\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 137, pp. 1--13.\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-2011/137\hfil Existence of solutions] {Existence of solutions for nonlocal elliptic systems with nonstandard growth conditions} \author[G. Dai\hfil EJDE-2011/137\hfilneg] {Guowei Dai} \address{Guowei Dai\newline Department of Mathematics, Northwest Normal University, Lanzhou, 730070, China} \email{daiguowei@nwnu.edu.cn} \thanks{Submitted July 9, 2011. Published October 19, 2011.} \thanks{Supported by grants 11061030 from the NSFC, and NWNU-LKQN-10-21} \subjclass[2000]{35D05, 35J60, 35J70} \keywords{Variational method; nonlinear elliptic systems; nonlocal condition} \begin{abstract} This article concerns the existence and multiplicity of solutions for a $p(x)$-Kirchhoff-type systems with Dirichlet boundary condition. By a direct variational approach and the theory of the variable exponent Sobolev spaces, under growth conditions on the reaction terms, we establish the existence and multiplicity of solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \allowdisplaybreaks \section{Introduction} In this article, we study the following nonlocal elliptic systems of gradient type with nonstandard growth conditions \begin{equation} \begin{gathered} -M_1\Big(\int_\Omega\frac{1}{p(x)}|\nabla u|^{p(x)}\,dx\Big)\operatorname{div}\big(|\nabla u|^{p(x)-2}\nabla u\big) = \frac{\partial F}{\partial u} (x,u,v)\quad \text{in } \Omega,\\ -M_2\Big(\int_\Omega\frac{1}{q(x)}|\nabla v|^{q(x)}\,dx\Big) \operatorname{div}\big(|\nabla v|^{q(x)-2}\nabla v\big) = \frac{\partial F}{\partial v} (x,u,v)\quad \text{in }\Omega,\\ u=0,\quad v=0\quad \text{on }\partial\Omega, \end{gathered} \label{e1.1} \end{equation} where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ with a smooth boundary $\partial\Omega$, $p(x), q(x)\in C_+(\overline{\Omega})$ with \begin{gather*} 1
1
\,\ \text{for}\,\ \text{any} \,\ x\in\overline{\Omega}\},
\\
h^{-}:=\min_{\overline{\Omega}}h(x),\quad
h^{+}:=\max_{\overline{\Omega}}h(x)\quad \text{for every }
h\in C_+(\overline{\Omega}).
\end{gather*}
Define
\begin{equation*}
L^{p(x)}( \Omega ) =\{u\in {\mathbf{S}}(\Omega
):\int_{\Omega }|u(x)|^{p(x)}\,dx<+\infty \text{ for }
p\in C_+ (\overline{\Omega})\}
\end{equation*}
with the norm
\begin{equation*}
|u|_{L^{p(x)}( \Omega ) }=|u|_{p(x)}
=\inf \{ \lambda >0:\int_{\Omega }
|\frac{ u(x)}{\lambda}|^{p(x)}\,dx\leq 1\},
\end{equation*}%
and
\begin{equation*}
W^{1,p(x) }( \Omega ) =\{ u\in
L^{p(x) }( \Omega ) :|\nabla
u|\in L^{p(x) }( \Omega ) \}
\end{equation*}
with the norm
\begin{equation*}
\| u\| _{W^{1,p(x)}(\Omega )}= |u|_{L^{p(x)}(\Omega )} +|\nabla u|
_{L^{p(x)}(\Omega )}.
\end{equation*}
Denote by $W_{0}^{1,p(x) }( \Omega ) $ the
closure of $C_{0}^{\infty }( \Omega ) $ in $W^{1,p(x) }( \Omega )$.
\begin{proposition}[\cite{f6}] \label{prop2.1}
The spaces $L^{p(x)}( \Omega)$, $W^{1,p(x) }( \Omega ) $ and
$W_{0}^{1,p(x) }( \Omega ) $ are separable and
reflexive Banach spaces.
\end{proposition}
\begin{proposition}[\cite{f6}] \label{prop2.2}
Set $\rho (u)=\int_{\Omega }|u(x)|^{p(x)}\,dx$.
