\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 139, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/139\hfil Existence of multiple solutions] {Existence of multiple solutions for a $p(x)$-biharmonic equation} \author[L. Li, L. Ding, W.-W. Pan \hfil EJDE-2013/139\hfilneg] {Lin Li, Ling Ding, Wen-Wu Pan} % in alphabetical order \address{Lin Li \newline School of Mathematics and Statistics, Southwest University, Chongqing 400715, China} \email{lilin420@gmail.com} \address{Ling Ding\newline School of Mathematics and Computer Science, Hubei University of Arts and Science, Hubei 441053, China} \email{591517149@qq.com} \address{Wen-Wu Pan\newline Department of Science, Sichuan University of Science and Engineering, Zigong 643000, China} \email{23973445@qq.com} \thanks{Submitted December 30, 2012. Published June 21, 2013.} \subjclass[2000]{35J65, 35J60, 47J30, 58E05} \keywords{$p(x)$-biharmonic equation; Navier boundary condition; \hfill\break\indent Multiple solutions; three critical points theorem; variational methods} \begin{abstract} In this article, we show the existence of at least three solutions to a Navier boundary problem involving the $p(x)$-biharmonic operator. The technical approach is mainly base on a three critical points theorem by Ricceri. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction and statement of the main result} In this article, we consider the fourth-order quasilinear elliptic equation \begin{equation}\label{p1} \begin{gathered} \Delta_{p(x)}^2u+|u|^{p(x)-2}u=\lambda f(x,u)+\mu g(x,u), \quad \text{in }\Omega, \\ u=0,\quad \Delta u=0, \quad \text{on } \partial\Omega, \end{gathered} \end{equation} where $\Delta_{p(x)}^2u=\Delta(|\Delta u|^{p(x)-2}\Delta u)$ is the $p(x)$-biharmonic operator of fourth order, $\lambda$, $\mu\in [0,\infty)$, $\Omega\subset\mathbb{R}^N(N > 1)$ is a nonempty bounded open set with a sufficient smooth boundary $\partial\Omega$. $f$, $g\colon\Omega\times\mathbb{R}\to\mathbb{R}$ are Carath\'{e}odory functions. Next, let $F(x,u)=\int_0^uf(x,s)ds$ and $G(x,u)=\int_0^ug(x,s)ds$. For $p\in C(\overline{\Omega})$, denote $1
0$ for a.e. $x\in\Omega$ and all $s\in ]0,\varrho]$;
\item[(I2)] there exist $p_1(x)\in C(\overline{\Omega})$ and
$p^+ 0,
\]
which means that $T$ is strictly monotone.
(2) Let $( u_n) _n$ be a sequence of $X$ such that
\[
u_n\rightharpoonup u \text{ weakly in }X\quad \text{and}\quad
\limsup_{ n\to +\infty }\langle T(u_n),u_n-u\rangle \leq 0.
\]
From Proposition \ref{prop1.3}, it suffices to shows that
\begin{equation}
\int_{\Omega }( | \Delta u_n-\Delta u| ^{p(x)}+ | u_n-u| ^{p(x)}) dx\to 0. \label{e15}
\end{equation}
In view of the monotonicity of $T$, we have
\[
\langle T(u_n)-T(u),u_n-u\rangle \geq 0,
\]
and since $u_n\rightharpoonup u $ weakly in $X$, it follows that
\begin{equation}
\limsup_{n\to +\infty } \langle T(u_n)-T(u),u_n-u\rangle =0. \label{e16}
\end{equation}
Put
\begin{gather*}
\varphi _n( x) =( | \Delta u_n| ^{p(x)-2}\Delta u_n-| \Delta u|
^{p(x)-2}\Delta u) ( \Delta u_n-\Delta u) , \\
\psi _n( x) =( | u_n| ^{p(x)-2}u_n-| u| ^{p(x)-2}u) ( u_n-u) .
