace operator is not trivial, since the $p(x)$-Laplacian
has a more complicated structure than the $p$-Laplace operator,
for example it is inhomogeneous.

\smallskip
 Finally, we mention that a similar study regarding the existence and
multiplicity of solutions for a system of equations involving the $p(x)$-Laplace
operator can be found in El Hamidi (2004). The arguments used by the
author rely on the Mountain Pass Theorem and Bartsch's Fountain Theorem.




\section{Proof of Theorem \ref{t1}}
Let $E$ denote the generalized Sobolev
space $W_0^{1,p(x)}(\Omega)$.

Define the energy functional
$I:E\rightarrow\RR$ by
$$I(u)=\int_\Omega A(x,\nabla u)\;dx-\frac{\lambda}
{\gamma}\int_\Omega u_+^\gamma\;dx+\frac{\lambda}{\beta}\int_\Omega
u_+^\beta\;dx\,,$$
where $u_+(x)=\max\{u(x),0\}$.

We first establish some basic properties of $I$.

\begin{prop}\label{p1}
The functional $I$ is well-defined on $E$ and $I\in C^1(E,\RR)$ with
the derivative given by
$$\langle I^{'}(u),\phi\rangle=\int_\Omega a(x,\nabla u)
\cdot\nabla\phi\;dx-\lambda\int_\Omega u_+^{\gamma-1}\phi\;dx
+\lambda\int_\Omega u_+^{\beta-1}\phi\;dx,$$
for all $u$, $\phi\in E$.
\end{prop}
To prove Proposition \ref{p1} we define the functional $\Lambda:E\rightarrow
\RR$ by
$$\Lambda(u)=\int_{\Omega}A(x,\nabla u)\;dx,\;\;\;\forall u\in E.$$
\begin{lemma}\label{l1}
(i) The functional $\Lambda$ is well-defined on $E$.\\
(ii) The functional $\Lambda$ is of class $C^1(E,\RR)$ and
$$\langle\Lambda^{'}(u),\phi\rangle=\int_\Omega a(x,\nabla u)\cdot
\nabla\phi\;dx,$$
for all $u,\phi\in E$.
\end{lemma}
\proof
(i) For any $x\in\Omega$ and $\xi\in\RR^N$ we have
$$A(x,\xi)=\int_0^1\frac{d}{dt}A(x,t\xi)\;dt=\int_0^1a(x,t\xi)\cdot
\xi\;dt.$$
Using hypotheses (A2) we get
\begin{equation}\label{ecc4}
\begin{array}{lll}
A(x,\xi)&\leq&c_1\di\int_0^1(1+|\xi|^{p(x)-1}t^{p(x)-1})|\xi|\;dt\\
&\leq&c_1|\xi|+\di\frac{c_1}{p(x)}|\xi|^{p(x)}\\
&\leq&c_1|\xi|+\di\frac{c_1}{p^-}|\xi|^{p(x)},\;\;\;\forall x\in
\overline\Omega,\;\xi\in\RR^N.
\end{array}
\end{equation}
The above inequality and (A5) imply
$$0\leq\int_\Omega A(x,\nabla u)\;dx\leq c_1\int_\Omega|\nabla u|
\;dx+\frac{c_1}{p^-}\int_\Omega|\nabla u|^{p(x)}\;dx,\;\;\;
\forall u\in E.$$
Using inequality \eq{Hol} and relations \eq{L4} and \eq{L5} we
deduce that $\Lambda$ is well defined on $E$.
\smallskip

\noindent (ii) {\bf Existence of the G\^ateaux derivative.} Let
$u$, $\phi\in E$. Fix $x\in\Omega$ and $0<|r|<1$. Then, by the mean 
value
theorem, there exists $\nu\in[0,1]$ such that
$$|A(x,\nabla u(x)+r\nabla\phi(x))-A(x,\nabla u)|/|r|=
|a(x,\nabla u(x)+\nu\;r\;\nabla\phi(x))||\nabla\phi(x)|.$$
Using condition (A2) we obtain
\begin{eqnarray*}
|A(x,\nabla u(x)+r\nabla\phi(x))-A(x,\nabla u)|/|r|&\leq&
[c_1+c_1(|\nabla u(u)|+|\nabla\phi(x)|)^{p(x)-1}]|\nabla\phi(x)|\\
&\leq&[c_1+c_12^{p^+}(|\nabla u(x)|^{p(x)-1}+|\nabla\phi(x)|^{p(x)
-1})]|\nabla\phi(x)|.
\end{eqnarray*}
Next, by inequality \eq{Hol}, we have
$$\int_\Omega c_1|\nabla\phi|\;dx\leq|c_1|_{\frac{p(x)}{p(x)-1}}
\cdot|\nabla\phi|_{p(x)}$$
and
$$\int_\Omega|\nabla u|^{p(x)-1}|\nabla\phi|\;dx\leq
||\nabla u|^{p(x)-1}|_{\frac{p(x)}{p(x)-1}}\cdot|\nabla\phi|_{p(x)}.$$
The above inequalities imply
$$c_1[1+2^{p^+}(|\nabla u(x)|^{p(x)-1}+|\nabla\phi(x)|^{p(x)-1})]
|\nabla\phi(x)|\in L^1(\Omega).$$
It follows from the Lebesgue theorem that
$$\langle\Lambda^{'}(u),\phi\rangle=\int_\Omega a(x,\nabla u)\cdot
\nabla\phi\;dx.$$
{\bf Continuity of the G\^ateaux derivative.} Assume $u_n\rightarrow u$
in $E$. Let us define $\theta(x,u)=a(x,\nabla u)$. Using hypotheses
(A2) and Proposition 2.2 in Fan {\it et al.} (2003) we deduce that $\theta(x,u_n)
\rightarrow\theta(x,u)$ in $(L^{q(x)}(\Omega))^N$, where $q(x)=
\frac{p(x)}{p(x)-1}$. By inequality \eq{Hol} we obtain
$$|\langle\Lambda^{'}(u_n)-\Lambda^{'}(u),\phi\rangle|\leq
|\theta(x,u_n)-\theta(x,u)|_{q(x)}|\nabla\phi|_{p(x)}$$
and so
$$\|\Lambda^{'}(u_n)-\Lambda^{'}(u)\|\leq|\theta(x,u_n)-
\theta(x,u)|_{q(x)}\rightarrow 0,\;\;\;{\rm as}\;n\rightarrow
\infty.$$
The proof of Lemma \ref{l1} is complete.  \qed