For any $u\in L^{p(x)}( \Omega ) $, then
\begin{itemize}
\item[(1)] for $u\neq 0$, $|u|_{p(x)}=\lambda$
if and only if $\rho (\frac{u}{\lambda })=1$;
\item[(2)] $|u|_{p(x)}<1$ $(=1;>1)$ if and only if
$\rho (u)<1$ $(=1;>1)$;
\item[(3)] if $|u|_{p(x)}>1$, then
$|u|_{p(x)}^{p^{-}}\leq \rho ( u)\leq |u|_{p(x)}^{p^{+}}$;
\item[(4)] if $|u|_{p(x)}<1$, then
$|u|_{p(x)}^{p^{+}}\leq \rho ( u)\leq |u|_{p(x)}^{p^{-}}$;
\item[(5)] $\lim_{k\to +\infty } |u_{k}| _{p(x)}=0$ if and only if
$\lim_{k\to +\infty } \rho (u_{k})=0$;
\item[(6)] $\lim_{k\to +\infty } |u_{k}|_{p(x)}= +\infty$
if and only if $\lim_{k\to +\infty } \rho(u_{k})= +\infty$.
\end{itemize}
\end{proposition}
\begin{proposition}[\cite{f6}] \label{prop2.3}
In $W_{0}^{1,p(x) }( \Omega ) $ the Poincar\'{e}
inequality holds; that is, there exists a positive constant
$C_0$ such that
\begin{equation*}
|u|_{L^{p(x)}(\Omega )}\leq C_0|\nabla
u|_{L^{p(x)}(\Omega )}, \quad \forall u\in W_{0}^{1,p(x) }( \Omega ).
\end{equation*}
\end{proposition}
So, $|\nabla u|_{L^{p(x)}(\Omega )}$ is a norm
equivalent to the norm $\| u\|$ in the space
$W_{0}^{1,p(x) }( \Omega )$. We will use the equivalent norm in
the following discussion and write
$\| u\|_p=|\nabla u|_{L^{p(x)}(\Omega )}$ for simplicity.
\begin{proposition}[\cite{f3,f6}] \label{prop2.4}
If $q\in C_+(\overline{\Omega})$
and $q(x)\leq p^{\ast }(x)$ ($ q(x)< p^{\ast }(x)$) for
$x\in \overline{\Omega}$, then there is a continuous (compact)
embedding $W_0^{1,p(x)}(\Omega )\hookrightarrow
L^{q(x)}(\Omega )$, where
\begin{equation*}
p^*(x)=\begin{cases}
\frac{Np(x)}{N-p(x)}& \text{if } p(x) 0$ such that
\begin{gather*}
|u|_{p^+}\leq c_8\| u\|_p \quad\text{for } u\in W_0^{1,p(x)}(\Omega) \\
|v|_{q^+}\leq c_9\| v\|_q\quad \text{for } v\in W_0^{1,q(x)}(\Omega),
\end{gather*}
where $|\cdot|_r$ denote the norm on $L^{r(x)}(\Omega)$ with
$r\in C_+(\overline{\Omega})$.
Let $\varepsilon>0$ be small enough such that
$\varepsilon c_8^{p^+} \leq\frac{m_0}{2p^+}$ and
$\varepsilon c_9^{q^+} \leq\frac{m_0}{2q^+}$.
By the assumptions (H1) and (H3), we have
\[ % \label{e3.4}
|F(x,s,t)|\leq\varepsilon\big(|s|^{p^+}+|t|^{q^+}\big)
+c(\varepsilon)(|s|^{p_1(x)}+|t|^{q_1(x)}+|
s|^{\alpha(x)}|t|^{\beta(x)})
\]
for all $(x,s,t)\in\Omega\times\mathbb{R}^2$.
