\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 317, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2015 Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2015/317\hfil Auasilinear problems with a $p(x)$-biharmonic operator] {Existence of infinitely many solutions for quasilinear problems with a $p(x)$-biharmonic operator} \author[G. A. Afrouzi, S. Shokooh \hfil EJDE-2015/317\hfilneg] {Ghasem A. Afrouzi, Saeid Shokooh} \address{Ghasem A. Afrouzi \newline Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran} \email{afrouzi@umz.ac.ir} \address{Saeid Shokooh \newline Department of Mathematics, Faculty of Sciences, Gonbad Kavous University, Gonbad Kavous, Iran} \email{shokooh@gonbad.ac.ir} \thanks{Submitted June 16, 2015. Published December 28, 2015.} \subjclass[2010]{35D05, 34B18, 35J60} \keywords{Ricceri's variational principle; infinitely many solutions; \hfill\break\indent Navier condition; $p(x)$-biharmonic operator} \begin{abstract} By using critical point theory, we establish the existence of infinitely many weak solutions for a class of Navier boundary-value problem depending on two parameters and involving the $p(x)$-biharmonic operator. Under an appropriate oscillatory behaviour of the nonlinearity and suitable assumptions on the variable exponent, we obtain a sequence of pairwise distinct solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} In this work we study the existence of infinitely many weak solutions for Navier boundary-value problem \begin{equation}\label{e1.1} \begin{gathered} \Delta_{p(x)}^2u=\lambda f(x,u)+\mu g(x,u),\quad x\in\Omega,\\ u=\Delta u=0,\quad x\in \partial \Omega, \end{gathered} \end{equation} where $\Omega \subset \mathbb{R}^N$ $(N\geq 2)$ is a bounded domain with boundary of class $C^1$, $\lambda$ is a positive parameter, $\mu$ is a non-negative parameter, $f,g\in C^0(\overline{\Omega}\times \mathbb{R})$, $p(\cdot)\in C^0(\overline{\Omega})$ with \[ \max \{2,\frac{N}{2}\}1$ is a constant). The study of differential equations and variational problems with variable exponents has attracted intense research interests in recent years. Such problems arise from the study of electrorheological fluids, image processing, and the theory of nonlinear elasticity. So the investigation of existence and multiplicity of solutions for problems involving biharmonic, $p$-biharmonic and $p(x)$-biharmonic operators has drawn the attention of many authors see \cite{CanLiLiv,dema, LT,BiRe,repovs, SZ, Holi, Home,yucedag}. In particular, in \cite{Holi}, the authors studied the following $p(x)$-biharmonic elliptic problem with Navier boundary conditions: \begin{equation}\label{e1.2} \begin{gathered} \Delta_{p(x)}^2u=\lambda a(x) f(x,u)+\mu g(x,u),\quad x\in\Omega,\\ u=\Delta u=0,\quad x\in \partial \Omega, \end{gathered} \end{equation} where $\Omega \subset \mathbb{R}^N$ $(N\geq 2)$ is a bounded domain with boundary of class $C^1$, $\lambda,\mu$ are non-negative parameters, $p(\cdot)\in C^0(\overline{\Omega})$ with \[ \max \{2,\frac{N}{2}\}\inf_X\Phi$, let \begin{gather*} \varphi(r):=\inf_{u\in\Phi^{-1}(-\infty,r)} \frac{\big(\sup_{v\in\Phi^{-1}(-\infty,r)}\Psi(v)\big)-\Psi(u)}{r-\Phi(u)}, \\ \gamma:=\liminf_{r\to +\infty}\varphi(r),\quad\text{and}\quad \delta:=\liminf_{r\to(\inf_X\Phi)^+}\varphi(r). \end{gather*} Then the following properties hold: \begin{itemize} \item[(a)] For every $r>\inf_X\Phi$ and every $\lambda\in (0,1/\varphi(r))$, the restriction of the functional $$ I_\lambda:=\Phi-\lambda\Psi $$ to $\Phi^{-1}(-\infty,r)$ admits a global minimum, which is a critical point (local minimum) of $I_\lambda$ in $X$. \item[$\rm(b)$] If $\gamma<+\infty$, then for each $\lambda\in (0,1/\gamma)$, the following alternative holds: either \begin{enumerate} \item $I_\lambda$ possesses a global minimum, or \item there is a sequence $\{u_n\}$ of critical points (local minima) of $I_\lambda$ such that $$ \lim_{n\to+\infty}\Phi(u_n)=+\infty. $$ \end{enumerate} \item[(c)] If $\delta<+\infty$, then for each $\lambda\in (0,1/\delta)$, the following alternative