\begin{lemma}\label{l2}
If $u\in E$ then $u_+$, $u_-\in E$ and
$$\nabla u_+=\left\{\begin{array}{lll}
0, &\mbox{if}& [u\leq 0]\\
\nabla u, &\mbox{if}& [u>0],
\end{array}\right. \qquad
\nabla u_-=\left\{\begin{array}{lll}
0, &\mbox{if}& [u\geq 0]\\
\nabla u, &\mbox{if}& [u<0]
\end{array}\right.$$
where $u_\pm=\max\{\pm u(x),0\}$ for all $x\in\Omega$.
\end{lemma}

\proof
Let $u\in E$ be fixed. Then there exists a sequence $(\phi_n)\in
C_0^\infty(\Omega)$ such that
$$|\nabla(\phi_n-u)|_{p(x)}\rightarrow 0.$$
Since $1<p^-\leq p(x)$ for all $x\in\Omega$, it follows  that
$L^{p(x)}$ is continuously embedded in $L^{p^-}(\Omega)$ and thus
$$|\nabla(\phi_n-u)|_{p^-}\rightarrow 0.$$
Hence $u\in W^{1,p^-}_0(\Omega)$. We obtain
\begin{equation}\label{ecca}
u_+,u_-\in W_0^{1,p^-}(\Omega)\subset W_0^{1,1}(\Omega).
\end{equation}
On the other hand, Theorem 7.6 in Gilbarg and Trudinger (1998) implies
$$\nabla u_+=\left\{\begin{array}{lll}
0, &\mbox{if}& [u\leq 0]\\
\nabla u, &\mbox{if}& [u>0],
\end{array}\right.
\nabla u_-=\left\{\begin{array}{lll}
0, &\mbox{if}& [u\geq 0]\\
\nabla u, &\mbox{if}& [u<0].
\end{array}\right.$$
By the above equalities we deduce that
\begin{equation}\label{ecc9}
|u_+(x)|^{p(x)}\leq|u(x)|^{p(x)},\;\;\;
|\nabla u_+(x)|^{p(x)}\leq|\nabla u|^{p(x)},\;\;\;{\rm a.e.}\;
x\in\Omega
\end{equation}
and
\begin{equation}\label{ecc10}
|u_-(x)|^{p(x)}\leq|u(x)|^{p(x)},\;\;\;
|\nabla u_-(x)|^{p(x)}\leq|\nabla u|^{p(x)},\;\;\;{\rm a.e.}\;
x\in\Omega.
\end{equation}
Since $u\in E$ we have
\begin{equation}\label{ecc11}
|u(x)|^{p(x)},\;|\nabla u(x)|^{p(x)}\in L^1(\Omega).
\end{equation}
By \eq{ecc9}, \eq{ecc10} and \eq{ecc11} and Lebesgue theorem we
obtain that $u_+$, $u_-\in L^{p(x)}(\Omega)$ and $\rho_{p(x)}
(|\nabla u_+|)<\infty$, $\rho_{p(x)}(|\nabla u_-|)<\infty$.
It follows that
\begin{equation}\label{eccb}
u_+,u_-\in W^{1,p(x)}(\Omega)
\end{equation}
where $W^{1,p(x)}(\Omega)=\{u\in L^{p(x)}(\Omega);\;\;|\nabla u|
\in L^{p(x)}(\Omega)\}$ (see Fan {\it et al.} (2001) for more details).

By \eq{ecca} and \eq{eccb} we conclude that
$$u_+,u_-\in W^{1,p(x)}(\Omega)\cap W_0^{1,1}(\Omega).$$
Since $p\in C^{0,\alpha}(\overline\Omega)$,
Theorem 2.6 and Remark 2.9 in Fan {\it et al.} (2001) show that $E=
W^{1,p(x)}(\Omega)\cap W_0^{1,1}(\Omega)$. Thus
$u_+$, $u_-\in E$ and
the proof of Lemma \ref{l2} is complete.  \qed

By Lemmas \ref{l1} and \ref{l2} it is clear that Proposition \ref{p1}
holds true.
\smallskip

\noindent{\bf Remark.} If $u$ is a critical point of $I$  then
using Lemma \ref{l2} and condition (A5) we have
\begin{eqnarray*}
0&=&\langle I^{'}(u),u_-\rangle=\int_\Omega a(x,\nabla u)
\cdot\nabla u_-\;dx-\lambda\int_\Omega(u_+)^{\gamma-1}u_-\;dx
+\lambda\int_\Omega(u_+)^{\beta-1}u_-\;dx\\
&=&\int_\Omega a(x,\nabla u)\cdot\nabla u_-\;dx
=\int_\Omega a(x,\nabla u_-)\cdot\nabla u_-\;dx
\geq\int_\Omega|\nabla u_-|^{p(x)}\;dx.
\end{eqnarray*}
Thus we deduce that $u\geq 0$. It follows that the nontrivial critical
points of $I$ are non-negative solutions of \eq{2}.