In view of (H4) and and the above inequality, for $\| (u, v)\|$ sufficiently small,
noting Proposition \ref{prop2.2}, we have
\begin{align*}
J(u,v)&\geq \frac{m_0}{p^+}\int_\Omega|\nabla
u|^{p(x)}\,dx+\frac{m_0}{q^+}\int_\Omega|\nabla
v|^{q(x)}\,dx-\varepsilon\int_\Omega|u|^{p^+}\,dx-\varepsilon\int_\Omega|
v|^{q^+}\,dx\\
&\quad -c(\varepsilon)\int_\Omega\Big(|u|^{p_1(x)}+|v|^{q_1(x)}+|
u|^{\alpha(x)}|v|^{\beta(x)}\Big)\,dx\\
&\geq \frac{m_0}{p^+}\| u\|_p^{p^+}-\varepsilon c_8^{p^+}\|
u\|_p^{p^+}+\frac{m_0}{q^+}\| v\|_q^{q^+}-\varepsilon c_9^{q^+}\|
v\|_q^{q^+}\\
&\quad -c(\varepsilon)\Big(\| u\|_p^{p_1^-}
+\| v\|_q^{q_1^-}+c_7\| u\|_p^{2\alpha^-}
+c_7\| v\|_q^{2\beta^-}\Big)\\
&\geq \frac{m_0}{2p^+}\| u\|_p^{p^+}+\frac{m_0}{2q^+}\| v\|_q^{q^+}
-c(\varepsilon)\Big(\| u\|_p^{p_1^-}+\| v\|_q^{q_1^-}
+c_7\| u\|_p^{2\alpha^-}+c_7\| v\|_q^{2\beta^-}\Big).
\end{align*}
Since $p_1^-, 2\alpha^->p^+$ and $q_1^-, 2\beta^->q^+$,
there exist $r>0$, $\delta>0$ such that
$J(u)\geq\delta>0$ for every $\| (u,v)\| = r$.
On the other hand, we have known that the assumption (H2)
implies the following assertion:
for every $x\in\overline{\Omega}$, $s, t\in \mathbb{R}$, the inequality
\begin{equation}
F(x,s,t)\geq c_{10}(|s|^{\theta_1}+|t|^{\theta_2}-1)
\end{equation}
holds; see \cite{h1}.
When $t>t_0$, from (H5) we can easily obtain that
\[
\widehat{M}(t)\leq \frac{\widehat{M}(t_0)}{t_0^{1/(1-\mu)}}
t^{1/(1-\mu)}:=c_{11}t^{1/(1-\mu)},
\]
where $t_0$ is an arbitrarily positive constant.
For $(\widetilde{u},\widetilde{v})\in W \setminus \{(0,0)\}$ and $t>1$, we have
\begin{align*}
J(t\widetilde{u},t\widetilde{v})
&= \widehat{M}\Big(\int_{\Omega}\frac{1}{p(x)}|t\nabla
\widetilde{u}|^{p(x)}\,dx\Big)
+\widehat{M}\Big(\int_{\Omega}\frac{1}{q(x)}|t\nabla
\widetilde{v}|^{q(x)}\,dx\Big)\\
&\quad -\int_\Omega F(x,t\widetilde{u},t\widetilde{v})\,dx\\
&\leq c_{12}(\int_\Omega|t\nabla
\widetilde{u}|^{p(x)}\,dx)^{1/(1-\mu)}-c_{10}t^{\theta_1}\int_\Omega
|\widetilde{u}|^{\theta_1}\,dx\\
& \quad +c_{13}\Big(\int_\Omega|t\nabla
\widetilde{v}|^{q(x)}\,dx\Big)^{1/(1-\mu)}
-c_{10}t^{\theta_2}\int_\Omega
|\widetilde{v}|^{\theta_2}\,dx-c_{14}\\
&\leq c_{12}t^{\frac{p^+}{1-\mu}}\Big(\int_\Omega|\nabla
\widetilde{u}|^{p(x)}\,dx\Big)^{1/(1-\mu)}
-c_{10}t^{\theta_1}\int_\Omega
|\widetilde{u}|^{\theta_1}\,dx\\
&\quad +c_{13}t^{\frac{q^+}{1-\mu}}\Big(\int_\Omega|\nabla
\widetilde{v}|^{q(x)}\,dx\Big)^{1/(1-\mu)}
-c_{10}t^{\theta_2}\int_\Omega
|\widetilde{v}|^{\theta_2}\,dx-c_{14} \\
&\to -\infty, \quad \text{as } t\to +\infty,
\end{align*}
due to $\theta_1 > \frac{p^+}{1-\mu}$ and
$\theta_2 > \frac{q^+}{1-\mu}$. Since $J(0,0)=0$, considering
Lemmas \ref{lem3.1} and \ref{lem3.2}, we see that $J$ satisfies
the conditions of Mountain Pass Theorem. So $J$ admits at least one
nontrivial critical point.
