\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 90, pp. 1--15.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2011 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2011/90\hfil Compactness results] {Compactness results for quasilinear problems with variable exponent on the whole space} \author[O. Allegue, A. K. Souayah\hfil EJDE-2011/90\hfilneg] {Olfa Allegue, Asma Karoui Souayah} % in alphabetical order \address{Olfa Allegue \newline Institut Pr\'eparatoire aux Etudes d'ing\'enieurs de Tunis\\ 2 Rue Jawaher Lel Nehru, 1008, Montfleury-Tunis, Tunisia} \email{allegue\_olfa@yahoo.fr} \address{Asma Karoui Souayah\newline Institut Pr\'eparatoire aux Etudes d'ing\'enieurs de Tunis\\ 2 Rue Jawaher Lel Nehru, 1008, Montfleury-Tunis, Tunisia} \email{asma.souayah@yahoo.fr} \thanks{Submitted May 2, 2011. Published July 6, 2011.} \subjclass[2000]{35J655, 35J60, 35J70, 58E05} \keywords{$p(x)$-Laplace operator; critical Sobolev exponent; compactness; \hfill\break\indent Palais-Smale condition} \begin{abstract} In this work we give a compactness result which allows us to prove the point-wise convergence of the gradients of a sequence of solutions to a quasilinear inequality and for an arbitrary open set. This result suggests solutions to many problems, notably nonlinear elliptic problems with critical exponent. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{proposition}[theorem]{Proposition} \allowdisplaybreaks \section{Introduction and preliminary results} In their recent work El Hamidi and Rakotoson \cite{hami} gave a compactness result to prove the point-wise convergence of the gradients of a sequence of solutions to a general quasilinear inequality and for an arbitrary open set. They proved the following result. \begin{lemma}\label{leml} Let $\hat{a}$ be a Carath\'eodory function from $ \mathbb{R}^N\times \mathbb{R} \times \mathbb{R}$ into $\mathbb{R}^N$ satisfying the usual Leray-Lions growth and monotonicity conditions. Let $(u_n)$ be a bounded sequence of $W^{1,p}_{\rm loc}(\mathbb{R}^N)=\{v \in L^{p}_{\rm loc}(\mathbb{R}^N), |\nabla v | \in L^{p}_{\rm loc}(\mathbb{R}^N) \}$, with $1
1
\text{ for all }x\in\overline\Omega\}.
$$
For any $h\in C_+(\overline{\Omega})$ we define
$$
h^+=\sup_{x\in\Omega}h(x)\quad\text{and}\quad
h^-=\inf_{x\in\Omega}h(x).
$$
For any $p\in C_+(\overline{\Omega})$, we define the variable
exponent Lebesgue space
\[
L^{p(x)}(\Omega)=\{u: \text{$u$ is a Borel real-valued function on }
\Omega, \int_\Omega|u(x)|^{p(x)}\,dx<\infty \}.
\]
We define on $L^{p(x)}$, the so-called \emph{Luxemburg norm},
by the formula
$$
|u|_{p(x)}:=\inf\big\{\mu>0:\int_\Omega|\frac{u(x)}{\mu}|^{p(x)}\,dx
\leq 1\big\}.
$$
Variable exponent Lebesgue spaces resemble classical Lebesgue
spaces in many aspects: they are separable and Banach spaces
\cite[Theorem 2.5; Corollary 2.7]{KR} and the H\"older inequality holds
\cite[Theorem 2.1]{KR}. The inclusions between Lebesgue spaces
are also naturally generalized \cite[Theorem 2.8]{KR}:
if $0 <|\Omega|<\infty$ and
$r_1$, $r_2$ are variable exponents so that
$r_1(x) \leq r_2(x)$ almost everywhere in $\Omega$ then there exists
the continuous embedding
$L^{r_2(x)}(\Omega)\hookrightarrow L^{r_1(x)}(\Omega)$.
We denote by $L^{p'(x)}(\Omega)$ the conjugate space of
$L^{p(x)}(\Omega)$, where $1/p(x)+1/p'(x)=1$. For any $u\in
L^{p(x)}(\Omega)$ and $v\in L^{p'(x)}(\Omega)$ the H\"older type
inequality
\begin{equation}\label{eq1}
\big|\int_\Omega uv\,dx\big|\leq\big(\frac{1}{p^-}+
\frac{1}{{p'}^-}\big)|u|_{p(x)}|v|_{p'(x)},
\end{equation}
is held.