The above remark shows that we can prove
Theorem \ref{t1} using the critical points theory. More exactly, we
first show that for $\lambda>0$ large enough, the functional $I$ has
a global minimizer $u_1\geq 0$ such that $I(u_1)<0$. Next,
by means of the Mountain
Pass Theorem, a second
critical point $u_2$ with $I(u_2)>0$ is obtained.


\begin{lemma}\label{l3}
The functional $\Lambda$ is weakly lower semi-continuous.
\end{lemma}

\proof
By Corollary III.8 in Brezis (1992), it is enough to show that $\Lambda$
is  lower semi-continuous. For this purpose, we fix
$u\in E$ and $\epsilon>0$. Since $\Lambda$ is convex
(by condition (A4)), we deduce that for any $v\in E$
the following inequality holds
$$\int_\Omega A(x,\nabla v)\;dx\geq\int_\Omega A(x,\nabla u)\;dx
+\int_\Omega a(x,\nabla u)\cdot(\nabla v-\nabla u)\;dx.$$
Using condition (A2) and inequality \eq{Hol} we have
\begin{eqnarray*}
\int_\Omega A(x,\nabla v)\;dx&\geq&\int_\Omega A(x,\nabla u)\;dx-
\int_\Omega|a(x,\nabla u)||\nabla v-\nabla u|\;dx\\
&\geq&\int_\Omega A(x,\nabla u)\;dx- c_1\int_\Omega|\nabla(v-u)|\;dx
-c_1\int_\Omega|\nabla u|^{p(x)-1}|\nabla(v-u)|\;dx\\
&\geq&\int_\Omega A(x,\nabla u)\;dx-c_2|1|_{q(x)}|\nabla(v-u)|_{p(x)}
-c_3||\nabla u|^{p(x)-1}|_{q(x)}|\nabla(v-u)|_{p(x)}\\
&\geq&\int_\Omega A(x,\nabla u)\;dx-c_4\|v-u\|\\
&\geq&\int_\Omega A(x,\nabla u)\;dx-\epsilon
\end{eqnarray*}
for all $v\in E$ with $\|v-u\|<\delta=\epsilon/c_4$, where $c_2$, 
$c_3$,
$c_4$ are positive constants, and
$q(x)=\frac{p(x)}{p(x)-1}$.
We conclude that $\Lambda$ is weakly lower semi-continuous. The proof
of Lemma \ref{l3} is complete.  \qed

\begin{lemma}\label{le4}
There exists $\lambda_1>0$ such that
$$\lambda_1=\inf\limits_{u\in E,\;\|u\|>1}\frac{\di\int_\Omega
\di\frac{1}{p(x)}|\nabla u|^{p(x)}\;dx}{\di\int_\Omega|u|^{p^-}\;dx}
.$$
\end{lemma}
\proof
We know that $E$ is continuously embedded in $L^{p^-}(\Omega)$. It
follows that there exists $C>0$ such that
$$\|u\|\geq C|u|_{p^-},\;\;\;\forall u\in E.$$
On the other hand, by \eq{L4} we have
$$\int_\Omega|\nabla u|^{p(x)}\;dx\geq\|u\|^{p^-},\;\;\;\forall
u\in E\;{\rm with}\;\|u\|>1.$$
Combining the above inequalities we obtain
$$\int_\Omega\frac{1}{p(x)}|\nabla u|^{p(x)}\;dx\geq\frac{C^{p^-}}
{p^+}\int_\Omega|u|^{p^-}\;dx,\;\;\;\forall u\in E\;{\rm with}\;
\|u\|>1.$$
The proof of Lemma \ref{le4} is complete.\qed


\begin{prop}\label{p2}
(i) The functional $I$ is bounded from below and coercive.\\
(ii) The functional $I$ is weakly lower semi-continuous.
\end{prop}
\proof
(i) Since $1<\beta<\gamma<p^-$ we have
$$\lim\limits_{t\rightarrow\infty}\frac{\di\frac{1}{\gamma}t^\gamma-
\di\frac{1}{\beta}t^\beta}{t^{p^-}}=0.$$
Then for any $\lambda>0$ there exists $C_\lambda>0$ such that
$$\lambda\left(\frac{1}{\gamma}t^\gamma-\frac{1}{\beta}t^\beta
\right)\leq\frac{\lambda_1}{2}t^{p^-}+C_\lambda,\;\;\;\forall
t\geq 0,$$
where $\lambda_1$ is defined in Lemma \ref{le4}.