\end{proof}
Next we will prove under some symmetry condition on the function
$F$ that \eqref{e1.1} possesses infinitely many nontrivial
weak solutions.
\begin{theorem} \label{thm3.3}
Assume {\rm (H1), (H2), (H4), (H5)},
and that $F(x,u,v)$ is even in $u$, $v$.
Then \eqref{e1.1} has a sequence of solutions
$\{(\pm u_k,\pm v_k)\}_{k=1}^{\infty}$ such that
$J(\pm u_k,\pm v_k)\to +\infty$ as
$k\to +\infty$.
\end{theorem}
Because $W_0^{1,p(x)}$ and $W_0^{1,q(x)}$ are a reflexive and
separable Banach space, then $W$ and $W^*$ are too. There exist
$\{e_j\}\subset W$ and $\{e_j^*\}\subset W^*$ such that
\[
W=\overline{\mathrm{span}\{e_j:j=1,2, \dots\}},\quad
W^*=\overline{\mathrm{span}\{e_j^*:j=1,2,\dots\}},
\]
and
\[
\langle e_i,e_j^*\rangle=\begin{cases}
1, & i=j,\\
0, & i\neq j,
\end{cases}
\]
where $\langle\cdot,\cdot\rangle$ denotes the duality product
between $W$ and $W^*$.
For convenience, we write $X_j = \operatorname{span}\{e_j\}$,
$Y_k = \oplus_{j=1}^kX_j , Z_k = \overline{\oplus_{j=k}^\infty X_j}$.
We will use the following ``Fountain
theorem'' to prove Theorem \ref{thm3.3}.
\begin{lemma}[\cite{w1}] \label{lem3.3}
Assume
\begin{itemize}
\item[(A1)] $X$ is a Banach space, $I\in C^1(X,\mathbb{R})$
is an even functional.
\item[(A2)]
For each $k = 1, 2, \dots$, there exist $\rho_k >r_k >0$ such
that
\item[(A2)] $\inf_{u\in Z_k, \| u\| =r_k} I(u)\to+\infty$
as $k\to+\infty$.
\item[(A3)] $\max_{u\in Y_k, \| u\| =\rho_k} I(u)\leq0$.
\item[(A4)] $I$ satisfies Palais-Smale condition for every $c>0$.
\end{itemize}
Then $I$ has a sequence of critical values tending to $+\infty$.
\end{lemma}
For every $a >1$, $u,v\in L^a(\Omega)$, we define
\[
|(u,v)|_a:=\max\{|u|_a,|v|_a\}.
\]
Set
\begin{gather*}
a:=\max_{x\in\overline{\Omega}}\{2\alpha(x),2\beta(x),p_1(x),q_1(x)\}
>\min\{p^-,q^-\},
\\
b:=\min_{x\in\overline{\Omega}}\{2\alpha(x),2\beta(x),p_1(x),q_1(x)\}>0.
\end{gather*}
Then we have the following result.
\begin{lemma}[\cite{h1}] \label{lem3.2b}
Denote
\[
\beta_k=\sup\{|(u,v)|_{a}:\| (u,v)\|=1, (u,v)\in Z_k\}.
\]
Then $\lim_{k\to+\infty} \beta_k=0$.