holds: either \begin{enumerate} \item there is a global minimum of $\Phi$ which is a local minimum of $I_\lambda$, or \item there is a sequence $\{u_n\}$ of pairwise distinct critical points (local minima) of $I_\lambda$ that converges weakly to a global minimum of $\Phi$. \end{enumerate} \end{itemize} \end{lemma} For the reader's convenience, we recall some background facts concerning the Lebesgue-Sobolev spaces variable exponent and introduce some notation. For more details, we refer the reader to \cite{Fazh,Kora,ouso,Radu1,Radu2,Ruz,San}. Set \[ C_{+}( \Omega ) :=\big\{ h\in C( \overline{ \Omega }) :h(x)>1,\,\forall x\in \overline{\Omega }\big\}. \] For $p(\cdot)\in C_{+}( \Omega )$, define \[ L^{p(\cdot)}( \Omega ) :=\big\{ u:\Omega \to \mathbb{R} \text{ measurable and } \int_{\Omega }| u( x) | ^{p(x)}dx<\infty \big\}. \] We introduce a norm on $L^{p(\cdot)}( \Omega ) $ by \[ | u| _{p(\cdot)}=\inf \big\{ \beta >0 : \int_{\Omega }| \frac{u( x) }{\beta } | ^{p(x)}dx\leq 1\big\}. \] The space $( L^{p(\cdot)}( \Omega ) ,|u |_{p(\cdot)}) $ is a Banach space called a variable exponent Lebesgue space. Define the Sobolev space with variable exponent \[ W^{m,p(\cdot)}( \Omega ) =\big\{ u\in L^{p(\cdot)}( \Omega ) :D^{\alpha }u\in L^{p(\cdot)}( \Omega ) ,| \alpha | \leq m\big\} , \] 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^{m,p(\cdot)}(\Omega ) $, equipped with the norm \[ \| u\| _{m,p(\cdot)}:=\sum_{| \alpha | \leq m}| D^{\alpha }u| _{p(\cdot)}, \] becomes a separable, reflexive and uniformly convex Banach space. We denote by $W_{0}^{m,p(\cdot)}( \Omega ) $ the closure of $C_{0}^{\infty }(\Omega )$ in $W^{m,p(\cdot)}(\Omega ) $. Now we denote \[ X:=W^{2,p(\cdot)}( \Omega ) \cap W_{0}^{1,p(\cdot)}( \Omega ). \] For $u\in X$, we define \[ \| u\| =\inf \Big\{ \beta >0:\int_{\Omega } |\frac{\Delta u( x) }{\beta } | ^{p(x)} dx\leq 1\Big\}. \] It is easy to see that $X$ endowed with the above norm is a separable and reflexive Banach space. We denote by $X^*$ its dual. \begin{remark}\label{rq1.1}\rm{ According to \cite{Zang2008}, the norm $\| u \| _{2,p(\cdot)}$ is equivalent to the norm $| \Delta u | _{p(\cdot)}$ in the space $X$. Consequently, the norms $\|u \|_{2,p(\cdot)},\| u \|$ and $| \Delta u | _{p(\cdot)}$ are equivalent.} \end{remark} For the rest of this article, we use $\| u \|$ instead of $\|u \|_{2,p(\cdot)}$ on $X$. \begin{proposition}[\cite{Fazh,Ruz}] \label{prop1.1} The conjugate space of $L^{p(\cdot)}( \Omega )$ is $L^{q(\cdot)}( \Omega ) $ where $q(\cdot)$ is the conjugate function of $p(\cdot)$; i.e., \[ \frac{1}{p(\cdot)}+\frac{1}{q(\cdot)}=1. \] For $u\in L^{p(\cdot)}( \Omega ) $ and $v\in L^{q(\cdot)}(\Omega ) $, we have \[ \Big| \int_{\Omega }u( x) v(x)dx\Big| \leq \Big(\frac{1}{p^{-}}+\frac{1}{q^{-}}\Big)| u| _{p(\cdot)}|v| _{q(\cdot)}\leq 2| u| _{p(\cdot)}| v| _{q(\cdot)}. \] \end{proposition} \begin{proposition}[\cite{Fazh,Ruz}]\label{prop1.3} Set $\rho(u) =\int_{\Omega }| u| ^{p(x)} dx$. For $u,u_n\in L^{p(\cdot)}(\Omega)$, we have \begin{itemize} \item[(1)] $| u|_{p(\cdot)} <( =;>) 1 \Leftrightarrow \rho( u) <( =;>)1 $, \item[(2)] $| u|_{p(\cdot)} > 1\Rightarrow | u|_{p(\cdot)}^{p^-} \leq \rho( u) \leq | u|_{p(\cdot)} ^{p^{+}}$, \item[(3)] $| u|_{p(\cdot)} < 1\Rightarrow | u|_{p(\cdot)} ^{p^{+}}\leq \rho( u) \leq | u|_{p(\cdot)} ^{p^{-}}$, \item[(4)] $| u_n|_{p(\cdot)} \to 0\Leftrightarrow \rho(u_n) \to 0$, \item[(5)] $| u_n|_{p(\cdot)} \to \infty \Leftrightarrow \rho( u_n) \to \infty $. \end{itemize} \end{proposition} From Proposition \ref{prop1.3}, for $u\in L^{p(\cdot)}(\Omega)$ the following inequalities hold: \begin{gather}\label{mol2} \|u\|^{p^-}\leq \int_\Omega |\Delta u(x)|^{p(x)}\;dx\leq \|u\|^{p^+} \quad \text{if }\|u\|>1; \\ \label{mol3} \|u\|^{p^+}\leq\int_\Omega |\Delta u(x)|^{p(x)}\;dx\leq\|u\|^{p^-} \quad \text{if }\|u\|<1. \end{gather} \begin{proposition}[\cite{Home}]\label{prop11.3} If $\Omega \subset \mathbb{R}^N$ is a bounded domain, then the embedding $X\hookrightarrow C^0(\overline{\Omega})$ is compact whenever $N/20. $$ Now fix $\overline{\mu}\in (0,\mu_{g,\overline{\lambda}})$ and set $$ J(x,\xi):=F(x,\xi)+\frac{\overline{\mu}}{\overline{\lambda}}G(x,\xi) $$ for all $(x,\xi)\in \Omega\times\mathbb{R}$. For each $u\in X$, we let the functionals $\Phi,\Psi:X\to\mathbb{R}$ be defined by \[ \Phi(u):=\int_\Omega\frac{1}{p(x)}|\Delta u(x)|^{p(x)}dx, \quad \Psi(u):=\int_\Omega J(x,u(x))\,dx, \] and put $$ I_{\overline{\lambda}}(u):=\Phi(u)-\overline{\lambda}\Psi(u),\quad u\in X. $$ Note that the weak solutions of \eqref{e1.1} are exactly the critical points of $I_{\overline{\lambda}}$. The functionals $\Phi,\Psi$ satisfy the regularity assumptions of Lemma \ref{lem2.1}. Indeed, by standard arguments, we have that $\Phi$ is G\^{a}teaux differentiable and sequentially weakly lower semicontinuous and its G\^{a}teaux derivative is the functional $\Phi'(u)\in X^*$, given by \begin{align*} \Phi'(u)(v)=\int_{\Omega }| \Delta u| ^{p(x)-2}\Delta u\Delta v dx \end{align*} for any $v\in X$. Furthermore, the differential $\Phi':X\to X^*$ admits a continuous inverse (see \cite[Lemma 3.1]{Holi}). On the other hand, the fact that $X$ is compactly embedded into $C^0([0,1])$ implies that the functional $\Psi$ is well defined, continuously G\^{a}teaux differentiable and with compact derivative, whose G\^{a}teaux derivative is given by $$ \Psi'(u)(v)=\int_\Omega f(x,u(x))v(x)\,dx+ \frac{\overline{\mu}}{\overline{\lambda}}\int_\Omega g(x,u(x))v(x)\,dx. $$ Furthermore, we have from \eqref{mol2} that \begin{equation} \Phi(u)\geq \frac{1}{p^+}\|u\|^{p^-} \end{equation} for all $u\in X$ such that $\|u\|>1$, and so $\Phi$ is coercive. First of all, we will show that $\overline{\lambda}<1/\gamma$. Hence, let $\{\xi_n\}$ be a sequence of positive numbers such that $\lim_{n\to +\infty}\xi_n=+\infty$ and $$ \lim_{n\to+\infty}\frac{ \int_\Omega\sup_{|t|\leq\xi_n}F(x,t)\,dx}{\xi_n^{p^-}}=\eta. $$ Put $$ r_n:=\frac{1}{p^+}\Big(\frac{\xi_n}{c}\Big)^{p^-} $$ for all $n\in\mathbb{N}$. Then, for all $v\in X$ with $\Phi(v)0$ such that $\lim_{n\to +\infty}\tau_n=+\infty$ and \begin{equation}\label{e3.3} \frac{1}{\overline{\lambda}}<\tau<\frac{p^-c^{p^-}}{\sigma}\frac{\int_{B(x^0,r_1)} F(x,\tau_n)\,dx}{\tau_n^{p^+}} \end{equation} for each $n\in\mathbb{N}$ large enough. For all $n\in\mathbb{N}$ define $w_n\in X$ by \begin{equation}\label{sasiras} w_n(x):=\begin{cases} 0, & x\in \overline{\Omega}\setminus B(x^0,r_2),\\[4pt] \frac{\tau_n[3(l^4-r_2^4)-4(r_1+r_2)(l^3-r_2^3) +6r_1r_2(l^2-r_2^2)]}{(r_2-r_1)^3(r_1+r_2)}, &x\in B(x^0,r_2)\setminus B(x^0,r_1),\\[4pt] \tau_n, &x\in B(x^0,r_1), \end{cases} \end{equation} where $l=\operatorname{dist}(x,x^0)=\sqrt{\sum_{i=1}^{N}(x_i-x_i^0)^2}$. Then \[ \frac{\partial w_n(x)}{\partial x_i} =\begin{cases} 0,\quad\text{if } x\in \overline{\Omega}\setminus B(x^0,r_2)\cup B(x^0,r_1),\\[4pt] \frac{12\tau_n[l^2(x_i-x_i^0)-l(r_1+r_2)(x_i-x_i^0) +r_1r_2(x_i-x_i^0)]}{(r_2-r_1)^3(r_1+r_2)}, \\ \quad\text{if } x\in B(x^0,r_2)\setminus B(x^0,r_1), \end{cases} \] \[ \frac{\partial^2 w_n(x)}{\partial x^2_i} =\begin{cases} 0, \quad\text{if } x\in \overline{\Omega}\setminus B(x^0,r_2)\cup B(x^0,r_1),\\[4pt] \frac{12\tau_n[r_1r_2+(2l-r_1-r_2)(x_i-x_i^0)^2/l -(r_1+r_2-l)l]}{(r_2-r_1)^3(r_1+r_2)}, \\ \quad\text{if } x\in B(x^0,r_2)\setminus B(x^0,r_1), \end{cases} \] \[ \sum_{i=1}^{N}\frac{\partial^2 w_n(x)}{\partial x^2_i} =\begin{cases} 0, \quad\text{if } x\in \overline{\Omega}\setminus B(x^0,r_2)\cup B(x^0,r_1),\\[4pt] \frac{12\tau_n[(N+2)l^2-(N+1)(r_1+r_2)l+Nr_1r_2]}{(r_2-r_1)^3(r_1+r_2)}, \\ \quad\text{if } x\in B(x^0,r_2)\setminus B(x^0,r_1). \end{cases} \] For any fixed $n\in\mathbb{N}$, one has \begin{equation}\label{e3.5} \Phi(w_n)= \int_{B(x^0,r_2)\setminus B(x^0,r_1)}\frac{1}{p(x)}| \Delta w_n(x)|^{p(x)}dx\leq \frac{\sigma \tau_n^{p^+}}{p^-c^{p^-}}. \end{equation} On the other hand, bearing (A1) in mind and since $G$ is non-negative, from the definition of $\Psi$, we infer \begin{equation}\label{e3.6} \Psi(w_n)=\int_\Omega\Big[F(x,w_n(x)) +\frac{\overline{\mu}}{\overline{\lambda}}G(x,w_n(x))\Big]\,dx \geq\int_{B(x^0,r_1)} F(x,\tau_n)\,dx. \end{equation} By \eqref{e3.3}, \eqref{e3.5} and \eqref{e3.6}, we observe that \begin{equation}\label{sds} I_{\overline{\lambda}}(w_n)\leq \frac{\sigma \tau_n^{p^+}}{p^-c^{p^-}} -\overline{\lambda}\int_{B(x^0,r_1)} F(x,\tau_n)\,dx < \frac{\sigma \tau_n^{p^+}}{p^-c^{p^-}}(1-\overline{\lambda}\tau) \end{equation} for every $n\in\mathbb{N}$ large enough. Since $\overline{\lambda}\tau>1$ and $\lim_{n\to +\infty}\tau_n=+\infty$, we have $$ \lim_{n\to +\infty}I_{\overline{\lambda}}(w_n)=-\infty. $$ Then, the functional $I_{\overline{\lambda}}$ is unbounded from below, and it follows that $I_{\overline{\lambda}}$ has no global minimum. Therefore, by Lemma \ref{lem2.1}(b), there exists a sequence $\{u_n\}$ of critical points of $I_{\overline{\lambda}}$ such that $\lim_{n\to +\infty}\|u_n\|=+\infty$, and the conclusion is achieved. \end{proof} \begin{remark}\label{rem3.2} \rm{ Under the conditions $\eta=0$ and $\theta=+\infty$, from Theorem \ref{the3.1} we see that for every $\lambda>0$ and for each $\mu\in\big{[}0,\frac{1}{p^+c^{p^-}g_\infty}\big{)}$, problem \eqref{e1.1} admits a sequence of weak solutions which is unbounded in $X$. Moreover, if $g_\infty=0$, the result holds for every $\lambda>0$ and $\mu\geq 0$.} \end{remark} \section{Applications} In this section, we point out some consequences and applications of the results previously obtained. First, we present the following consequence of Theorem \ref{the3.1} with $\mu=0$. \begin{theorem}\label{the3.3} Assume that all the assumptions in the Theorem \ref{the3.1} hold. Then, for each $$ \lambda \in \Big(\frac{\sigma}{p^-c^{p^-}\theta},\frac{1}{p^+c^{p^-}\eta}\Big), $$ the problem \begin{equation}\label{e.4.1} \begin{gathered} \Delta_{p(x)}^2u=\lambda f(x,u),\quad x\in\Omega,\\ u=\Delta u=0,\quad x\in \partial \Omega \end{gathered} \end{equation} has an unbounded sequence of weak solutions in $X$. \end{theorem} Here we point out the following consequence of Theorem \ref{the3.1}. \begin{corollary}\label{core3.3.1} Assume that the assumption {\rm (A1)} in Theorem \ref{the3.1} holds. Suppose that $$ \eta<\frac{1}{p^+c^{p^-}},\quad \theta>\frac{\sigma}{p^-c^{p^-}}. $$ Then, the problem \begin{equation}\label{e.4.2} \begin{gathered} \Delta_{p(x)}^2u= f(x,u)+\mu g(x,u),\quad x\in\Omega,\\ u=\Delta u=0,\quad x\in \partial \Omega \end{gathered} \end{equation} has an unbounded sequence of weak solutions in $X$. \end{corollary} \begin{corollary}\label{cor3.6} Let $g_1:\mathbb{R}\to\mathbb{R}$ be a non-negative continuous function. Put $G_1(\xi):=\int_0^\xi g_1(t)\,dt$ for all $\xi\in\mathbb{R}$ and assume that \begin{itemize} \item[(A3)] $\liminf_{\xi\to+\infty}\frac{G_1(\xi)}{\xi^{p^-}}<+\infty$; \item[(A4)] $\limsup_{\xi\to+\infty}\frac{G_1(\xi)}{\xi^{p^+}}=+\infty$. \end{itemize} Then, for every $\alpha_i\in L^1(\Omega)$ for $1\leq i\leq n$, with $\min_{x\in \Omega}\{\alpha_i(x): 1\leq i\leq n\}\geq 0$ and with $\alpha_1\neq 0$, and for every non-negative continuous $g_i:\mathbb{R} \to \mathbb{R}$ for $2\leq i\leq n$, satisfying \begin{gather*} \max\Big\{\sup_{\xi\in \mathbb{R}}G_i(\xi): 2\leq i\leq n\Big\}\leq 0,\\ \min\Big\{\liminf_{\xi\to +\infty}\frac{G_i(\xi)}{\xi^{p^-}}: 2\leq i\leq n\Big\} > -\infty, \end{gather*} where $G_i(\xi):=\int_0^\xi g_i(t)\,dt$ for all $\xi\in \mathbb{R}$ for $2\leq i\leq n$, for each $$ \lambda\in \Big]0,\frac{1}{p^+c^{p^-} \liminf_{\xi\to+\infty}\frac{G_1(\xi)}{\xi^{p^-}}\int_\Omega\alpha_1(x)\,dx}\Big[, $$ the problem \begin{gather*} \Delta_{p(x)}^2u=\lambda\sum _{i=1}^{n}\alpha_i(x)g_i(u),\quad x\in\Omega,\\ u=\Delta u=0,\quad x\in \partial \Omega \end{gather*} has an unbounded sequence of weak solutions in $X$. \end{corollary} \begin{proof} Set $f(x,t)=\sum_{i=1}^{n}\alpha_i(x)g_i(t)$ for all $(x,t)\in \Omega\times\mathbb{R}$. From the assumption (A4) and the condition $\min\{\liminf_{\xi\to +\infty}\frac{G_i(\xi)}{\xi^{p^-}}: 2\leq i\leq n\}> -\infty$, we have $$ \limsup_{\xi\to +\infty}\frac{\int_\Omega F(x,\xi)\,dx}{\xi^{p^+}} =\limsup_{\xi\to +\infty}\frac{\sum_{i=1}^{n}\left(G_i(\xi) \int_\Omega\alpha_i(x)\,dx\right)}{\xi^{p^+}}=+\infty. $$ Moreover, from the assumption $({\rm A_3})$ and the condition $\max\{\sup_{\xi\in \mathbb{R}}G_i(\xi): 2\leq i\leq n\}\leq 0$, we have $$ \liminf_{\xi\to +\infty}\frac{\int_\Omega \sup_{|t|\leq \xi}F(x,t)\,dx}{\xi^{p^-}} \leq \Big(\int_\Omega\alpha_1(x)\,dx\Big) \liminf_{\xi\to +\infty}\frac{G_1(\xi)}{\xi^2}<+\infty. $$ Hence, applying Theorem \ref{the3.1} the desired conclusion follows. \end{proof} Let us observe that the function $s:\overline{\Omega}\to \mathbb{R}_0^+$ defined by $$ s(x)=d(x,\partial\Omega)\quad \forall x\in \overline{\Omega} $$ is Lipschitz continuous. Hence, there exists $y^0\in \Omega$ such that $$ \overline{s}=s(y^0)=\max_{x\in \Omega}s(x). $$ Moreover, put \begin{equation}\label{eqsi1} \begin{aligned} \sigma':&= \frac{|\Omega|c^{p^-}(1-\overline{\mu}^N)}{\overline{\mu}^N}\\ &\quad \times \max\Big\{\big[\frac{12(N+2)^2(1+\overline{\mu})} {\overline{s}^2(1-\overline{\mu})^3}\big]^{p^-} , \big[\frac{12(N+2)^2(1+\overline{\mu})}{\overline{s}^2(1-\overline{\mu})^3} \big]^{p^+} \Big\}, \end{aligned} \end{equation} where $\overline{\mu}\in ]0,1[$. The following is an autonomous version of Theorem \ref{the3.1}. \begin{theorem}\label{arasi3.1} Let $h:\mathbb{R}\to \mathbb{R}$ be a continuous function such that: \begin{itemize} \item[(A1')] $H(t)=\int_0^th(\xi)d\xi\geq 0$ for every $t\in [0,+\infty[$. \item[(A2')] Putting $$ \eta':=\liminf_{t\to +\infty}\frac{ \max_{|\xi|\leq t}H(\xi)}{t^{p^-}}, \quad \theta':=\limsup_{t\to +\infty}\frac{ H(t)}{t^{p^+}}, $$ one has $\eta'<\frac{p^-}{p^+\sigma'}\theta'$, where $\sigma'$ is defined in \eqref{eqsi1}. \end{itemize} Then, for each $\lambda\in \frac{1}{c^{p^-}|\Omega|} (\frac{\sigma'}{p^-\theta'},\frac{1}{p^+\eta'})$ and for every $q\in C^0(\mathbb{R})$ such that \begin{gather} Q(t)=\int_0^tq(\xi)d\xi\geq 0\quad\text{for every }t\in [0,+\infty[, \label{q1} \\ Q_\infty:=\limsup_{\xi\to+\infty} \frac{\max_{|\xi|\leq t}Q(\xi)}{{t^{p^-}}}<+\infty, \label{q2} \end{gather} if we put $\mu^*:=\frac{1}{p^+c^{p^-}|\Omega|Q_{\infty}}(1-\lambda \eta'p^+c^{p^-})$, for every $\mu \in [0,\mu^*[$ the problem \begin{equation}\label{eqras1.1} \begin{gathered} \Delta_{p(x)}^2u=\lambda h(u)+\mu q(u),\quad x\in\Omega,\\ u=\Delta u=0,\quad x\in \partial \Omega \end{gathered} \end{equation} admits an unbounded sequence of weak solutions. \end{theorem} \begin{proof} Put $x^0=y^0$, $s_2=\overline{s}$, $s_1=\bar{\mu}\bar{s}$, $f(x,t)=h(t)$ and $g(x,t)=q(t)$ for every $(x,t)\in \overline{\Omega}\times \mathbb{R}$. obviously (A1') implies (A1). Moreover, $$ \eta=|\Omega|\eta',\quad \theta=\frac{\pi^{N/2}}{\Gamma(1+N/2)} (\bar{s}\bar{\mu})^N \theta', \quad \sigma=\frac{(\bar{s}\bar{\mu})^N\pi^{N/2}}{|\Omega|\Gamma(1+N/2)}\sigma'. $$ Hence, in view of (A2'), one has $$ \eta<\frac{p^-}{p^+\sigma'}|\Omega|\theta'=\frac{p^-}{p^+\sigma}\theta; $$ that is (A2) holds and the conclusion follows directly from Theorem \ref{the3.1} upon observing that $G(x,t)=Q(t)$ for every $(x,t)\in \Omega\times \mathbb{R}$ and $g_{\infty}=|\Omega|Q_{\infty}$. \end{proof} \begin{corollary}\label{corarsi} Let $h:\mathbb{R}\to \mathbb{R}$ be a continuous and non-negative function such that $$ \liminf_{t\to+\infty}\frac{H(t)}{t^{p^-}} <\frac{p^-}{p^+\sigma'}\limsup_{t\to+\infty}\frac{H(t)}{t^{p^+}}, $$ where $\sigma'$ is defined by \eqref{eqsi1}. Then, for every $$ \lambda\in\frac{1}{c^{p^-}|\Omega|}\Big(\frac{\sigma'} {p^-\limsup_{t\to+\infty}\frac{H(t)}{t^{p^+}}} ,\frac{1}{p^+\liminf_{t\to+\infty}\frac{H(t)}{t^{p^-}}}\Big), $$ for every $q\in C^0(\mathbb{R})$ such that: \begin{gather} tq(t)\geq 0\quad\text{for every } t\in \mathbb{R}, \label{q1'} \\ \lim_{|t|\to+\infty} \frac{q(t)}{{|t|^{p^--1}}}=0 \label{q2'} \end{gather} and for every $\mu\geq 0$, problem \eqref{eqras1.1} admits an unbounded sequence of weak solutions. \end{corollary} \begin{proof} It follows from Theorem \ref{arasi3.1} on observing that, in view of the non-negativity of $h$, (A1') holds and $\eta'=\liminf_{t\to+\infty}\frac{H(t)}{t^{p^-}}$, and also \eqref{q1'} implies \eqref{q1}. Moreover, by \eqref{q1'} one has $$ 0\leq \limsup_{t\to+\infty}\frac{\max_{|\xi|\leq t}Q(\xi)}{t^{p^-}} =\limsup_{t\to+\infty}\frac{\{Q(t),Q(-t)\}}{t^{p^-}}. $$ Exploiting, (2) and owing to the H\^{o}pital rule we have $$ \lim_{t\to+\infty}\frac{Q(t)}{t^{p^-}}= \lim_{t\to+\infty}\frac{Q(-t)}{t^{p^-}} =\pm\lim_{t\to+\infty}\frac{q(\pm t)}{t^{p^--1}}=0. $$ Hence $Q_{\infty}=0$ and our conclusion follows. \end{proof} Now, put \begin{gather}\label{eqsi2} \begin{aligned} \sigma^0:&=\frac{2c^{p^+}\pi^{\frac{N}{2}}(r_2^N-r_1^N)} {N\Gamma(\frac{N}{2})}\times\\ &\max \Big\{\big[\frac{12(N+2)^2(r_1+r_2)}{(r_2-r_1)^3}\big]^{p^-} , \big[\frac{12(N+2)^2(r_1+r_2)}{(r_2-r_1)^3}\big]^{p^+} \Big\}, \end{aligned} \\ \eta^0:=\liminf_{\xi\to 0^+}\frac{\int_\Omega \sup_{|t|\leq\xi}F(x,t)\,dx}{\xi^{p^+}}, \nonumber \\ \theta^0:=\limsup_{\xi\to 0^+}\frac{\int_{B(x^0,r_1)} F(x,\xi)\,dx}{\xi^{p^-}}, \nonumber \\ \lambda_3:=\frac{\sigma^0}{p^-c^{p^+}\theta^0},\quad \lambda_4:=\frac{1}{p^+c^{p^+}\eta^0}. \nonumber \end{gather} Using Lemma \ref{lem2.1}(c) and arguing as in the proof of Theorem \ref{the3.1}, we can obtain the following result. \begin{theorem}\label{the3.6} Assume that (A1) holds and \begin{itemize} \item[(A5)] $\eta^0<\frac{p^-}{p^+\sigma}\theta^0$. \end{itemize} Then, for every $\lambda\in(\lambda_3,\lambda_4)$ and for every $g\in C^0(\bar{\Omega}\times \mathbb{R})$, such that \begin{gather} % [${\rm(k_1)}$] \text{There exists $\tau>0$ such that $G(x,t)\geq 0$ for every $(x,t)\in \bar{\Omega}\times [0,\tau]$}, \label{k1} \\ g_0:=\limsup_{t\to 0^+} \frac{\int_{\Omega}\max_{|\xi|\leq t}G(x,\xi)dx}{{t^{p^+}}}<+\infty, \label{k2} \end{gather} if we put $$ \mu'_{g,\lambda}:=\frac{1}{p^+c^{p^+}g_0}\left(1-\lambda p^+c^{p^+}\eta^0\right), $$ where $\mu'_{g,\lambda}=+\infty$ when $g_0=0$, then for every $\mu\in[0,\mu'_{g,\lambda})$ problem \eqref{e1.1} has a sequence of weak solutions, which converges strongly to zero in $X$. \end{theorem} \begin{proof} Fix $\overline{\lambda}\in (\lambda_3,\lambda_4)$ and let $g$ be a function that satisfies the condition \eqref{k2}. Since $\overline{\lambda}<\lambda_4$, we obtain \[ \mu'_{g,\overline{\lambda}}:=\frac{1}{p^+c^{p^-} g_0}\big(1-\overline{\lambda}p^+c^{p^-}\eta^0\big)>0. \] Now fix $\overline{\mu}\in (0,\mu'_{g,\overline{\lambda}})$ and set \[ J(x,t):=F(x,\xi)+\frac{\overline{\mu}}{\overline{\lambda}}G(x,\xi), \] for all $(x,t)\in \Omega\times\mathbb{R}$. We take $\Phi, \Psi$ and $I_{\overline{\lambda}}$ as in the proof of Theorem \ref{the3.1}. Now, as it has been pointed out before, the functionals $\Phi$ and $\Psi$ satisfy the regularity assumptions required in Lemma \ref{lem2.1}. As first step, we will prove that $\overline{\lambda}<1/\delta$. Then, let $\{\xi_n\}$ be a sequence of positive numbers such that $\lim_{n\to +\infty}\xi_n=0$ and $$ \lim_{n\to+\infty}\frac{\int_\Omega\sup_{|t|\leq\xi_n}F(x,t)\,dx}{\xi_n^{p^+}}=\eta^0. $$ By the fact that $\inf_X\Phi=0$ and the definition of $\delta$, we have $\delta=\liminf_{r\to{0^+}}\varphi(r)$. Putting $r_n=\frac{1}{p^+}\big(\frac{\xi_n}{c}\big)^{p^+}$. Then, as in showing \eqref{e3.2} in the proof of Theorem \ref{the3.1}, we can prove that $\delta<+\infty$. From $\overline{\mu}\in (0,\mu'_{g,\overline{\lambda}})$, the following inequalities hold $$ \delta\leq p^+c^{p^+}\Big(\eta^0+\frac{\overline{\mu}}{\overline{\lambda}}g_0\Big) 0$ such that $\lim_{n\to +\infty}\tau_n=0^+$ and \[ \frac{1}{\overline{\lambda}}<\zeta<\frac{p^-c^{p^+}}{\sigma^0} \frac{\int_{B(x^0,r_1)} F(x,\tau_n)\,dx}{\tau_n^{p^-}} \] for each $n\in\mathbb{N}$ large enough. Let $\{w_n\}$ be the sequence in $X$ defined in \eqref{sasiras}. From ${\rm(k_1)}$ and ${\rm(A_1)}$ one has \eqref{e3.6} holds. Note that $\overline{\lambda}\zeta>1$. Then, as in showing \eqref{sds}, we can obtain \begin{align*} I_{\overline{\lambda}}(w_n) < \frac{\tau_n^{p^-}\sigma^0}{p^-c^{p^+}} (1-\overline{\lambda}\zeta)<0=\Phi(0)-\overline{\lambda}\Psi(0) \end{align*} for every $n\in\mathbb{N}$ large enough. Then, we see that zero is not a