Condition (A5) and the above inequality show that for any $u\in E$
with $\|u\|>1$ we have
\begin{eqnarray*}
I(u)&\geq&\int_\Omega\frac{1}{p(x)}|\nabla u|^{p(x)}\;dx-
\frac{\lambda_1}{2}\int_\Omega|u|^{p^-}\;dx-C_\lambda\mu(\Omega)\\
&\geq&\frac{1}{2}\int_\Omega\frac{1}{p(x)}|\nabla u|^{p(x)}\;dx-
C_\lambda\mu(\Omega)\\
&\geq&\frac{1}{2p^+}\|u\|^{p^-}-C_\lambda\mu(\Omega).
\end{eqnarray*}
This shows that $I$ is bounded from below and coercive.
\smallskip

\noindent (ii) Using Lemma \ref{l3} we deduce that
$\Lambda$ is weakly lower semi-continuous.
We show that $I$ is weakly lower semi-continuous. Let
$(u_n)\subset E$ be a sequence which converges weakly to $u$ in $E$.
Since $\Lambda$ is weakly lower semi-continuous we have
\begin{equation}\label{3}
\Lambda(u)\leq\liminf\limits_{n\rightarrow\infty}\Lambda(u_n).
\end{equation}
On the other hand, since $E$ is compactly embedded in
$L^\gamma(\Omega)$ and $L^\beta(\Omega)$ it follows that $({u_n}_+)$
converges strongly to $u_+$ both in $L^\gamma(\Omega)$ and in
$L^\beta(\Omega)$. This fact together with relation \eq{3} imply
$$I(u)\leq\liminf\limits_{n\rightarrow\infty}I(u_n).$$
Therefore, $I$ is weakly lower semi-continuous.
The proof of Proposition \ref{p2} is complete.  \qed

By Proposition \ref{p2} and Theorem 1.2 in Struwe (1996) we deduce that
there exists $u_1\in E$ a global minimizer of $I$. The following result
implies that
$u_1\neq 0$, provided that $\lambda$ is sufficiently large.

\begin{prop}\label{p3}
There exists $\lambda^\star>0$ such that $\inf_E I<0$.
\end{prop}

\proof
Let $\Omega_1\subset\Omega$ be a compact subset, large enough and
$u_0\in E$ be such that $u_0(x)=t_0$ in $\Omega_1$ and $0\leq u_0(x)
\leq t_0$ in $\Omega\setminus\Omega_1$, where $t_0>1$ is chosen
such that
$$\frac{1}{\gamma}t_0^\gamma-\frac{1}{\beta}t_0^\beta>0.$$
We have
\begin{eqnarray*}
\frac{1}{\gamma}\int_\Omega u_0^\gamma\;dx-\frac{1}{\beta}\int_\Omega
u_0^\beta\;dx&\geq&\frac{1}{\gamma}\int_{\Omega_1}u_0^\gamma\;dx-
\frac{1}{\beta}\int_{\Omega_1}u_0^\beta\;dx-\frac{1}{\beta}
\int_{\Omega\setminus\Omega_1}u_0^\beta\;dx\\
&\geq&\frac{1}{\gamma}
\int_{\Omega_1}u_0^\gamma\;dx-\frac{1}{\beta}\int_{\Omega_1}u_0^\beta
\;dx-\frac{1}{\beta}\;t_0^\beta\;\mu(\Omega\setminus\Omega_1)>0
\end{eqnarray*}
and thus $I(u_0)<0$ for $\lambda>0$ large enough.
The proof of Proposition \ref{p3} is complete.  \qed

Since Proposition \ref{p3} holds true it follows that $u_1\in E$
is a nontrivial weak solution of problem \eq{2}.
\bigskip

Fix $\lambda\geq\lambda^\star$. Set
$$
g(x,t)=\left\{\begin{array}{lll}
0, &\mbox{for}& t<0\\
t^{\gamma-1}-t^{\beta-1}, &\mbox{for}& 0\leq t\leq u_1(x)\\
u_1(x)^{\gamma-1}-u_1(x)^{\beta-1}, &\mbox{for}& t>u_1(x)
\end{array}\right.
$$
and
$$G(x,t)=\int_0^tg(x,s)\;ds.$$
Define the functional $J:E\rightarrow\RR$ by
$$J(u)=\int_\Omega A(x,\nabla u)\;dx-\lambda \int_\Omega G(x,u)
\;dx.$$
The same arguments as those used for functional $I$ imply that
$J\in C^1(E,\RR)$ and
$$\langle J^{'}(u),\phi\rangle=\int_\Omega a(x,\nabla u)
\cdot\nabla\phi\;dx-\lambda\int_\Omega g(x,u)\phi\;dx,$$
for all $u$, $\phi\in E$.

On the other hand, we point out that if $u\in E$ is a
critical point of $J$ then $u\geq 0$. The proof can be carried out
as in the case of functional $I$.

Next, we prove
\begin{lemma}\label{l5}
If $u$ is a critical point of $J$ then $u\leq u_1$.
\end{lemma}
\proof
We have
\begin{eqnarray*}
0&=&\langle J^{'}(u)-I^{'}(u_1),(u-u_1)_+\rangle\\
&=&\int_\Omega(a(x,\nabla u)-a(x,\nabla u_1))
\cdot\nabla(u-u_1)_+\;dx-\lambda\int_\Omega[g(x,u)-(u_1^
{\gamma-1}-u_1^{\beta-1})](u-u_1)_+\;dx\\
&=&\int_{[u>u_1]}(a(x,\nabla u)-a(x,\nabla u_1))
\cdot\nabla(u-u_1)\;dx.
\end{eqnarray*}
By condition (A3) we deduce that the above equality holds if and
only if $\nabla u=\nabla u_1$. It follows that $\nabla u(x)=
\nabla u_1(x)$ for all $x\in\omega:=\{y\in\Omega;\;\;u(y)>u_1(y)\}$.
Hence
$$\int_\omega|\nabla(u-u_1)|^{p(x)}\;dx=0$$
and thus
$$\int_\Omega|\nabla(u-u_1)_+|^{p(x)}\;dx=0.$$
By relation \eq{L5} we obtain
$$\|(u-u_1)_+\|=0.$$
Since $u-u_1\in E$ by Lemma \ref{l2} we have that $(u-u_1)_+\in E$.
Thus we obtain that $(u-u_1)_+=0$ in $\Omega$, that is, $u\leq u_1$ in
$\Omega$.
The proof of Lemma \ref{l5} is complete.  \qed