\end{lemma}
\begin{proof}[Proof of Theorem \ref{thm3.3}]
According to the assumptions on $F$, Lemmas \ref{lem3.1} and
\ref{lem3.2}, $J$
is an even functional and satisfies Palais-Smale condition.
We will prove that if $k$ is large enough, then there exist
$\rho_k>r_k>0$ such that ($A_2$) and ($A_3$) holding.
Thus, the conclusion can be obtained from
Fountain theorem.
(A2): For any $(u_k,v_k)\in Z_k$, $\| u_k\|_p\geq1$,
$\| v_k\|_q\geq1$ and $\| (u_k,v_k)\| = r_k$ ($r_k$ will be
specified below),
we have
\begin{align*}
&J(u_k,v_k)\\
&= \widehat{M}\Big(\int_{\Omega}\frac{1}{p(x)}|\nabla
u_k|^{p(x)}\,dx\Big)+\widehat{M}
\Big(\int_{\Omega}\frac{1}{q(x)}|\nabla v_k|^{q(x)}\,dx\Big)
-\int_{\Omega}F(x,u_k,v_k)\,dx\\
&\geq m_0\int_{\Omega}\frac{1}{p(x)}|\nabla u_k|^{p(x)}\,dx
+m_0\int_{\Omega}\frac{1}{q(x)}|\nabla
v_k|^{q(x)}\,dx -\int_{\Omega}F(x,u_k,v_k)\,dx\\
&\geq \frac{m_{0}}{p^+}\int_{\Omega}|\nabla
u_k|^{p(x)}\,dx+\frac{m_{0}}{q^+}\int_{\Omega}|\nabla
v_k|^{q(x)}\,dx\\
&\quad -c\int_\Omega\Big(1+|u_k|^{p_1(x)}+|v_k|^{q_1(x)}
+|u_k|^{\alpha(x)}|v_k|^{\beta(x)}\Big)\,dx\\
&\geq \frac{m_{0}}{p^+}\| u_k\|_p^{p^-}
+\frac{m_{0}}{q^+}\| v_k\|_q^{q^-}
-c|u_k|_{p_1(x)}^{p_1(\xi_1^k)}-c|v_k|_{q_1(x)}^{q_1(\xi_2^k)}\\
&\quad -c_{15}|u_k|_{2\alpha(x)}^{2\alpha(\eta_1^k)}
-c_{15}|v_k|_{2\beta(x)}^{2\beta(\eta_2^k)}-c|\Omega|,
\end{align*}
where $\xi_1^k,\xi_2^k,\eta_1^k,\eta_2^k\in\Omega$. Therefore,
\begin{align*}
&J(u_k,v_k)\\
&\geq \frac{m_{0}}{\max\{p^+,q^+\}}\| (u_k,v_k)\|^{\min\{p^-,q^-\}}
-c|u_k|_{a}^{p_1(\xi_1^k)}-c|v_k|_{a}^{q_1(\xi_2^k)}\\
&\quad -c|u_k|_{a}^{2\alpha(\eta_1^k)}
-c|v_k|_{a}^{2\beta(\eta_2^k)}-c|\Omega|\\
&\geq \frac{m_{0}}{\max\{p^+,q^+\}}\| (u_k,v_k)\|^{\min\{p^-,q^-\}}
-c(\beta_k\| (u_k,v_k)\|)^{p_1(\xi_1^k)}
-c(\beta_k\| (u_k,v_k)\|)^{q_1(\xi_2^k)}\\
&\quad -c(\beta_k\| (u_k,v_k)\|)^{2\alpha(\eta_1^k)}
-c(\beta_k\| (u_k,v_k)\|)^{2\beta(\eta_2^k)}-c|\Omega|\\
&\geq \frac{m_{0}}{\max\{p^+,q^+\}}\| (u_k,v_k)\|^{\min\{p^-,q^-\}}
-c_{16}\beta_k^b\| (u_k,v_k)\|^a-c|\Omega|,
\end{align*}
where $a$, $b$ are defined above. At this stage, we fix $r_k$
as follows:
\[
r_k:=\Big(\frac{m_0}{2c_{16}\max\{p^+,q^+\}\beta_k^b}\Big)
^{1/(a-\min\{p^-,q^-\})}\to +\infty\quad \text{as }k\to+\infty.