\end{gather*}
By the compact embedding of $X$ into $L^{p(x)}( \Omega ) $, it follows that
\begin{gather*}
u_n\to u\quad \text{in }L^{p(x)}( \Omega ), \\
| u_n| ^{p(x)-2}u_n\to | u| ^{p(x)-2}u\quad \text{in }L^{q(x)}(\Omega ),
\end{gather*}
where $1/q(x)+1/p(x)=1$ for all $x\in \Omega $. It results
that
\begin{equation}
\int_{\Omega }\psi _n( x) dx\to 0. \label{e21}
\end{equation}
It follows by \eqref{e16}\ and \eqref{e21} that
\begin{equation}
\limsup_{n\to +\infty } \int_{\Omega }\varphi _n( x) dx=0. \label{e18}
\end{equation}
Thanks to the above inequalities,
\begin{gather*}
\int_{U_p}| \Delta u_n-\Delta u_{k}| ^{p(x)}dx\leq
2^{p^{+}}\int_{U_p}\varphi _n( x) dx, \\
\int_{U_p}| u_n-u_{k}| ^{p(x)}dx\leq 2^{p^{+}}\int_{U_p}\psi _n( x)
dx.
\end{gather*}
Then
\begin{equation}
\int_{U_p}\left( | \Delta u_n-\Delta u| ^{p(x)}+ | u_n-u| ^{p(x)}\right)
dx\to 0\quad \text{as }n\to +\infty .
\label{e19}
\end{equation}
On the other hand, in $V_p$, setting $\delta _n=| \Delta u_n| +|
\Delta u| $, we have
\[
\int_{V_p}| \Delta u_n-\Delta u| ^{p(x)}dx\leq \frac{1 }{p^{-}-1}%
\int_{V_p}( \varphi _n) ^{\frac{p(x)}{2}}( \delta _n)
^{\frac{p(x)}{2}( 2-p(x)) }dx\,.
\]
For $d > 0$, by Young's inequality,
\begin{equation}
\begin{aligned} d\int_{V_p}| \Delta u_n-\Delta u| ^{p(x)}dx
&\leq \int_{V_p}[ d( \varphi _n) ^{\frac{p(x)}{2}}] ( \delta _n)
^{\frac{p(x)}{2}( 2-p(x)) }dx, \\
&\leq \int_{V_p}\varphi _n( d)
^{\frac{2}{p(x)} }dx+\int_{V_p}( \delta _n) ^{p(x)}dx. \end{aligned}
\label{e17}
\end{equation}
From \eqref{e18} and since $\varphi _n\geq 0$, one can consider that
\[
0\leq \int_{V_p}\varphi _ndx<1.
\]
If $\int_{V_p}\varphi _ndx=0$ then $\int_{V_p}| \Delta u_n-\Delta u|
^{p(x)}dx=0.$\ If $0<\int_{V_p}\varphi _ndx<1$, we choose
\[
d=\Big( \int_{V_p}\varphi _n( x) dx\Big) ^{-1/2}>1,
\]
and the fact that $2/p(x)<2$, inequality \eqref{e17} becomes
\begin{align*}
\int_{V_p}| \Delta u_n-\Delta u| ^{p(x)}dx
&\leq \frac{1}{d}
\Big( \int_{V_p}\varphi _nd^2dx+\int_{\Omega }\delta _n^{p(x)}dx\Big) , \\
&\leq \Big( \int_{V_p}\varphi _ndx\Big) ^{1/2} \Big(1+\int_{\Omega
}\delta _n^{p(x)}dx\Big) .
\end{align*}
Note that, $\int_{\Omega }\delta _n^{p(x)}dx$ is bounded, which implies
\[
\int_{V_p}| \Delta u_n-\Delta u| ^{p(x)}dx\to 0\quad \text{as }n\to
+\infty .
\]
A similar method gives
\[
\int_{V_p}| u_n-u| ^{p(x)}dx\to 0\quad \text{as }n\to +\infty .