An important role in manipulating the generalized Lebesgue-Sobolev
spaces is played by the \emph{modular} of the $L^{p(x)}(\Omega)$
space, which is the mapping
$\rho_{p(x)}:L^{p(x)}(\Omega)\to \mathbb{R}$ defined by
$$
\rho_{p(x)}(u)=\int_\Omega|u|^{p(x)}\,dx.
$$
The space $W^{1,p(x)}(\Omega)$ is equipped by the norm
$$
\|u\|=| u|_{p(x)}+|\nabla u|_{p(x)}.
$$
We recall that if $(u_n), u, \in W^{1,p(x)}(\Omega)$ and
$p^+<\infty$ then the following relations hold:
\begin{gather}\label{eq2}
\min(|u|_{p(x)}^{p^{-}},|u|_{p(x)}^{p^{+}})\leq
\rho_{p(x)}(u)\leq \max(|u|_{p(x)}^{p^{-}},|u|_{p(x)}^{p^{+}}),\\
\label{eq3}
\min(|\nabla u|_{p(x)}^{p^{-}},|\nabla u|_{p(x)}^{p^{+}})\leq
\rho_{p(x)}(|\nabla u|)\leq \max(|\nabla u|_{p(x)}^{p^{-}},
|\nabla u|_{p(x)}^{p^{+}}),\\
\label{eq4}
\begin{gathered}
|u|_{p(x)}\to 0\;\Leftrightarrow\;\rho_{p(x)}
(u)\to 0,\\
\lim_{n\to\infty}|u_n-u|_{p(x)}=0\;
\Leftrightarrow\;\lim_{n\to\infty}\rho_{p(x)}(u_n-u)=0,\\
|u_n|_{p(x)}\to \infty\;\Leftrightarrow\;\rho_{p(x)} (u_n)\to \infty.
\end{gathered}
\end{gather}
We define also $W_{0}^{1,p(x)}(\Omega)$ as the closure of
$C_{0}^{\infty}(\Omega)$ under the norm
$$
\|u\|_{p(x)}=|\nabla u|_{p(x)}.
$$
The space $(W_{0}^{1,p(x)}(\Omega),\|\cdot\|)$ is a separable and
reflexive Banach space.
Next, we recall some embedding results regarding variable exponent
Lebesgue-Sobolev spaces. We note that if $s(x)\in
C_+(\overline{\Omega})$ and $s(x)< p^*(x)$ for all $x\in
\overline{\Omega} $ then the embedding
$W_0^{1,p(x)}(\Omega)\hookrightarrow L^{s(x)}(\Omega)$ is compact
and continuous, where $p^*(x)=Np(x)/(N-p(x))$ if $p(x) 0$:
$$
\limsup_{n\to +\infty}\int_{\Omega} \hat{a}(x,u_n(x),
\nabla u_n(x))\cdot \nabla(\phi S_{\epsilon}(u_n-u^k))
\leq o(1)
$$
as $\epsilon\to 0$ then there exists a subsequence still denoted
$(u_n)$ such that
$$
\nabla u_n(x) \to \nabla u(x)\quad
\text{ a.e. in } \Omega \text{ as } n \to +\infty.
$$
\end{itemize}
\end{theorem}
\begin{remark} \label{rmk1}\rm
\begin{enumerate}
\item The term $o(1)$ in (ii) might depend on $k$ and $\phi$.
\item (L2) is satisfied if for all $\omega\subset\subset\Omega$,
there is a constant $c_{\omega}>0$ and a function
$a_0\in L^{p'(x)}(\omega)$ such that for almost every
$x\in\omega$, for all $(\sigma,\xi)\in\mathbb{R}\times\mathbb{R}^N$:
$$
|\hat{a}(x,\sigma,\xi)|\leq c_{\omega}[|\sigma|^{p(x)-1}
+|\xi|^{p(x)-1}+a_0(x)].
$$
and (L4) is true if $\hat{a}(x,\sigma,\xi)\cdot\xi
\geq c_{\omega}^1|\xi|^{p(x)}-c_{\omega}^2$, $c_{\omega}^1>0$.