\]
Consequently, if $\|(u_k, v_k)\|=r_k$ then
\[
J(u_k,v_k)\geq\frac{m_{0}}{2\max\{p^+,q^+\}}\|
(u_k,v_k)\|^{\min\{p^-,q^-\}}-c|\Omega|\to+\infty\quad
\text{as }k\to+\infty.
\]
(A3): From (H2), we have
$F(x, u,v)\geq c_{10}(|u|^{\theta_1}+|v|^{\theta_2}-1)$
for every $x\in\Omega$ and $u,v\in \mathbb{R}$.
Therefore, for any $(u,v)\in Y_k$ with $\| (u,v)\|=1$ and
$1<\rho_k=t_k$ with $t_k\to +\infty$, we have
\begin{align*}
&J(t_ku,t_kv)\\
&= \widehat{M}\Big(\int_{\Omega}\frac{1}{p(x)}|t_k\nabla
u|^{p(x)}\,dx\Big)
+\widehat{M}\Big(\int_{\Omega}\frac{1}{q(x)}|t_k\nabla v|^{q(x)}\,dx\Big)
-\int_\Omega F(x,t_ku,t_kv)\,dx. \\
&\leq c_{17}\Big(\int_\Omega|t_k\nabla u|^{p(x)}\,dx\Big)^{1/(1-\mu)}
+c_{18}\Big(\int_\Omega|t_k\nabla v|^{q(x)}\,dx\Big)^{1/(1-\mu)}\\
&\quad -c_{10}t_k^{\theta_1}\int_\Omega |u|^{\theta_1}\,dx
-c_{10}t_k^{\theta_2}\int_\Omega |v|^{\theta_2}\,dx+c_{19},\\
&\leq c_{17}t_k^{\frac{p^+}{1-\mu}}
\Big(\int_\Omega|\nabla u|^{p(x)}\,dx\Big)^{1/(1-\mu)}
-c_{10}t_k^{\theta_1}\int_\Omega |u|^{\theta_1}\,dx\\
&\quad +c_{18}t_k^{\frac{q^+}{1-\mu}}
\Big(\int_\Omega|\nabla v|^{q(x)}\,dx\Big)^{1/(1-\mu)}
-c_{10}t_k^{\theta_2}\int_\Omega |v|^{\theta_2}\,dx+c_{19}.
\end{align*}
By $\theta_1>\frac{p^+}{1-\mu}$, $\theta_2>\frac{q^+}{1-\mu}$ and $\text{dim} Y_k =k$, it is easy to
see that $J(t_ku,t_kv)\to-\infty$ as $\|
(t_ku,t_kv)\|\to+\infty$ for $(u,v) \in Y_k$.
The proof of Theorem \ref{thm3.3} is completed by the Fountain theorem.
\end{proof}
\subsection*{Acknowledgments}
The author wishes to express his gratitude to the anonymous referee for
reading the original manuscript carefully and making several
corrections and remarks.
\begin{thebibliography}{99}
\bibitem{a1} E. Acerbi and G. Mingione;
\emph{Regularity results for stationary electro-rheologicaluids},
Arch. Ration. Mech. Anal., 164 (2002), 213--259.
\bibitem{a2} E. Acerbi, G. Mingione;
\emph{Gradient estimate for the $p(x)$-Laplacean system},
J. Reine Angew. Math., 584 (2005), 117--148.
\bibitem{a3} A. Arosio, S. Panizzi;
\emph{On the well-posedness of the Kirchhoff
string}, Trans. Amer. Math. Soc. 348 (1996) 305--330.
\bibitem{a4} C. O. Alves, F. J. S. A. Corr\^{e}a, T. F. Ma;
\emph{Positive solutions for a quasilinear elliptic equation
of Kirchhoff type}, Comput. Math. Appl. 49 (2005) 85--93.