\]
Hence, it result that
\begin{equation}
\int_{V_p}( | \Delta u_n-\Delta u| ^{p(x)}+ | u_n-u| ^{p(x)}) dx\to 0\quad \text{as \ }n\to +\infty
. \label{e20}
\end{equation}
Finally, \eqref{e15} is given by combining \eqref{e19} and \eqref{e20}.
(3) Note that the strict monotonicity of $T$ implies its injectivity.
Moreover, $T$ is a coercive operator. Indeed, since $p^{-}-1>0$,
for each $u\in X$ such that $\| u\| \geq 1$ we have
\[
\frac{\langle T(u),u\rangle }{\| u\| }=\frac{ \Phi( u) }{\| u\| }\geq \| u\|
^{p^{-}-1}\to \infty \quad \text{as } \| u\|\to \infty .
\]
Consequently, thanks to Minty-Browder theorem \cite{Zeidler1990},
the operator $T$ is an surjection and admits an inverse mapping.
It suffices then to show the continuity of $T^{-1}$. Let $(f_n)_n$ be
a sequence of $X'$ such that $f_n\to f$ in $X'$. Let $u_n$ and $u$ in $X$
such that
\[
T^{-1}( f_n) =u_n\quad \text{and}\quad T^{-1}( f) =u.
\]
By the coercivity of $T$, one deducts that the sequence $( u_n)$ is
bounded in the reflexive space $X$. For a subsequence, we have
$u_n\rightharpoonup \widehat{u}$ in $X$, which implies
\[
\lim_{n\to +\infty } \langle T(u_n)-T(u),u_n- \widehat{u}\rangle
=\lim_{n\to +\infty }\langle f_n-f,u_n-\widehat{u}\rangle =0.
\]
It follows by the second assertion and the continuity of $T$ that
\[
u_n\to \widehat{u}\quad \text{in } X\quad \text{and}\quad T(u_n)\to
T(\widehat{u})=T(u)\quad \text{in } X'.
\]
Moreover, since $T$ is an injection, we conclude that $u=\widehat{u}$.
\end{proof}
\section{Proof of main theorem}
For the reader's convenience, we recall the revised form of Ricceri's
three critical points theorem
\cite[Theorem 1]{Ricceri2009} and \cite[Proposition 3.1]{Ricceri2000a}.
\begin{theorem}[{\cite[Theorem 1]{Ricceri2009}}] \label{thm:ricceri}
Let $X$ be a reflexive real Banach space.
$ \Phi\colon X \to \mathbb{R} $ is a continuously
G\^{a}teaux differentiable and sequentially weakly lower semicontinuous
functional whose G\^{a}teaux
derivative admits a continuous inverse on $ X'$ and $\Phi$ is
bounded on each bounded subset of
$X$; $\Psi\colon X \to \mathbb{R} $ is a continuously G\^{a}teaux
differentiable functional whose G\^{a}teaux derivative is compact;
$ I \subseteq \mathbb{R} $ an interval. Assume that
\begin{equation}\label{qiangzhi}
\lim_{\|x\| \to +\infty } (\Phi(x)+\lambda \Psi (x))=+\infty
\end{equation}
for all $ \lambda \in I $, and that there exists $h\in \mathbb{R}$ such that
\begin{equation}\label{t2}
\sup_{\lambda \in I} \inf_{x \in X} (\Phi (x)+ \lambda (\Psi (x)+h))
< \inf_{x \in X} \sup_{\lambda \in I} (\Phi (x)+ \lambda (\Psi (x)+ h)).
\end{equation}
Then, there exists an open interval $ \Lambda \subseteq I $ and a
positive real number $ \rho $ with
the following property: for every $ \lambda \in \Lambda $ and every $ C^1 $
functional $ J\colon X \mapsto \mathbb{R} $ with compact derivative,
there exists $ \delta > 0 $ such that, for each $ \mu \in [0,\delta] $
the equation
\[
\Phi '(x)+\lambda \Psi '(x)+\mu J'(x)=0
\]
has at least three solutions in $X$ whose norms are less than $ \rho $.