\item Bounded sets in $W_{\rm loc}^{1,p(x)}(\Omega)$ will be
bounded in
$$
W^{1,p(x)}(\omega)=\{v\in L^{p(x)}(\omega),
\nabla v\in L^{p(x)}(\omega)\},\quad\text{for every }
\omega \subset\subset \Omega.
$$
\end{enumerate}
\end{remark}
\begin{proof}[Proof of theorem \ref{thm1}]
(i) Let $(w_j)_{j\geq 0}$ be a sequence of bounded relatively
compact subsets of $\Omega$ such that
$\overline{\omega}_j\subset \omega_{j+1}$ and
$ \cup_{j=0}^{+\infty}\omega_j=\Omega$.
Since $(u_n)_n$ is bounded in $W^{1,p(x)}(\omega_j)$,
by the usual embeddings, we deduce that there is a subsequence
$u_{n_{j}}$ and a function $u$ in $W^{1,p(x)}(\omega_j)$
such that $u_{n_{j}}(x)\to u(x)$ as $n\to \infty$.
We conclude with the usual diagonal Cantor process.
(ii) Let $\phi\in C_c^{\infty}(\Omega), 0\leq \phi\leq 1, \phi=1$
on $\omega_j$ and supp$(\phi)\subset\omega_{j+1}$, and set
$$
\Delta(u_n,u)(x)=[\hat{a}(x,u_n(x),\nabla u_n(x))
-\hat{a}(x,u_n(x),\nabla u(x))]\nabla (u_n-u)(x).
$$
Then one has:
\begin{itemize}
\item[(ii.1)] $\Delta(u_n,u)(x)\geq 0$ a.e. on $\Omega$ (due to (L3)).
\item[(ii.2)] $ \sup_n \int_{\omega_{j+1}} \Delta (u_n,u)dx$ is finite
(since $(u_n)$ is in a bounded set of $W^{1,p(x)}_{\rm loc}(\Omega)$
and the growth condition (L2)).
\end{itemize}
Let us show that
$ \lim_n\int_{\Omega}\phi\Delta (u_n,u)^{\frac{1}{p(x)}}dx=0$.
On one hand,
\begin{equation}\label{R1}
\int_{\Omega}\phi\Delta (u_n,u)^{\frac{1}{p(x)}}dx
=\int_{\{|u|>k\}}\phi\Delta (u_n,u)^{\frac{1}{p(x)}}dx
+\int_{\{|u|\leq k\}}\phi\Delta (u_n,u)^{\frac{1}{p(x)}}dx.
\end{equation}
By the H\"older inequality
\[
\int_{\{|u|>k\}}\Delta (u_n,u)^{\frac{1}{p(x)}}\phi\,dx
\leq |\Delta(u_n,u)^{\frac{1}{p(x)}}|_{p(x)}
|\phi|_{\frac{p(x)}{p(x)-1}}
\leq a_1(j)|\phi|_{\frac{p(x)}{p(x)-1}},
\]
where $a_s(j)$ are different constants depending on $j$ but
independent of $n$, $\epsilon$ and $k$.
Noticing that
$$
\operatorname{meas}\{x\in w_{j+1}:|u|>k\}\leq\frac{c_1(j)}{k^{p^-}},
$$
one deduces that
\begin{equation}\label{R2}
\rho_{\frac{p(x)}{p(x)-1}}(\phi)
=\int_{\{|u|>k\}}\phi^{\frac{p(x)}{p(x)-1}}dx
\leq \frac{c_1(j)}{k^{p^-}}
\end{equation}
where $c_m(j)$ are different constants depending on $j$ and
$\phi$ but independent of $n$, $\epsilon$ and $k$.