\bibitem{a5} S. N. Antontsev, S. I. Shmarev;
\emph{A Model Porous Medium Equation with Variable Exponent
of Nonlinearity: Existence, Uniqueness and
Localization Properties of Solutions}, Nonlinear Anal. 60 (2005),
515--545.
\bibitem{a6} S. N. Antontsev , J. F. Rodrigues;
\emph{On Stationary Thermo-rheological Viscous Flows},
Ann. Univ. Ferrara, Sez. 7, Sci. Mat. 52 (2006), 19--36.
\bibitem{b1} S. Bernstein;
\emph{Sur une classe d'\'{e}quations fonctionnelles aux
d\'{e}riv\'{e}es partielles}, Bull. Acad. Sci. URSS. Ser. Math.
[Izv. Akad. Nauk SSSR] 4 (1940) 17--26.
\bibitem{c1} M. M. Cavalcanti, V. N. Cavalcanti, J.A. Soriano;
\emph{Global existence and uniform decay rates for the
Kirchhoff-Carrier equation with nonlinear dissipation},
Adv. Differential Equations 6 (2001) 701--730.
\bibitem{c2} Y. Chen, S. Levine, M. Rao;
\emph{Variable exponent, linear growth
functionals in image restoration}, SIAM J. Appl.Math. 66 (4) (2006),
1383--1406.
\bibitem{c3} F. J. S. A. Corr\^{e}a, G. M. Figueiredo;
\emph{On a elliptic equation of $p$-kirchhoff type via variational
methods}, Bull. Austral. Math. Soc. Vol. 74 (2006) 263--277.
\bibitem{c4} F. J. S. A. Corr\^{e}a, R.G. Nascimento;
\emph{On a nonlocal elliptic system of $p$-Kirchhoff-type
under Neumann boundary condition}, Mathematical and Computer
Modelling 49 (2009) 598--604.
\bibitem{d1} G. Dai, R. Hao;
\emph{Existence of solutions for a $p(x)$-Kirchhoff-type equation},
J. Math. Anal. Appl. 359 (2009) 275--284.
\bibitem{d2} L. Diening, P. H\"{a}st\"{o}, A. Nekvinda;
\emph{Open problems in variable exponent Lebesgue and Sobolev spaces},
in: P. Dr\'{a}bek, J. R\'{a}kosn\'{\i}k, FSDONA04 Proceedings,
Milovy, Czech Republic, 2004, pp.38--58.
\bibitem{d3} G. Dai, D. Liu;
\emph{Infinitely many positive solutions for a $p(x)$-Kirchhoff-type
equation}, J. Math. Anal. Appl. 359 (2009) 704--710.
\bibitem{d4} G. Dai, J. Wei;
\emph{Infinitely many non-negative solutions for a $p(x)$-Kirchhoff-type
problem with Dirichlet boundary condition}, Nonlinear Analysis
73 (2010) 3420--3430.
\bibitem{d5} P. D'Ancona, S. Spagnolo;
\emph{Global solvability for the degenerate Kirchhoff equation
with real analytic data}, Invent. Math. 108 (1992) 247--262.
\bibitem{d6} D. G. De Figueiredo;
\emph{Semilinear elliptic systems: a survey of superlinear problems},
Resenhas 2 (1996) 373--391.
\bibitem{d7} M. Dreher;
\emph{The Kirchhoff equation for the $p$-Laplacian}, Rend. Semin. Mat.
Univ. Politec. Torino 64 (2006) 217--238.
\bibitem{d8} M. Dreher;
\emph{The wave equation for the $p$-Laplacian},
Hokkaido Math. J. 36 (2007) 21--52.
\bibitem{f1} X. L. Fan;
\emph{On the sub-supersolution methods for $p(x)$-Laplacian equations},
J. Math. Anal. Appl. 330 (2007), 665--682.
\bibitem{f2} X. L. Fan and X. Y. Han;
\emph{Existence and multiplicity of solutions for $p(x)$-Laplacian
equations in $\mathbb{R}^{N}$},
Nonlinear Anal. 59 (2004), 173--188.