local minimum of $I_{\overline{\lambda}}$. This, together with the fact that zero is the only global minimum of $\Phi$, we deduce that the energy functional $I_{\overline{\lambda}}$ has not a local minimum at the unique global minimum of $\Phi$. Therefore, by Lemma \ref{lem2.1}(c), there exists a sequence $\{u_n\}$ of critical points of $I_{\overline{\lambda}}$ which converges weakly to zero. In view of the fact that the embedding $X\hookrightarrow C^0(\overline{\Omega})$ is compact, we know that the critical points converge strongly to zero, and the proof is complete. \end{proof} \begin{remark}\label{rem3.2.1} \rm{ Under the conditions $\eta^0=0$ and $\theta^0=+\infty$, Theorem \ref{the3.6} ensures that for every $\lambda>0$ and for each $\mu\in\big{[}0,\frac{1}{p^+c^{p^+}g_0}\big{)}$, problem \eqref{e1.1} admits a sequence of weak solutions which strongly converges to 0 in $X$. Moreover, if $g_0=0$, the result holds for every $\lambda>0$ and $\mu\geq 0$.} \end{remark} \begin{remark}\label{rem3.7}\rm{ Applying Theorem \ref{the3.6}, results similar to Theorem \ref{the3.3} Corollaries \ref{core3.3.1} and \ref{cor3.6}, can be obtained. We omit the discussions here.} \end{remark} We conclude this article with the following example that illustrates our results. \begin{example} \rm Let $\Omega=\{(x,y)\in \mathbb{R}^2; x^2+y^2<3\}$. Then consider the problem \begin{equation} \label{arasjan} \begin{gathered} \Delta_{p(x,y)}^2u=\lambda f(x,y,u)+\mu g(x,y,u),\quad x\in \Omega,\\ u=\Delta u=0,\quad x\in \partial \Omega, \end{gathered} \end{equation} where $p(x,y)=x^2+y^2+3$ for all $(x,y)\in \Omega$, \[ f(x,y,t)= \begin{cases} f^{*}(x,y)t^6\big(7+\sin(\ln(|t|))-7\cos(\ln(|t|))\big), & (x,y,t)\in \Omega\times (\mathbb{R}-\{0\}),\\ 0,& (x,y,t)\in \Omega \times \{0\}, \end{cases} \] where $f^{*}:\Omega \to \mathbb{R}$ is a non-negative continuous function, and $$ g(x,y,t)=e^{x+y-t^{+}}(t^{+})^{\varsigma-1}(\varsigma-t^{+}) $$ for all $(x,y)\in \Omega$ and $t\in \mathbb{R}$, where $t^+=\max\{t,0\}$ and $\varsigma$ is a positive real number. It is obvious that $p^-=3$ and $p^+=6$. A direct calculation shows that \[ F(x,y,t)=\begin{cases} f^{*}(x,y)t^7\big(1-\cos(\ln(|t|))\big), &(x,y,t)\in \Omega\times (\mathbb{R}-\{0\}), \\ 0, &(x,y,t)\in \Omega \times \{0\}. \end{cases} \] So, \begin{gather*} \liminf_{\xi\to+\infty}\frac{\int_{\Omega} \max_{|t|\leq\xi}F(x,y,t)\,d\sigma}{\xi^{3}}=0, \\ \limsup_{\xi\to+\infty}\frac{\int_{B((0,0),1)}F(x,y,\xi)\,dx}{\xi^{6}}=+\infty. \end{gather*} Hence, using Theorem \ref{the3.1}, since $g_\infty=0$, the problem \eqref{arasjan} for every $(\lambda,\mu)\in ]0,+\infty[\times [0,+\infty[$ admits infinitely many weak solutions in $X$. \end{example} \begin{thebibliography}{99} \bibitem{AfrHeiSho} G. A. Afrouzi, S. Heidarkhani, S. Shokooh; Infinitely many solutions for Steklov problems associated to non-homogeneous differential operators through Orlicz-Sobolev spaces, \emph{Complex Var. Elliptic Equ.}, \textbf{60}(11) (2015), 1505-1521. \bibitem{BonaMolica} G. Bonanno, G. Molica Bisci; Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, \emph{Bound. Value Probl.,} \textbf{2009} (2009), 1-20. \bibitem{BonaMolica1} G. Bonanno, G. Molica Bisci; Infinitely many solutions for a Dirichlet problem involving the $p$-Laplacian, \emph{Proc. Roy. Soc. Edinburgh Sect. A,} \textbf{140} (2010), 737-752. \bibitem{BonaMolicaOr} G. Bonanno, G. Molica Bisci, D. O'Regan; Infinitely many weak solutions for a class of quasilinear elliptic systems, \emph{Math. Comput. Modelling,} \textbf{52} (2010), 152-160. \bibitem{BonaMolicaRad0} G. Bonanno, G. Molica Bisci, V. R\u{a}dulescu; Infinitely many solutions for a class of nonlinear eigenvalue problem in Orlics-Sobolev spaces, \emph{C. R. Acad. Sci. Paris. Ser. I,} \textbf{349} (2011), 263-268. \bibitem{BonaMolicaRad00} G. Bonanno, G. Molica Bisci, V. R\u{a}dulescu; Infinitely many solutions for a class of nonlinear elliptic problems on fractals, \emph{C. R. Acad. Sci. Paris. Ser. I,} \textbf{350} (2012), 387-391. \bibitem{BonaMolicaRad} G. Bonanno, G. Molica