In the following we determine a critical point $u_2\in E$ of $J$
such that $J(u_2)>0$ via the Mountain Pass Theorem. By the above lemma
we will deduce that $0\leq u_2\leq u_1$ in $\Omega$. Therefore
$$g(x,u_2)=u_2^{\gamma-1}-u_2^{\beta-1}\;\;\;{\rm and}\;\;\;
G(x,u_2)=\frac{1}{\gamma}u_2^\gamma-\frac{1}{\beta}u_2^\beta$$
and thus
$$J(u_2)=I(u_2)\;\;\;{\rm and}\;\;\;J^{'}(u_2)=I^{'}(u_2).$$
More exactly we find
$$I(u_2)>0=I(0)>I(u_1)\qquad\mbox{and}\qquad I^{'}(u_2)=0\,.$$
This shows that $u_2$ is a weak solution of problem \eq{2} such
that $0\leq u_2\leq u_1$, $u_2\neq 0$ and $u_2\neq u_1$.

In order to find $u_2$ described above we prove
\begin{lemma}\label{l6}
There exists $\rho\in(0,\|u_1\|)$ and $a>0$ such that
$J(u)\geq a$, for all $u\in E$ with $\|u\|=\rho.$
\end{lemma}

\proof
Let $u\in E$ be fixed, such that $\|u\|<1$. It is clear that there
exists $\delta>1$ such that
$$\frac{1}{\gamma}t^\gamma-\frac{1}{\beta}t^\beta\leq 0,\;\;\;
\forall t\in[0,\delta].$$
For $\delta$ given above we define
$$\Omega_u=\{x\in\Omega;\;u(x)>\delta\}.$$
If $x\in\Omega\setminus\Omega_u$ with $u(x)<u_1(x)$ we have
$$G(x,u)=\frac{1}{\gamma}u_+^\gamma-\frac{1}{\beta}u_+^\beta\leq 0.$$
If $x\in\Omega\setminus\Omega_u$ with $u(x)>u_1(x)$ then
$u_1(x)\leq\delta$ and we have
$$G(x,u)=\frac{1}{\gamma}u_1^\gamma-\frac{1}{\beta}u_1^\beta\leq 0.$$
Thus we deduce that
$$G(x,u)\leq 0,\;\;\;{\rm on}\;\Omega\setminus\Omega_u.$$
Provided that $\|u\|<1$ by condition (A5) and relation \eq{L5} we get
\begin{equation}\label{s1}
\begin{array}{lll}
J(u)&\geq&\di\int_\Omega\di\frac{1}{p(x)}|\nabla u|^{p(x)}\;dx-
\lambda\di\int_{\Omega_u}G(x,u)\;dx\\
&\geq&\di\frac{1}{p^+}\|u\|^{p^+}-\lambda\di\int_{\Omega_u}G(x,u)\;dx
\end{array}
\end{equation}
Since $p^+<\min\{N, \frac{Np^-}{N-p^-}\}$ it follows that
$p^+<p^\star(x)$ for all $x\in\overline\Omega$. Then there exists
$q\in(p^+,\frac{Np^-}{N-p^-})$ such that $E$ is continuously
embedded in $L^q(\Omega)$. Thus there exists a positive constant
$C>0$ such that
$$|u|_q\leq C\|u\|,\;\;\;\forall u\in E.$$
Using the definition of $G$, H\"older's inequality and the
above estimate, we obtain
\begin{equation}\label{s2}
\begin{array}{lll}
\lambda\di\int_{\Omega_u} G(x,u)\;dx&=&\lambda\di\int_{\Omega_u\cap
[u<u_1]}\left(\di\frac{1}{\gamma}u_+^\gamma-\di\frac{1}{\beta}
u_+^\beta\right)\;dx+\lambda\di\int_{\Omega_u\cap[u>u_1]}\left(
\di\frac{1}{\gamma}u_1^\gamma-\di\frac{1}{\beta}u_1^\beta\right)
\;dx\\
&\leq&\di\frac{2\lambda}{\gamma}\di\int_{\Omega_u}u_+^\gamma\;dx\\
&\leq&\di\frac{2\lambda}{\gamma}\di\int_{\Omega_u}u_+^{p^+}\;dx\\
&\leq&\di\frac{2\lambda}{\gamma}\left(\di\int_{\Omega_u}u_+^{q}
\;dx\right)^{p^+/q}[\mu(\Omega_u)]^{1-p^+/q}\\
&\leq&C\di\frac{2\lambda}{\gamma}[\mu(\Omega_u)]^{1-p^+/q}\|u\|^{p^+}.
\end{array}
\end{equation}
By \eq{s1} and \eq{s2} we infer that it is enough to show that
$\mu(\Omega_u)\rightarrow 0$ as $\|u\|\rightarrow 0$ in order to
prove Lemma \ref{l6}.

