%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% %% This is the mc_template.tex file for the mc document class. %% %% It is used to prepare s manuscript for Mathematical %% %% Communications journal. %% %% %% %% The mc.cls class works only with a pdflatex engine. %% %% The file newmc.cls should be placed where LaTeX %% %% can find it, e.g. in the current working directory. %% %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass{mc} %%===============================================================%% %% Please add here your own packages, macros and enviroments. %% %% It is not necessary to include ams* and graphicx packages %% %% since they are automatically included by the mc class. %% %% Avoid defining your own environments and use the already %% %% defined ones (e.g.~theorem, lemma etc.) %% %%===============================================================%% %\usepackage{enumerate} % uncomment to use this package %\newcommand{\E}{\mathbb{E}} % example of a macro %%===============================================================%% %%===============================================================%% %% Journal info will be edited by the typesetter %% %% DO NOT CHANGE THIS PART %% %%===============================================================%% \setcounter{page}{1} \renewcommand\thisnumber{x} \renewcommand\thisyear {2025} \renewcommand\thismonth{xxx} \renewcommand\thisvolume{30} \renewcommand\datereceived{November 14, 2024} \renewcommand\dateaccepted{July 29, 2025} \renewcommand\doinum{10.1000/100} %%===============================================================%% \begin{document} % \linenumbers %%===============================================================%% %% TITLE %% %% Please add the title with \title[Short title]{Title} %% %% Short title is a running head apearing in the header. %% %%===============================================================%% \title[Three solutions to a ${p(\cdot)}$-biharmonic problem] % at most 50 characters including spaces {Existence of three solutions to a ${p(\cdot)}$-biharmonic problem via a local mountain pass theorem} % at most 150 characters including spaces () %%===============================================================%% %%===============================================================%% %% AUTHOR(S) %% %% %% %% Add author's details in the following format. For each author %% %% provide the affiliation, address and Orcid identifier. %% %% Mark the corresponding author with \comma\corrauth. %% %%===============================================================%% \author[G.~Afrouzi, J.~R.~Graef, and A.~R.~Jalali] % put here short author names for header {Ghasem Afrouzi\affil1\orcidnumber{0000-0001-8794-3594}, John R. Graef\affil2\comma\corrauth\orcidnumber{0000-0002-8149-4633}, and % A. R. Jalali\affil1\orcidnumber{0000-0001-7079-0659} % if Second and Third authors share the same affiliation } \address{\affilnum1 Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran \\ \affilnum2 University of Tennessee at Chattanooga, Chattanooga, 37\,403 TN, USA} \emails{% \email{afrouzi@umz.ac.ir} (G.~Afrouzi), \email{john-graef@utc.edu, johngraef9@gmail.com} (J.~R.~Graef), \email{jalali.atieh@yahoo.com} (A.~R.~Jalali) } %% For single author use the following format. %% %\author[F.~Author]% % {First Author\orcidnumber{0000-0000-0000-0000} % } % %\address{Affiliation 1} % %\emailsingle{% % \email{fauthor@mathos.hr} % } %%===============================================================%% %%===============================================================%% %% ABSTRACT %% %%===============================================================%% \begin{abstract} The authors consider the problem of the existence of multiple weak solutions to $p(x)$-biharmonic equations with Navier boundary conditions. Using Ricceri's variational principle and a local mountain pass theorem, and without requiring the Palais-Smale condition, the authors establish sufficient conditions for the existence of at least three solutions to the problem. \end{abstract} %%===============================================================%% %%===============================================================%% %% KEYWORDS %% %%===============================================================%% \keywords{p(x)-biharmonic; Neumann problem; embedding theorem; variational methods} %%===============================================================%% %%===============================================================%% %% AMS subject classification %% %%===============================================================%% \ams{35J60, 35J48} %%===============================================================%% %%===============================================================%% \maketitle %%===============================================================%% %%===============================================================%% %% MAIN BODY %% %%===============================================================%% \section{Introduction} In this work, we are interested in the existence of at least three weak solutions to the Navier boundary value problem \begin{equation}\label{e1.1} \begin{cases} \Delta^{2}_{p(x)}u=\lambda t(x,u(x))+\mu k(x,u(x)), &\mbox{in } \Omega,\\ u=\Delta u =0,&\mbox{on } \partial \Omega. \end{cases} \end{equation} The setting for the problem is as follows: \begin{enumerate} \item[$*$] $\lambda$, $\mu\geq 0$, and $\Omega$ is a bounded open domain in $\mathbb{R}^{N}$ with a smooth boundary $\partial\Omega$; \item[$*$] $p\in C(\bar{\Omega})$ with $p(x)>1$, $p^{-}>1$, and $p^{+}<\infty$, where $p^{-}:=\inf_{x\in\Omega}p(x)$ and $p^{+}:=\sup_{x\in\Omega}p(x)$; \item[$*$] the functions $t$ and $k$ belong to $C(\bar{\Omega}\times \mathbb{R})$. \end{enumerate} Here, $\Delta^{2}_{p(x)}u:= \Delta(\mid\Delta\mid^{p(x)-2}\Delta u)$ is the fourth order operator known as the $p(x)$-biharmonic operator. In the last two decades, the literature on various mathematical problems with variable exponents has been increasing, especially the study of elastic mechanics, electro-rheological fluids, image processing, micro electro-mechanical systems, surface diffusion on solids, and flows in Hele-Shaw cells. One reason for such an increase in interest is the fact that different areas of applied mathematics and physical phenomena can be modeled by such equations. For example, applications in nonlinear elasticity and electro-rheological fluids can be found in \cite{new1, new5, new2, new3, new88, new22, new21}. Many authors have considered differential equations with variable exponents (see \cite{new32, new31, new16}) and have pointed out applications involving the $p(x)$-biharmonic operator. This operator allows growth conditions that involve more complicated nonlinearities than the constant order case. We refer the reader to \cite{newN, new4, new5, new6, new7, new8, new9, new10, new11, new12, new13, new14, new15, new16, new17, new18, new19, new20} and the references therein for details. Some classical tools such as the three critical points theorem of Ricceri \cite{new43, new43a} have been used in solving such problems. Additional background on variational approaches to obtaining multiple solutions to boundary value problems can be found in the monograph by Graef and Kong \cite{GK}. Kong \cite{new12} used Ekeland's variational principle and some recent results on the generalized Lebesgue-Sobolev spaces $L^{p(x)}(\Omega)$ and $W^{h,p(x)}(\Omega)$ to prove the existence of solutions to \begin{equation*} \begin{cases} \Delta^{2}_{p(x)}u + a(x)|u|^{p(x)-2} u = \lambda w(x)f(u), &\mbox{in } \Omega,\\ u=\Delta u =0, &\mbox{on } \partial \Omega. \end{cases} \end{equation*} A class of quasilinear elliptic equations involving the $p(x)$-biharmonic operator with Navier boundary conditions was recently analyzed by Yin and Xu in \cite{new19}. Deng \cite{new25} applied a local mountain pass theorem, without the Palais-Smale condition, and Ricceri's variational principle to obtain the existence of multiple solutions to the $p(x)$-Laplacian doubly perturbed Neumann problem \begin{equation*} \begin{cases} \Delta^{2}_{p(x)}u + a(x)|u|^{p(x)-2} u = f(x,u) + \lambda h_1(x,u) &\mbox{in } \Omega,\\ |\nabla u|^{p(x)-2}\frac{\partial u}{\partial y} = g(x,u) + \mu h_2(x,u), &\mbox{on } \partial \Omega. \end{cases} \end{equation*} In the present paper, we discuss the existence of at least three weak solutions to problem \eqref{e1.1}. We first review some basic facts on variable exponent Lebesgue and Sobolev spaces and then describe Ricceri's variational principle (see Theorem \ref{r-th} below). We will also make use of a local mountain pass theorem, without requiring the Palais Smale condition (Proposition \ref{j10}), to prove our main results. \section{Preliminaries} We first review some definitions and facts about variable exponent Lebesgue and Sobolev spaces; detailed descriptions can be found in \cite{new28,new30, new29}. Set \[ C_{+}(\Omega):= \{ r : r\in C(\bar{\Omega})\quad \mbox{and} \quad r(x)> 1 \mbox{ for all } x\in \bar{\Omega}\}.