\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 $$\label{p1} \begin{gathered} \Delta_{p(x)}^2u+|u|^{p(x)-2}u=\lambda f(x,u)+\mu g(x,u), \quad \text{in }\Omega, \\ u=0,\quad \Delta u=0, \quad \text{on } \partial\Omega, \end{gathered}$$ 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 $p^+0$ such that, for each $\mu\in[0,\delta]$, problem \eqref{p1} has at least three weak solutions whose norms in $X$ are less than $\rho$. \end{theorem} \begin{remark} \rm The conclusion of Theorem \ref{thm} gives a precise information about the $p(x)$-biharmonic equation \eqref{p1} with parameter, namely, one can see that \eqref{p1} is stable with respect to small perturbations. \end{remark} This article is divided into four sections. In Section 2, we recall some basic facts about the variable exponent Lebesgue and Sobolev spaces. In the third section, we present some important properties of the $p(x)$-biharmonic operator. In section 4, we recall B. Ricceri's three critical points theorem at first, then prove our main result. \section{Preliminaries} To study $p(x)$-biharmonic problems, we need some results on the spaces $L^{p(x)}( \Omega )$ and $W^{k,p(x)}( \Omega )$, and properties of $p(x)$-biharmonic operator, which we will use later. Define the generalized Lebesgue space by $L^{p(x)}( \Omega ) :=\big\{ u:\Omega \to \mathbb{R} \text{ measurable and } \int_{\Omega }| u( x) | ^{p(x)}dx<\infty \big\},$ where $p(x)\in C_{+}( \overline{\Omega })$ and $C_{+}( \overline{\Omega }) :=\big\{ p\in C( \overline{ \Omega }) :p(x)>1\big\}, \text{ for any } x\in \overline{\Omega }.$ Denote $p^{+}=\max_{x\in \overline{\Omega }} p(x),\quad p^{-}=\min_{x\in \overline{\Omega }} p(x),$ and for any $x\in \overline{\Omega }$, $k\geq 1$, \begin{gather*} p^{\ast }( x) := \begin{cases} \frac{Np(x)}{N-p(x)} & \text{if }p(x)0 : \int_{\Omega }| \frac{u( x) }{\alpha } | ^{p(x)}dx\leq 1\big\}\,. \] The space $( L^{p(x)}( \Omega ) ,|\cdot |_{p(x)})$ is a Banach space. \begin{proposition}[\cite{Fan2001}] \label{prop1.1} The space $(L^{p(x)}( \Omega ) ,| \cdot| _{p(x)})$ is separable, uniformly convex, reflexive and its conjugate space is $L^{q(x)}( \Omega )$ where $q(x)$ is the conjugate function of $p(x)$; i.e., $\frac{1}{p(x)}+\frac{1}{q(x)}=1,$ for all $x\in \Omega$. For $u\in L^{p(x)}( \Omega )$ and $v\in L^{q(x)}(\Omega )$ we have $\Big| \int_{\Omega }u( x) v(x)dx\Big| \leq \Big(\frac{1}{p^{-}}+\frac{1}{q^{-}}\Big)| u| _{p(x)}|v| _{q(x)}.$ \end{proposition} The Sobolev space with variable exponent $W^{k,p(x)}( \Omega )$ is defined as $W^{k,p(x)}( \Omega ) =\big\{ u\in L^{p(x)}( \Omega ) :D^{\alpha }u\in L^{p(x)}( \Omega ) ,| \alpha | \leq k\big\} ,$ where $D^{\alpha }u=\frac{\partial ^{| \alpha | }}{ \partial x_{1}^{\alpha _{1}}\partial x_{2}^{\alpha _{2}}\dots \partial x_{N}^{\alpha _{N}}}u$ with $\alpha =(\alpha _{1},\dots ,\alpha_{N})$ is a multi-index and $| \alpha| =\sum_{i=1}^{N}\alpha _{i}$. The space $W^{k,p(x)}(\Omega )$, equipped with the norm $\| u\| _{k,p(x)}:=\sum_{| \alpha | \leq k}| D^{\alpha }u| _{p(x)},$ also becomes a Banach, separable and reflexive space. For more details, we refer the reader to \cite{Edmunds1999,Edmunds2000,Fan2001a,Fan2001}. \begin{proposition}[\cite{Fan2001}] \label{prop1.2} For $p,r\in C_{+}( \overline{\Omega })$ such that $r(x)\leq p_{k}^{\ast }( x)$ for all $x\in \overline{\Omega }$, there is a continuous and compact embedding $W^{k,p(x)}( \Omega ) \hookrightarrow L^{r(x)}( \Omega) .$ \end{proposition} We denote by $W_{0}^{k,p(x)}( \Omega )$ the closure of $C_{0}^{\infty }(\Omega )$\ in $W^{k,p(x)}(\Omega )$. \section{Properties of the $p(x)$-biharmonic operator} Note that the weak solutions of \eqref{p1} are considered in the generalized Sobolev space $X:=W^{2,p(x)}( \Omega ) \cap W_{0}^{1,p(x)}( \Omega ),$ equipped with the norm $\| u\| =\inf \Big\{ \alpha >0:\int_{\Omega } \Big(| \frac{\Delta u( x) }{\alpha } | ^{p(x)}+| \frac{u( x) }{\alpha } | ^{p(x)}\Big) dx\leq 1\Big\}.$ \begin{remark}\label{rq1.1} \rm (1) According to \cite{Zang2008}, the norm $\| \cdot \| _{2,p(x)}$, cited in the preliminaries, is equivalent to the norm $| \Delta \cdot | _{p(x)}$ in the space $X$. Consequently, the norms $\|\cdot \|_{2,p(x)},\| \cdot \|$ and $| \Delta \cdot | _{p(x)}$ are equivalent. (2) By the above remark and Proposition \ref{prop1.2}, there is a continuous and compact embedding of $X$ into $L^{q(x)}( \Omega )$, where $q(x)) 1 \Leftrightarrow \Phi( u) <( =;>)1$, \item[(2)] $\| u\| \leq 1\Rightarrow \| u\| ^{p^{+}}\leq \Phi( u) \leq \| u\| ^{p^{-}}$, \item[(3)] $\| u\| \geq 1\Rightarrow \| u\| ^{p^{-}}\leq \Phi( u) \leq \| u\| ^{p^{+}}$, for all $u_n\in X$ we have \item[(4)] $\| u_n\| \to 0\Leftrightarrow \Phi(u_n) \to 0$, \item[(5)] $\| u_n\| \to \infty \Leftrightarrow \Phi( u_n) \to \infty$. \end{itemize} \end{proposition} The proof of this proposition is similar to the proof in \cite[Theorem 1.3]{Fan2001}. Moreover, the operator $T:=\Phi ':X\to X'$ defined as $\langle T(u),v\rangle =\int_{\Omega }( | \Delta u| ^{p(x)-2}\Delta u\Delta v + | u| ^{p(x)-2}uv) dx \quad \text{for any } u,v\in X,$ satisfies the assertions of the following theorem. \begin{theorem} \label{thm1.4} The following statements hold: \begin{itemize} \item[(1)] $T$ is continuous, bounded and strictly monotone. \item[(2)] $T$ is of $(S_{+})$ type. \item[(3)] $T$ is a homeomorphism. \end{itemize} \end{theorem} \begin{proof} (1) Since $T$ is the Fr\'{e}chet derivative of $\Phi$, it follows that $T$ is continuous and bounded. Let us define the sets $U_p=\{ x\in \Omega :p(x)\geq 2\} ,\quad V_p=\{ x\in \Omega :10,$ 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 $$\int_{\Omega }( | \Delta u_n-\Delta u| ^{p(x)}+ | u_n-u| ^{p(x)}) dx\to 0. \label{e15}$$ 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 $$\limsup_{n\to +\infty } \langle T(u_n)-T(u),u_n-u\rangle =0. \label{e16}$$ 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 $$\int_{\Omega }\psi _n( x) dx\to 0. \label{e21}$$ It follows by \eqref{e16}\ and \eqref{e21} that $$\limsup_{n\to +\infty } \int_{\Omega }\varphi _n( x) dx=0. \label{e18}$$ 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 $$\int_{U_p}\left( | \Delta u_n-\Delta u| ^{p(x)}+ | u_n-u| ^{p(x)}\right) dx\to 0\quad \text{as }n\to +\infty . \label{e19}$$ 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{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} 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 $$\int_{V_p}( | \Delta u_n-\Delta u| ^{p(x)}+ | u_n-u| ^{p(x)}) dx\to 0\quad \text{as \ }n\to +\infty . \label{e20}$$ 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 $$\label{qiangzhi} \lim_{\|x\| \to +\infty } (\Phi(x)+\lambda \Psi (x))=+\infty$$ for all $\lambda \in I$, and that there exists $h\in \mathbb{R}$ such that $$\label{t2} \sup_{\lambda \in I} \inf_{x \in X} (\Phi (x)+ \lambda (\Psi (x)+h)) < \inf_{x \in X} \sup_{\lambda \in I} (\Phi (x)+ \lambda (\Psi (x)+ h)).$$ 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 $$\label{eq:3.1} \frac{1}{p^+}\|u\|^{p^-}\leq\Phi(u)\leq\frac{1}{p^-}\|u\|^{p^+}.$$ 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)p^+$, it follows that $$\label{eq:lim} \lim_{r\to 0^+}\frac{\sup_{\|u\|^{p^+}/p^+\leq r}-\Psi(u)}{r}=0.$$ Let$u_1 \in C^2(\Omega)$be a function positive in$\Omega$, with$u_1|_{\partial \Omega} = 0$and$\max_{\overline{\Omega}} u_1 \leq d$. Then, of course,$u_1 \in X$and$\Phi(u_1) > 0$. In view of$(i_1)$we also have$-\Psi(u_1) = \int_{\Omega} F(x,u_1(x)) dx > 0$. Therefore, from \eqref{eq:lim}, we can find$r \in \big( 0, \min\{ \Phi(u_1), \frac{1}{p^+} \} \big)$such that \begin{equation*} \sup_{\|u\|^{p^+}/p^+ \leq r} (-\Psi(u)) < r\frac{-\Psi(u_1)}{\Phi(u_1)}. \end{equation*} Now, let$u \in \Phi^{-1} ((-\infty, r])$. Then,$\int_{\Omega} (|\Delta u|^{p(x)} + |u|^{p(x)} ) dx \leq rp^+ <1$which, by Proposition \ref{prop1.3}, implies$\|u\|<1$. Consequently, \begin{equation*} \frac{1}{p^+}\|u\|^{p^+} \leq \int_{\Omega} \frac{1}{p(x)} (|\Delta u|^{p(x)} + |u|^{p(x)}) dx < r. \end{equation*} Therefore, we infer that$\Phi^{-1} ((-\infty, r]) \subset \left\{ u \in X : \frac{1}{p^+}\|u\|^{p^+} < r \right\}$, and so \begin{equation*} \sup_{ u \in \Phi^{-1} ( ]-\infty ,r ] ) } -\Psi(u) < r \frac{-\Psi(u_1)}{\Phi(u_1)}. \end{equation*} At this point, conclusion follows from Proposition \ref{propo} and Theorem \ref{thm:ricceri}. \end{proof} \subsection*{Acknowledgments} The authors are very grateful to the anonymous referees for their knowledgeable reports, which helped us to improve our manuscript. The first and the third author were supported by grant XDJK2013D007 from the Fundamental Research Funds for the Central Universities, grant 2011KY03 from the Scientific Research Fund of SUSE, and grant 12ZB081 from the Scientific Research Fund of SiChuan Provincial Education Department. 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