Let $\epsilon>0$. We choose $\Omega_\epsilon\subset\Omega$ a compact
subset, such that $\mu(\Omega\setminus{\Omega_\epsilon})<\epsilon$.
We denote by $\Omega_{u,\epsilon}:=\Omega_u\cap\Omega_\epsilon$. Then
it is clear that
$$C[\mu(\Omega)]^{1-p^+/q}\|u\|^{p^+}\geq\left(\int_\Omega|u|^{q}
\;dx\right)^{p^+/q}\geq\left(\int_{\Omega_{u,\epsilon}}|u|^{q}\;dx
\right)^{p^+/q}\geq\delta^{p^+}[\mu(\Omega_{u,\epsilon})]^{p^+/q}.$$
The above inequality implies that $\mu(\Omega_{u,\epsilon})
\rightarrow 0$ as $\|u\|\rightarrow 0$.

Since $\Omega_u\subset\Omega_{u,\epsilon}\cup(\Omega\setminus
\Omega_\epsilon)$ we have
$$\mu(\Omega_u)\leq\mu(\Omega_{u,\epsilon})+\epsilon$$
and $\epsilon>0$ is arbitrary. We find that $\mu(\Omega_u)\rightarrow
0$ as $\|u\|\rightarrow 0$.
This concludes the proof of Lemma \ref{l6}. \qed

\begin{lemma}\label{l7}
The functional $J$ is coercive.
\end{lemma}
\proof
For each $u\in E$ with $\|u\|>1$ by condition (A5), relation \eq{L4}
and inequality \eq{Hol} we have
\begin{eqnarray*}
J(u)&\geq&\int_\Omega\frac{1}{p(x)}|\nabla u|^{p(x)}\;dx-\lambda
\int_{[u>u_1]}G(x,u)\;dx-\lambda\int_{[u<u_1]}G(x,u)\;dx\\
&\geq&\frac{1}{p^+}\|u\|^{p^-}-\frac{\lambda}{\gamma}\int_{[u>u_1]}
u_1^\gamma\;dx+\frac{\lambda}{\beta}\int_{[u>u_1]}u_1^\beta\;dx-
\frac{\lambda}{\gamma}\int_{[u<u_1]}u_+^\gamma\;dx+\frac{\lambda}
{\beta}\int_{[u<u_1]}u_+^\beta\;dx\\
&\geq&\frac{1}{p^+}\|u\|^{p^-}-\frac{\lambda}{\gamma}\int_{\Omega}
u_1^\gamma\;dx-\frac{\lambda}{\gamma}\int_{\Omega}u_+^\gamma\;dx\\
&\geq&\frac{1}{p^+}\|u\|^{p^-}-\frac{\lambda}{\gamma}[\mu(\Omega)]^
{1-\gamma/p^-}C_1\|u\|^\gamma-C_2\\
&\geq&\frac{1}{p^+}\|u\|^{p^-}-C_23\|u\|^\gamma-C_2,
\end{eqnarray*}
where $C_1$, $C_2$ and $C_3$ are positive constants. Since
$\gamma<p^-$ the above inequality implies that $J(u)\rightarrow
\infty$ as $\|u\|\rightarrow\infty$, that is, $J$ is coercive.
The proof of Lemma \ref{l7} is complete.  \qed

The following result yields a sufficient condition which ensures that a
weakly convergent sequence
in $E$ converges strongly, too.

\begin{lemma}\label{l8}
Assume that the sequence $(u_n)$ converges weakly to $u$ in $E$ and
$$\limsup\limits_{n\rightarrow\infty}\int_\Omega a(x,\nabla u_n)
\cdot(\nabla u_n-\nabla u)\;dx\leq 0.$$
Then $(u_n)$ converges strongly to $u$ in $E$.
\end{lemma}

\proof
Using relation \eq{ecc4} we have that there exists a positive
constant $c_5$ such that
$$A(x,\xi)\leq c_5(|\xi|+|\xi|^{p(x)}),\;\;\;\forall x\in
\overline\Omega,\;\xi\in\RR^N.$$
The above inequality implies
\begin{equation}\label{ecc5}
A(x,\nabla u_n)\leq c_5(|\nabla u_n|+|\nabla u_n|^{p(x)}),\;\;\;
\forall x\in\overline\Omega,\;n.
\end{equation}
The fact that $u_n$ converges weakly to $u$ in $E$ implies that
there exists $R>0$ such that $\|u_n\|\leq R$ for all $n$. By
relation \eq{ecc5}, inequalities \eq{Hol}, \eq{L4} and \eq{L5}
we deduce that $\{\int_\Omega A(x,\nabla u_n)\;dx\}$ is
bounded. Then, up to to a subsequence, we deduce that
$\int_\Omega A(x,\nabla u_n)\;dx\rightarrow c$. By Lemma \ref{l3}
we obtain
$$\int_\Omega A(x,\nabla u)\;dx
\leq\liminf\limits_{n\rightarrow\infty}\int_\Omega
A(x,\nabla u_n)\;dx=c.$$
On the other hand, since $\Lambda$ is convex, we have
$$\int_\Omega A(x,\nabla u)\;dx\geq\int_\Omega A(x,\nabla u_n)
\;dx+\int_\Omega a(x,\nabla u_n)\cdot(\nabla u-\nabla u_n)
\;dx.$$
Next, by the hypothesis  $\limsup\limits_{n\rightarrow\infty}
\int_\Omega a(x,\nabla u_n)\cdot(\nabla u_n-\nabla u)\;dx\leq 0$,
we conclude that $\int_\Omega A(x,\nabla u)\;dx=c$.