We conclude that
\begin{equation}\label{asm}
\limsup_{n\to \infty}\int_{\{|u|>k\}}\Delta (u_n,u)^{\frac{1}{p(x)}}
\phi\,dx\leq o(1)\quad \text{as } k\to \infty
\end{equation}
While for the second integral, we have
\begin{equation}\label{R3}
\begin{split}
\int_{\{|u|\leq k\}}\Delta (u_n,u)^{\frac{1}{p(x)}}\phi\,dx
&=\int_{\{|u|\leq k\}\cap \{|u_n-u|\leq \epsilon|\}}
\Delta (u_n,u)^{\frac{1}{p(x)}}\phi\,dx\\
&\quad +\int_{\{|u|\leq k\}
\cap \{|u_n-u|>\epsilon\}}\Delta (u_n,u)^{\frac{1}{p(x)}}\phi\,dx.
\end{split}
\end{equation}
Moreover, the second term in the right hand side in the last
inequality satisfies
\[
\int_{\{|u|\leq k\}\cap \{|u_n-u|>\epsilon\}}
\Delta (u_n,u)^{\frac{1}{p(x)}}\phi\,dx
\leq |\Delta(u_n,u)^{\frac{1}{p(x)}}|_{p(x)}
|\phi|_{\frac{p(x)}{p(x)-1}}\\
\leq a_2(j) |\phi|_{\frac{p(x)}{p(x)-1}}
\]
and
$$
\rho_{\frac{p(x)}{p(x)-1}}(\phi)\leq a_2(\phi)
\operatorname{meas} \{x\in w_{j+1}: |u_n-u|>\epsilon\}.
$$
Since $(u_n)$ converges to $u$ in measure, we deduce that,
for $n$ sufficiently large,
$\operatorname{meas} \{x\in w_{j+1}: |u_n-u|>\epsilon\}\leq \epsilon$.
It follows that
\begin{equation}\label{R4}
\limsup_{n\to +\infty}\int_{\{|u|\leq k\}\cap
\{|u_n-u|>\epsilon\}}\Delta (u_n,u)^{\frac{1}{p(x)}}\phi\,dx
\leq o(1) \quad \text{as }\epsilon\to 0.
\end{equation}
Setting $A_{n,k}^{\epsilon}
=w_{j+1}\cap\{|u|\leq k\}\cap \{|u_n-u|\leq\epsilon\}$,
we obtain from the H\"older inequality
\begin{equation} \label{R5}
\int_{A_{n,k}^{\epsilon}}\Delta (u_n,u)^{\frac{1}{p(x)}}\phi\,dx
\leq c_2(j) |\Delta(u_n,u)^{\frac{1}{p(x)}}\phi^{\frac{1}{p(x)}}
|_{p(x)},
\end{equation}
and
\[
\rho_{p(x)}(\Delta(u_n,u)^{\frac{1}{p(x)}}\phi^{\frac{1}{p(x)}})
=I_{n,k}^1(\epsilon)-I_{n,k}^2(\epsilon),
\]
with
\begin{gather*}
I_{n,k}^1(\epsilon)=\int_{A_{n,k}^{\epsilon}}
\hat{a}(x,u_n,\nabla u_n)\cdot\nabla(u_n-u)\phi\,dx,\\
I_{n,k}^2(\epsilon)=\int_{\{|u|\leq k\}} \hat{a}(x,u_n,\nabla u)
\cdot\nabla S_{\epsilon}(u_n-u)\phi\,dx.
\end{gather*}
Since $\hat{a}(x,u_n,\nabla u)\to \hat{a}(x,u,\nabla u)$
strongly in $L^{p'(x)}(w_{j+1})$ (by the last statement of (L2))
and $\nabla S_{\epsilon}(u_n-u)\rightharpoonup 0$ in
$L^{p(x)}(w_{j+1})$-weak, we deduce that
\begin{equation}\label{R6}
\lim_{n\to +\infty} I_{n,k}^2(\epsilon)=0,
\end{equation}
while for the term $I_{n,k}^1(\epsilon)$, we obtain
\begin{equation}
I_{n,k}^1(\epsilon)\leq\int_{\Omega}\hat{a}(x,u_n,\nabla u_n)
\cdot\nabla(\phi S_{\epsilon}(u_n-u^k))-\int_{\Omega}
\hat{a}(x,u_n,\nabla u_n)\cdot\nabla\phi S_{\epsilon}(u_n-u^k)dx.