Bisci, V. R\u{a}dulescu; Arbitrarily small weak solutions for a nonlinear eigenvalue problem in Orlicz-Sobolev spaces, \emph{Monatsh. Math.,} \textbf{165} (2012), 305-318. \bibitem{BonaMolicaRad1} G. Bonanno, G. Molica Bisci, V. R\u{a}dulescu; Variational analysis for a nonlinear elliptic problem on the Sierpinski gasket, \emph{ESAIM Control Optim. Calc. Var.,} \textbf{18} (2012), 941-953. \bibitem{CanLiLiv} P. Candito, L. Li, R. Livrea; Infinitely many solutions for a perturbed nonlinear Navier boundary value problem involving the $p$-biharmonic, \emph{Nonlinear Anal.,} \textbf{75} (2012), 6360-6369. \bibitem{dema} R. Demarque, O. Miyagaki; Radial solutions of inhomogeneous fourth order elliptic equations and weighted Sobolev embeddings, \emph{Advances in Nonlinear Analysis}, \textbf{4}(2), 135-151. \bibitem{Fazh} X. L. Fan, D. Zhao; On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$, \emph{J. Math. Anal. Appl.,} \textbf{263} (2001), 424-446. \bibitem{GrKon} J. R. Graef, S. Heidarkhani, L. Kong; Infinitely many solutions for systems of multi-point boundary value problems using variational methods, \emph{Topol. Methods Nonlinear Anal.,} \textbf{42} (2013), 105-118. \bibitem{Kora} O. Kov\'{a}cik, J. R\'{a}kos\'{i}k; On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$, \emph{Czechoslovak Math. J.,} \textbf{41} (1991), 592-618. \bibitem{LT} C. Li, C. L. Tang; Three solutions for a Navier boundary value problem involving the $p$-biharmonic operator, \emph{Nonlinear Anal.}, {\bf72} (2010), 1339-1347. \bibitem{BiRe} G. Molica Bisci, D. Repov\v{s}; Multiple solutions of $p$-biharmonic equations with Navier boundary conditions, \emph{Complex Var. Elliptic Equ.}, \textbf{59}(2) (2014), 271-284. \bibitem{ouso} S. Ouaro, A. Ouedraogo, S. Soma; Multivalued problem with Robin boundary condition involving diffuse measure data and variable exponent, \emph{Adv. Nonlinear Anal.}, \textbf{3}(4) (2014), 209-235. \bibitem{Radu1} V. R\u{a}dulescu; Nonlinear elliptic equations with variable exponent: old and new, \emph{Nonlinear Anal.}, \textbf{121} (2015), 336-369. \bibitem{Radu2} V. R\u{a}dulescu, D. Repov\v{s}; Partial Differential Equations with Variable Exponents. Variational Methods and Qualitative Analysis, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2015. \bibitem{repovs} D. Repov\v{s}; Stationary waves of Schr\"{o}dinger-type equations with variable exponent, \emph{Anal. Appl. (Singap.)}, \textbf{13}(6) (2015), 645-661. \bibitem{Ricceri1} B. Ricceri; A general variational principle and some of its applications, \emph{J. Comput. Appl. Math.,} \textbf{113} (2000), 401-410. \bibitem{Ricceri2} B. Ricceri; A three critical points theorem revisited, \emph{Nonlinear Anal.,} \textbf{9} (2009), 3084-3089. \bibitem{Ruz} M. Ru\v{z}i\v{c}ka; Electro-rheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math., 1784, Springer, Berlin, 2000. \bibitem{San} S. G. Samko; Denseness of $C^{\infty}_0(\mathbb{R}^N)$ in the generalized Sobolev spaces $W^{m,p(x)}(\mathbb{R}^N)$, \emph{Dokl. Ross. Akad. Nauk.,} \textbf{369} (1999), 451-454. \bibitem{SZ} Y. Shen, J. Zhang; Existence of two solutions for a Navier boundary value Problem involving the p-biharmonic, \emph{Differential Equations and Applications}, {\bf3} (2011), 399-414. \bibitem{Holi} H. Yin, Y. Liu; Existence of three solutions for a Navier boundary value problem involving the $p(x)$-biharmonic, \emph{Bull. Korean. Math. Soc.,} \textbf{6} (2013), 1817-1826. \bibitem{Home} H. Yin, M. Xu; Existence of three solutions for a Navier boundary value problem involving the $p(x)$-biharmonic operator, \emph{Ann. Polon. Math.,} \textbf{109} (2013), 47-54. \bibitem{yucedag} Z. Y\"{u}ceda\u{g}; Solutions of nonlinear problems involving $p(x)$-Laplacian operator, \emph{Advances in Nonlinear Analysis}, \textbf{4}(4), 285-293. \bibitem{Zang2008} A. Zang, Y. Fu; Interpolation inequalities for derivatives in variable exponent Lebesgue-Sobolev spaces, \emph{Nonlinear Anal.,} \textbf{69} (2008), 3629-3636. \end{thebibliography} \end{document}