\bibitem{f3} X. L. Fan, J. S. Shen, D. Zhao;
\emph{Sobolev embedding theorems for spaces $W^{k,p(x)}( \Omega )$},
J. Math. Anal. Appl. 262 (2001), 749--760.
\bibitem{f4} X. L. Fan, Q. H. Zhang;
\emph{Existence of solutions for $p(x)$-Laplacian Dirichlet problems},
Nonlinear Anal. 52 (2003), 1843--1852.
\bibitem{f5} X. L. Fan, Q. H. Zhang and D. Zhao;
\emph{Eigenvalues of $p(x)$-Laplacian Dirichlet problem},
J. Math. Anal. Appl. 302 (2005), 306--317.
\bibitem{f6} X.L. Fan, D. Zhao;
\emph{On the Spaces $L^{p(x)}$ and $W^{m,p(x)}$},
J. Math. Anal. Appl. 263 (2001), 424--446.
\bibitem{f7} X. L. Fan, Y. Z. Zhao, Q. H. Zhang;
\emph{A strong maximum principle for $p(x)$-Laplace equations},
Chinese J. Contemp. Math. 24 (3) (2003) 277--282.
\bibitem{f8} X. L. Fan;
\emph{On nonlocal $p(x)$-Laplacian Dirichlet problems},
Nonlinear Anal. 72 (2010) 3314--3323.
\bibitem{h1} A. El Hamidi;
\emph{Existence results to elliptic systems with nonstandard growth
conditions}, J. Math. Anal. Appl. 300 (2004) 30--42.
\bibitem{h2} P. Harjulehto, P. H\"{a}st\"{o};
\emph{An overview of variable exponent Lebesgue and Sobolev
spaces}, in Future Trends in Geometric Function Theory (D. Herron (ed.),
RNC Work-shop), Jyv\"{a}skyl\"{a}, 2003, 85--93.
\bibitem{k1} G. Kirchhoff;
\emph{Mechanik}, Teubner, Leipzig, 1883.
\bibitem{l1} J. L. Lions;
\emph{On some equations in boundary value problems of mathematical
physics}, in: Contemporary Developments in Continuum Mechanics and
Partial Differential Equations (Proc. Internat. Sympos., Inst. Mat.
Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977), North-Holland
Math. Stud., vol. 30, North-Holland, Amsterdam, 1978, pp. 284--346.
\bibitem{m1} T. F. Ma, J. E. Mu\~{n}oz Rivera;
\emph{Positive solutions for a nonlinear nonlocal elliptic
transmission problem}, Appl. Math. Lett. 16 (2003) 243--248.
\bibitem{p1} S. I. Poho\v{z}aev;
\emph{A certain class of quasilinear hyperbolic equations},
Mat. Sb. (N.S.) 96 (1975) 152--166, 168.
\bibitem{r1} M. R\.{u}\u{z}i\u{c}ka;
\emph{Electrorheological Fluids: Modeling and Mathematical Theory},
Springer-Verlag, Berlin, 2000.
\bibitem{s1} S. Samko;
\emph{On a progress in the theory of Lebesgue spaces with variable
exponent Maximal and singular operators},
Integral Transforms Spec. Funct. 16 (2005), 461--482.
\bibitem{w1} M. Willem;
\emph{Minimax Theorems}, Birkh\"{a}user, Boston, 1996.
\bibitem{z1} V. V. Zhikov;
\emph{Averaging of functionals of the calculus of
variations and elasticity theory}, Math. USSR. Izv. 9 (1987), 33--66.
\bibitem{z2} V. V. Zhikov, S. M. Kozlov, O. A. Oleinik;
\emph{Homogenization of Differential Operators and Integral
Functionals}, Translated from the Russian by G.A. Yosifian,
Springer-Verlag, Berlin, 1994.
\bibitem{z3} V. V. Zhikov;
\emph{On some variational problems},
Russian J. Math. Phys. 5 (1997), 105--116.
\end{thebibliography}
\end{document}