\end{equation}
Since
\begin{equation}
\big|\int_{\Omega}\hat{a}(x,u_n,\nabla u_n)\cdot\nabla
\phi S_{\epsilon}(u_n-u^k)dx\big|\leq c_3(j)\epsilon;
\end{equation}
then assumption (ii) implies
\begin{equation}\label{R9}
\limsup_{n\to +\infty} I_{n,k}^1(\epsilon)
\leq c_3(j)\epsilon+\circ (1) \quad \text{as } \epsilon\to 0.
\end{equation}
Combining relations \eqref{R5}, \eqref{R6} and \eqref{R9}, it follows
that
\begin{equation}\label{R10}
\limsup_{n\to +\infty}\int_{A_{n,k}^{\epsilon}}
\Delta (u_n,u)^{\frac{1}{p(x)}}\phi\,dx\leq o(1)\quad
\text{as }\epsilon\to 0.
\end{equation}
Letting first $\epsilon\to 0$ and then $k$ to infinity,
by relations \eqref{R1}, \eqref{asm}, \eqref{R3}, \eqref{R4}
and \eqref{R10}, we deduce
$$
\lim_{n\to +\infty}\int_{\Omega}\Delta (u_n,u)^{\frac{1}{p(x)}}
\phi\,dx=0.
$$
We then obtain that for a subsequence $(u_{j_n})$,
$$
\Delta (u_{j_n},u)(x)\to 0 \quad \text{a.e. on } w_j.
$$
Arguing as Leray-Lions \cite{JLL, jll}, we deduce from (L4) that
$\nabla u_{j_n}(x)\to \nabla u(x)$ a.e. on $w_j$.
The proof is achieved by the diagonal process of Cantor.
\end{proof}
\begin{proof}[Proof of lemma \ref{lem1}]
Since $(u_n)$ belongs to a bounded set of $W_{\rm
loc}^{1,p(x)}(\mathbb{R}^N)$, statement (i) of Theorem \ref{thm1}
implies that there is a function $u$ and a subsequence still
denoted by $(u_n)$ such that
$$
u_n(x)\to u(x)\quad \text{a.e. in $\mathbb{R}^N$, as } n\to \infty
$$
and
$$
u\in W_{\rm loc}^{1,p(x)}(\mathbb{R}^N).
$$
Then for all $\phi\in C_c^{\infty}(\mathbb{R}^N)$,
$\phi S_{\epsilon}(u_n-u^k)$ is an element of
$W_{\rm comp}^{1,p(x)}(\mathbb{R}^N)$ and
\begin{equation}
\big|\int_{\mathbb{R}^N}f_n\phi S_{\epsilon}(u_n-u^k)dx\big|
\leq \epsilon |\phi|_{\infty}|f_n|_{L^1(\omega)}\leq c(\phi)\epsilon,
\end{equation}
(for every $\phi$ such that $\operatorname{supp}(\phi)\subset\omega$,
$\overline{\omega}$ is a compact of $\mathbb{R}^N$), and
$$
|\langle g_n,\phi S_{\epsilon}(u_n-u^k)\rangle|
\leq|g_n|_{W^{-1,p'(x)}(\omega)}|\phi S_{\epsilon}
(u_n-u^k)|_{W^{1,p(x)}(\mathbb{R}^N)}.
$$
Using the fact that
$|\phi S_{\epsilon}(u_n-u^k)|_{W^{1,p(x)}(\mathbb{R}^N)}$
is bounded independently of $\epsilon$, $n$, $k$ and that
$|g_n|_{W^{-1,p'(x)}(\omega)}\to 0$, it holds:
$$
\limsup_n\int_{\mathbb{R}^N}\hat{a}(x,u_n,\nabla u_n)\cdot
\nabla(\phi S_{\epsilon}(u_n-u^k))dx\leq O(\epsilon).
$$
Finally, by Theorem \ref{thm1} we complete the proof.
\end{proof}
\section{Examples of applications}
In this section, we are interested in the existence of solutions
to the problem
\begin{equation}\label{P1}
\begin{gathered}
-\mathop{\rm div}\big(\left( |\nabla u(x)|^{p(x)-2}\right)
\nabla u(x)\big) = \lambda f(u)+g(u) \quad \text{for }x\in\Omega,\\
u\geq 0 \quad \text{for } x\in\Omega,\\
u=0 \quad \text{for } x\in\partial\Omega,
\end{gathered}
\end{equation}
where $\Omega \subset\mathbb{R}^N$, ($N\geq3$) is a
bounded domain with smooth boundary, $\lambda $ is a positive real
number and $p$ is
a continuous function on $\overline{\Omega}$ with $p^+ 0$ be such
that $q^-+\epsilon_0 0$.