Taking into account that $(u_n+u)/2$ converges weakly to $u$
in $E$ and using Lemma \ref{l3} we have
\begin{equation}\label{ecc6}
c=\int_\Omega A(x,\nabla u)\;dx
\leq\liminf\limits_{n\rightarrow\infty}
\int_\Omega A\left(x,\nabla \frac{u_n+u}{2}\right)\;dx.
\end{equation}
We assume by contradiction that $u_n$ does not converge to $u$
in $E$. Then by \eq{L6} it follows that there exist $\epsilon>0$ and
a subsequence $(u_{n_m})$ of $(u_n)$ such that
\begin{equation}\label{ecc7}
\int_\Omega|\nabla(u_{n_m}-u)|^{p(x)}\;dx\geq\epsilon,\;\;\;
\forall m.
\end{equation}
By condition (A4) we have
\begin{equation}\label{ecc8}
\frac{1}{2}A(x,\nabla u)+\frac{1}{2}A(x,\nabla u_{n_m})-A\left(x,\nabla
\frac{u+u_{n_m}}{2}\right)\geq k|\nabla(u_{n_m}-u)|^{p(x)}.
\end{equation}
Relations \eq{ecc7} and \eq{ecc8} yield
$$\frac{1}{2}\int_\Omega A(x,\nabla u)\;dx+\frac{1}{2}\int_\Omega
A(x,\nabla u_{n_m})\;dx-\int_\Omega A\left(x,\nabla
\frac{u+u_{n_m}}{2}\right)\;\geq k\int_\Omega|\nabla(u_{n_m}-u)|^{p(x)}
\;dx\geq k\epsilon.$$
Letting $m\rightarrow\infty$ in the above inequality we obtain
$$c-k\epsilon\geq\limsup\limits_{m\rightarrow\infty}
\int_\Omega A\left(x,\nabla\frac{u+u_{n_m}}{2}\right)\;dx$$
and that is a contradiction with \eq{ecc6}. It follows that $u_n$
converges strongly to $u$ in $E$ and Lemma \ref{l8} is
proved.  \qed

\medskip
{\sc Proof of Theorem \ref{t1} completed.}
Using Lemma \ref{l6} and the Mountain Pass Theorem (see Ambrosetti and Rabinowitz (1973)
with the variant given by Theorem 1.15 in Willem (1996)) we deduce that
there exists a sequence $(u_n)\subset E$ such that
\begin{equation}\label{MPT}
J(u_n)\rightarrow c>0\;\;\;{\rm and}\;\;\;J^{'}(u_n)\rightarrow 0
\end{equation}
where
$$c=\inf\limits_{\gamma\in\Gamma}\max\limits_{t\in[0,1]}
J(\gamma(t))$$
and
$$\Gamma=\{\gamma\in C([0,1],E);\;\gamma(0)=0,\;\gamma(1)=u_1\}.$$
By relation \eq{MPT} and Lemma \ref{l7} we obtain that $(u_n)$ is
bounded and thus passing eventually to a subsequence, still denoted
by $(u_n)$, we may assume that there exists $u_2\in E$ such that
$u_n$ converges weakly to $u_2$. Since $E$ is compactly embedded
in $L^i(\Omega)$ for any $i\in[1,p^-]$, it follows that $u_n$
converges strongly to $u_2$ in $L^i(\Omega)$ for all $i\in[1,p^-]$.
Hence
$$\langle\Lambda^{'}(u_n)-\Lambda^{'}(u_2),u_n-u_2\rangle=
\langle J^{'}(u_n)-J^{'}(u_2),u_n-u_2\rangle+\lambda\int_\Omega
[g(x,u_n)-g(x,u_2)](u_n-u_2)\;dx=o(1),$$
as $n\ri\infty$.
By Lemma \ref{l8}  we deduce that $u_n$ converges strongly
to $u_2$ in $E$ and using relation \eq{MPT} we find
$$J(u_2)=c>0\;\;\;{\rm and}\;\;\;J^{'}(u_2)=0.$$
Therefore, $J(u_2)=c>0$ and $J^{'}(u_2)=0$. By Lemma \ref{l5} we
deduce that $0\leq u_2\leq u_1$ in $\Omega$. Therefore
$$g(x,u_2)=u_2^{\gamma-1}-u_2^{\beta-1}\;\;\;{\rm and}\;\;\;
G(x,u_2)=\frac{1}{\gamma}u_2^\gamma-\frac{1}{\beta}u_2^\beta$$
and thus
$$J(u_2)=I(u_2)\;\;\;{\rm and}\;\;\;J^{'}(u_2)=I^{'}(u_2).$$
We conclude that $u_2$ is a critical point of $I$ and thus a
solution of \eq{2}. Furthermore, $I(u_2)=c>0$ and $I(u_2)>0>I(u_1)$.
Thus $u_2$ is not trivial and $u_2\neq u_1$.
The proof of Theorem \ref{t1} is now complete.
\qed











\begin{thebibliography}{99}
{\footnotesize
\bibitem{AM} Acerbi, E. \& Mingione, G. 2001 Regularity results for a
class of functionals with nonstandard growth, {\it Arch. Rational
Mech. Anal.} {\bf 156}, 121-140.

\bibitem{AR} Ambrosetti, A. \& Rabinowitz, P. H. 1973 Dual variational
methods in critical point theory,
{\it J. Funct. Anal.} {\bf 14}, 349-381.