\end{theorem}
\begin{proposition}[{\cite[Proposition 3.1]{Ricceri2000a}}] \label{propo}
Let $X$ be a non-empty set and $\Phi, \Psi$ two real functions on $X$.
Assume that there are $r > 0$ and
$ x_0, x_1 \in X$ such that
\[
\Phi(x_0)=-\Psi(x_0)=0, \quad \Phi(x_1)>r, \quad
\sup_{ x \in \Phi^{-1} ( ]-\infty ,r ] ) }
-\Psi(x) < r \frac{-\Psi(x_1)}{\Phi(x_1)}.
\]
Then, for each $ h $ satisfying
\[
\sup_{ x \in \Phi^{-1} ( ]-\infty ,r ] ) } -\Psi(x) < h
< r \frac{-\Psi(x_1)}{\Phi(x_1)},
\]
one has
\[
\sup_{\lambda \geq 0} \inf_{x \in X}(\Phi(x)+\lambda(h +\Psi(x)))
< \inf_{x\in X} \sup_{\lambda \geq 0} (\Phi(x)+\lambda(h +\Psi(x))).
\]
\end{proposition}
Now we can give the proof of our main result.
\begin{proof}[Proof Theorem \ref{thm}]
Set $\Phi(u)$, $\Psi(u)$ and $J(u)$ as \eqref{phi}, \eqref{psi} and \eqref{:J}.
So, for each $u$, $v\in X$, one has
\begin{gather*}
\langle\Phi '(u),v\rangle =\int_{\Omega}(|\Delta u|^{p(x)-2}\Delta u\Delta v
+ | u|^{p(x)-2}uv)\,dx, \\
\langle\Psi '(u),v\rangle =-\int_{\Omega}f(x,u)v\,dx, \\
\langle J'(u),v\rangle =-\int_{\Omega}g(x,u)v\,dx.
\end{gather*}
From Theorem \ref{thm1.4}, of course, $\Phi$ is a continuous
G\^{a}teaux differentiable and
sequentially weakly lower semicontinuous functional whose G\^{a}teaux
derivative admits a continuous
inverse on $X'$, moreover, $\Psi$ and $J$ are continuously
G\^{a}teaux differentiable functionals
whose G\^{a}teaux derivative is compact. Obviously, $\Phi$ is bounded
on each bounded subset of $X$ under our assumptions.
From Proposition \ref{prop1.3}, we have:
if $\|u\|\geq 1$, then
\begin{equation}\label{eq:3.1}
\frac{1}{p^+}\|u\|^{p^-}\leq\Phi(u)\leq\frac{1}{p^-}\|u\|^{p^+}.
\end{equation}
Meanwhile, for each $\lambda\in\Lambda$,
\begin{align*}
\lambda\Psi(u) & =-\lambda\int_{\Omega}F(x,u)dx\\
& \geq -\lambda\int_{\Omega}\vartheta(1+|u|^{\gamma(x)})dx\\
& \geq -\lambda \vartheta(|\Omega|+|u|_{\gamma(x)}^{\gamma^+})\\
& \geq -C_2(1+|u|_{\gamma(x)}^{\gamma^+})\\
& \geq -C_3(1+\|u\|^{\gamma^+})
\end{align*}
for any $u\in X$, where $C_2$ and $C_3$ are positive constants.
Here, we use condition (I3) and
(ii) of Proposition \ref{prop1.1}.
Combining the two inequalities above, we obtain
\[
\Phi(u)+\lambda\Psi(u)\geq \frac{1}{p^+}\|u\|^{p^-}-C_3(1+\|u\|^{\gamma^+}),
\]
because of $\gamma^+ 0$, such that
\[
F(x,s)