In fact if
$$
\int_{\Omega}|\nabla
\phi|^{p(x)}dx+C_3\int_{\Omega}|\phi|^{p^*(x)}dx =0,
$$
then $\int_{\Omega}|\phi|^{p^*(x)}dx =0$. Using relation \eqref{eq2},
we deduce that $|\phi|_{p^*(x)}=0$ and consequently $\phi=0$
in $\Omega$ which is a contradiction. The proof is complete.
\end{proof}
\subsection*{Proof of theorem \ref{thm2}}
Let $\lambda^*$ be defined as in \eqref{eq25} and $\lambda
\in (0,\lambda^{*})$. By Lemma \ref{lem2} it follows that on the
boundary of the ball centered at the origin and of radius $\rho$ in
$E$, denoted by $B_{\rho}(0)$, we have
\begin{equation}\label{eq26}
\inf_{\partial B_{\rho}(0)}J_{\lambda}>0.
\end{equation}
On the other hand, by Lemma \ref{lem3}, there exists $\phi \in E$
such that $J_{\lambda}(t\phi)<0$, for all $t>0$ small enough.
Moreover, relations \eqref{eq2} and \eqref{eq23} imply,
that for any $u \in B_{\rho}(0)$, we have
\[
J_{\lambda}(u)\geq\frac{1}{p^{+}}\|u\|^{p^{+}}
-\frac{\lambda}{q^{-}}C^{q^-}_2M^{q^-}_1\|u\|^{q^{-}}
-\frac{C^{p^{*-}}_3}{q^{-}}M^{q^-}_2\|u\|^{p^{*-}}.
\]
It follows that
$$
-\infty0$ problem \eqref{P1} has infinitely many weak
solutions provided that $p^{*-}>max(p^{+},q^{+})$.
\end{theorem}
\subsection*{Proof of Theorem \ref{thm2}}
Let $E$ denote the generalized Sobolev space
$W^{1,p(x)}_{0}(\Omega)$. The energy functional corresponding to
\eqref{P1} is $J_{\lambda}: E \to \mathbb{R}$, defined as
$$
J_{\lambda}(u):=\int_{\Omega}\frac{1}{p(x)}|\nabla
u|^{p(x)}dx-\lambda\int_{\Omega}F(u_{+})dx- \int_{\Omega}G(u_{+})dx,
$$
where $u_{+}(x)=\max\{u(x),0\} $ and $F$ is defined by
$F(t)=\int^{t}_{0}f(s)ds$.
\begin{remark} \label{rmk2} \rm
Assume that condition (G1) is fulfilled, it is clear that for
every $t\geq 0$, we obtain
$$
-\frac{C_3}{p^{*-}}t^{p^{*}(x)}\leq G(t)\leq
\frac{C_3}{p^{*-}}t^{p^{*}(x)}
$$
\end{remark}
\begin{proposition} \label{prop1}
The functional $J_{\lambda}$ is well-defined on $E$ and
$J_{\lambda} \in C^{1}(E,\mathbb{R})$.
\end{proposition}
\begin{proof}
We have the following continuous embedding
(see \cite[Theorem 2.8]{KR})
$$
W^{1,p(x)}_{0}(\Omega)\hookrightarrow L^{p^{*}(x)}(\Omega)
$$
using the fact that $\Omega$ is bounded, we obtain the continuous
embedding
$$
W^{1,p(x)}_{0}(\Omega)\hookrightarrow L^{s(x)}(\Omega),
\quad s\in [1,p^{*}],
$$
which implies that $J_{\lambda}$ is well-defined on $E$
and $J_{\lambda}\in C^{1}(E,\mathbb{R})$, with the derivative
given by
$$
\langle dJ_{\lambda}(u),v\rangle=\int_{\Omega}
\big(|\nabla u|^{p(x)-2}\nabla u \nabla v-\lambda f(u)
v -g(u) v \big)dx,\quad \forall v \in E.