\bibitem{B} Brezis, H. 1992 {\it Analyse fonctionnelle: th\'eorie et
applications}, Masson, Paris.

\bibitem{CF} Chabrowski, J. \& Fu, Y. 2005 Existence of solutions for
$p(x)$-Laplacian problems on bounded domains, {\it J. Math. Anal.
Appl.}, in press (doi:10.1016/j.jmaa.2004.10.028).

\bibitem{D} Diening, L. 2002 {\it Theorical and numerical results for
electrorheological fluids}, Ph.D. thesis, University of Freiburg,
Germany.

\bibitem{D1} Diening, L. H\"ast\"o, P. \& Nekvinda, A. 2004 Open problems
in variable exponent Lebesgue and Sobolev spaces, 
{\it http://www. math.cas.cz/fsdona2004/diening.pdf}.

\bibitem{edm} Edmunds, D. E. Lang, J. \& Nekvinda, A. 1999 On $L^{p(x)}$
norms, {\it Proc. Roy. Soc. London Ser.~A} {\bf 455}, 219-225.

\bibitem{edm2} Edmunds, D. E. \& R\'akosn\'{\i}k, J. 1992 Density of smooth
functions in $W^{k,p(x)}(\Omega)$, {\it Proc. Roy. Soc. London Ser.~A}
{\bf 437}, 229-236.

\bibitem{edm3} Edmunds, D. E. \&  R\'akosn\'{\i}k, J. 2000 Sobolev embedding
with variable exponent, {\it Studia Math.} {\bf 143}, 267-293.

\bibitem{H} El Hamidi, A. 2004 Existence results to elliptic systems with
nonstandard growth conditions, {\it Publ. D\'epartment de Math\'ematiques},
Universit\'e de La Rochelle, {\bf 12}, 1-15.

\bibitem{FZh} Fan, X. L. \& Zhang, Q. H. 2003 Existence of solutions for
$p(x)$-Laplacian Dirichlet problem, {\it Nonlinear Anal} {\bf 52}, 1843-1852.

\bibitem{FZZ} Fan, X. L. Zhang, Q. H. \& Zhao, D. 2005 Eigenvalues of
$p(x)$-Laplacian Dirichlet problem, {\it J. Math. Anal. Appl.}
{\bf 302}, 306-317.

\bibitem{FZ1} Fan, X. L. \& Zhao, D. 2001  On the Spaces $L^{p(x)}(\Omega)$
and $W^{m,p(x)}(\Omega)$, {\it J. Math. Anal. Appl.}, {\bf 263}, 424-446.

\bibitem{GT} Gilbarg, D. \& Trudinger, N. S. 1998 {\it Elliptic Partial
Differential Equations of Second Order}, Springer, Berlin.

\bibitem{hal} Halsey, T. C. 1992 Electrorheological fluids, {\it Science} 
{\bf 258} , 761-766.

\bibitem{KR} Kov\'a\v cik, O. \& R\'akosn\'{\i}k, J. 1991 On spaces 
$L^{p(x)}$ and $W^{1,p(x)}$, {\it Czechoslovak Math. J.} {\bf 41} , 592-618.

\bibitem{M} Musielak, J. 1983 {\it Orlicz Spaces and Modular  Spaces},
Lecture Notes in Mathematics, Vol. 1034, Springer, Berlin.

\bibitem{nak} Nakano, H. 1950 {\it Modulared Semi-ordered Linear Spaces},
Maruzen Co., Ltd., Tokyo.

\bibitem{orl} Orlicz, W. 1931 \"Uber konjugierte Exponentenfolgen, {\it 
Studia Math.} {\bf 3}, 200-212.

\bibitem{pfe} Pfeiffer, C. Mavroidis, C. Bar-Cohen, Y. \& Dolgin, B. 1999 
Electrorheological fluid based force feedback device, in {\it 
Proceedings of the 1999 SPIE Telemanipulator and Telepresence Technologies VI
Conference (Boston, MA)}, Vol. 3840, pp. 88-99.

\bibitem{P} Perera, K. 2003 Multiple positive solutions for a class
of quasilinear elliptic boundary-value problems, {\it Electronic
Journal of Differential Equations} {\bf 7}, 1-5.

\bibitem{RR} Rajagopal, K. R. \& Ruzicka, M. 2001 Mathematical modelling
of electrorheological fluids, {\it Cont. Mech. Term.} {\bf 13}, 59-78.

\bibitem{R} Ruzicka, M. 2002 {\it Electrorheological Fluids Modeling
and Mathematical Theory}, Springer-Verlag, Berlin.

\bibitem{sha} Sharapudinov, I. 1978 On the topology of the space
$L^{p(t)}([0;1])$, {\it Matem. Zametki} {\bf 26},  613-632.

\bibitem{S} Struwe, M. 1996  {\it Variational Methods: Applications to
Nonlinear Partial Differential Equations and Hamiltonian Systems},
Springer, Heidelberg.

\bibitem{tse} Tsenov, I. 1961 Generalization of the problem of best 
approximation of a function in the space $L^s$, {\it Uch. Zap. Dagestan Gos. Univ.} 
{\bf 7}, 25-37.

\bibitem{W} Willem, M. 1996 {\it Minimax Theorems}, Birkh\"auser, Boston.

\bibitem{Z1} Zhikov, V. 1987 Averaging of functionals of the calculus of
variations and elasticity theory, {\it Math. USSR Izv.} {\bf 29}, 33-66.

}
\end{thebibliography}
\end{document}