$$
\end{proof}
The proof of Theorem \ref{thm2} is related to Ekeland's variational
principle. In order to apply it we need the following lemmas:
\begin{lemma}\label{lem2}
Under hypotheses of theorem \ref{thm2}, there
exists $M_2>0$ such that for all $\rho \in(0,1)$ for all
$C_3<\frac{q^-}{p^+ M^{p^{*-}}_2}\rho^{p^+-q^-}$, there exists
$\lambda^{*}>0$ and $r>0$ such that, for all $u\in E$ with
$\|u\|=\rho$,
$J_{\lambda}(u)\geq r>0$ for all $\lambda \in(0,\lambda^{*})$.
\end{lemma}
\begin{proof}
Since $E\hookrightarrow L^{q(x)}(\Omega)$ and
$E\hookrightarrow L^{p^{*}(x)}(\Omega)$ are continuous, there
exists $M_1>0$ and $M_2>0$ such that
\begin{equation}\label{eq23}
|u|_{{q(x)}}\leq M_1 \|u\| \quad \text{and} \quad
|u|_{{p^{*}(x)}}\leq M_2 \|u\|, \quad \forall u\in E.
\end{equation}
We fix $\rho \in (0,1)$ such that $ \rho<\min (1,1/M_1,1/M_2)$.
Then for all $u \in E$, with $\|u\|=\rho $, we deduce that
$$
|u|_{{q(x)}}<1 \quad \text{and} \quad |u|_{{p^*(x)}}<1.
$$
Furthermore, by \eqref{eq2} for all $u \in E$ with
$\|u\|=\rho $, we have
$$
\int_{\Omega}|u|^{q(x)}\,dx\leq|u|^{q^-}_{q(x)},\quad \text{and} \quad
\int_{\Omega}|u|^{p^*(x)}\,dx\leq|u|^{p^{*-}}_{p(x)}.
$$
The above inequality and relation \eqref{eq23} imply that
for all $u \in E$ with $\|u\|=\rho $,
\begin{equation}\label{eq24}
\int_{\Omega}|u|^{q(x)}\,dx\leq M_1^{q^-}\|u\|^{q^-}, \quad and \quad
\int_{\Omega}|u|^{p^*(x)}\,dx\leq M_2^{p^{*-}}\|u\|^{p^{*-}}.
\end{equation}
Using relation \eqref{eq24} we deduce that, for any $u \in E$
with $\|u\|=\rho$, the following inequalities hold:
\begin{align*}
J_{\lambda}(u)
&\geq \frac{1}{p^{+}}\|u\|^{p^{+}}-
\frac{\lambda}{q^{-}}C_2M_1^{q^{-}}\|u\|^{q^{-}}
-\frac{C_3}{p^{*-}}M_2^{p^{*-}}\|u\|^{p^{*-}},\\
&\geq \frac{1}{p^{+}}\rho^{p^{+}}-\frac{\lambda}{q^{-}}C_2
M_1^{q^{-}}\rho^{q^{-}}-\frac{C_3}{p^{*-}}M_2^{p^{*-}}\rho^{p^{*-}}.
\end{align*}
By the above inequality we remark that if we define for
all $C_3<\frac{q^-}{p^+ M^{p^{*-}}_2}\rho^{p^+-q^-}$
\begin{equation} \label{eq25}
\lambda^{*}=\frac{q^{-}}{2C_2M_1^{q^{-}}}
\big[ \frac{1}{p^+}\rho^{p^{+}-q^{-}}-\frac{C_3}{q^-} M^{p^{*-}}_2\big],
\end{equation}
then for any $\lambda \in (0,\lambda^*)$, there exists
$ r>0$ such that
$J_{\lambda}(u)\geq r>0$ for all $u\in E$ with
$\|u\|=\rho$.
The proof is complete.
\end{proof}
\begin{lemma}\label{lem3}
There exists $\phi \in E$ such that $\phi \geq0$, $\phi \neq 0$
and $J_{\lambda}(t\phi)<0$, for $t>0$ small enough.
\end{lemma}
\begin{proof}
Since $q^-