\] Define the Lebesgue space with variable exponent $p(\cdot)\in C(\bar{\Omega})$ by \[ L^{p(\cdot)}(\Omega):= \left\{ u: \Omega\to \mathbb{R} \ \mbox{is measurable and} \ \int_{\Omega} \left|u(x) \right|^{p(x)}dx < \infty \right\}. \] This is a separable and reflexive Banach space if endowed with the Luxemburg norm \[ \parallel u \parallel_{L^{p(\cdot)}}= \inf \left\{ \gamma>0 : \int_{\Omega} \left| \frac{u(x)}{\gamma} \right|^{p(x)}dx \leq 1 \right\}. \] Next, we recall the mapping $ \rho: L^{p(x)}\to \mathbb{R}$ called the modular function for the space $L^{p(x)}(\Omega)$, as defined by \[ \rho(u)=\int_{\Omega}\mid u(x) \mid^{p(x)}dx. \] The relationship between the Luxemburg norm and the modular function is given in the following proposition. \begin{proposition} {\rm (\cite{new35a, rep})} If $u\in L^{p(x)}$, then the following relationships hold: \begin{itemize} \item[$(1)$] \ $\parallel u \parallel_{L^{p(\cdot)}(\Omega)} < 1 \ (=1; >1) \quad \mbox{if and only if} \quad \rho(u)< 1 \ (=1; > 1)$; \item[$(2)$] \ $\parallel u \parallel_{L^{p(\cdot)}(\Omega)}> 1$ \quad implies \quad $\parallel u \parallel^{p^{-}}_{L^{p(\cdot)}(\Omega)} \leq \rho(u) \leq \parallel u \parallel^{p^{+}}_{L^{p(\cdot)}(\Omega)}$; \item[$(3)$] \ $\parallel u \parallel_{L^{p(\cdot)}(\Omega)}< 1$ \quad implies \quad $\parallel u \parallel^{p^{+}}_{L^{p(\cdot)}(\Omega)} \leq \rho(u) \leq \parallel u \parallel^{p^{-}}_{L^{p(\cdot)}(\Omega)}$; \item[$(4)$] \ $\parallel u \parallel_{L^{p(\cdot)}} \to 0 \ (\to \infty) \quad \mbox{if and only if} \quad \rho(u)\to 0 \ (\to \infty)$; \item[$(5)$] \ if $(u_{n})_{n} \subset L^{p(\cdot)}(\Omega)$, then $\lim_{n\to\infty}\|u_{n}- u\|_{L^{p(\cdot)}(\Omega)} =0 \ \mbox{if and only if} \ \lim_{n\to\infty}\rho (u_{n}-u)=0 \ \mbox{if and only if } (u_{n})_{n}$ converges to $u$ in measure and $\lim_{n\to\infty}\rho(u_{n})= \rho(u)$. \end{itemize} \end{proposition} From \cite{new29, new33}, for any positive integer $h$, the Sobolev space with variable exponent $W^{h,p(x)}(\Omega)$ is defined as \[ W^{h,p(x)}(\Omega)=\{ u \in L^{p(x)}(\Omega): D^{\eta}u\in L^{p(x)}(\Omega), \ \mid\eta\mid\leq h\}, \] equipped with the norm \[ \parallel u \parallel_{W^{h,p(x)}(\Omega)}=\sum_{\mid\eta\mid\leq h}\parallel D^{\eta}u \parallel_{L^{p(\cdot)}(\Omega)}, \] where $D^{\eta}u=\frac{\partial^{\mid\eta\mid}}{\partial x^{\eta_{1}}_{1}\partial x^{\eta_{2}}_{2}...\partial x^{\eta_{N}}_{N}}u$ and $\eta=(\eta_{1}, ... , \eta_{N})$ is a multi-index such that $\mid \eta\mid=\sum_{i=1}^{N}\eta_{i}$. We note that $W^{h,p(x)}(\Omega)$ is a separable, reflexive, and uniformly convex Banach space. Let $q(x)$ be the conjugate exponent of $p(x)$, i.e., $\frac{1}{p(x)}+ \frac{1}{q(x)}=1$. Then the H\"older type inequality \[\int_{\Omega}|uv|dx \leq \left(\frac{1}{p^{-}}+\frac{1}{q^{-}}\right)|u|_{p(x)}\, |v|_{q(x)}, \quad u\in L^{p(x)}(\Omega) \ \mbox{and} \ v\in L^{q(x)}(\Omega), \] holds. \begin{proposition} {\rm (\cite[Theorem $2.3$]{new29})} \label{j4} Let $q\in C(\bar{\Omega};\mathbb{R})$ satisfy $1< q^{-}\leq q^{+} <\infty $ and $q(x)< p^{*}_{h}(x)$ for all $x\in \bar{\Omega}$, where \begin{equation*} p^{*}_{h}(x) = \begin{cases} \frac{Np(x)}{N-hp(x)}, &\mbox{if } \ hp(x)< N,\\ +\infty, &\mbox{if } \ hp(x)\geq N, \end{cases} \end{equation*} for any $x\in \bar{\Omega}$ and $h\geq 1$. Then there is a continuous and compact embedding $W^{h,p(\cdot)}(\Omega)$ into $L^{q(\cdot)}(\Omega)$. \end{proposition} We denote by $W_{0}^{h,p(x)}(\Omega)$ the closure of $C^{\infty}_{0}(\Omega)$ in $W^{h,p(x)}(\Omega)$. We next recall some properties of the spaces $W^{2,p(\cdot)}(\Omega)$, $W_{0}^{1,p(\cdot)}(\Omega)$, and $W^{2,p(\cdot)}(\Omega)\cap W_{0}^{1,p(\cdot)}(\Omega)$. \begin{remark} {\rm (\cite{new4444})} {\rm \begin{itemize} \item[(a)] For $u \in X = W^{2,p(x)}(\Omega)\cap W_{0}^{1,p(x)}(\Omega)$, we define $\parallel \cdot \parallel_{X}$ by \[ \parallel u \parallel_{X} = \parallel u \parallel_{W_{0}^{1, p(x)}} + \parallel u \parallel_{W^{2, p(x)}}. \] Note that $X$ is a separable and reflexive Banach space. In \cite{new34}, Zanga and Fu proved that the norms $\parallel \Delta u \parallel_{L^{p(\cdot)}}$ and $\parallel u \parallel_{X}$ are equivalent. \item[(b)] The space $W^{2,p(x)}(\Omega)$ or $W^{2,p(x)}(\Omega)\cap W_{0}^{1,p(x)}(\Omega)$ is equipped with the norm \[\parallel u \parallel_{a} = \inf \left\{n > 0 : \int_{\Omega}\left[\left|\frac{\Delta u}{n}\right|^{p(x)}+ a(x)\left| \frac{u}{n}\right|^{p(x)}\right]dx \leq 1, \ a\in L^{\infty}\right\}. \] This norm is equivalent to the norm $\parallel \Delta u \parallel_{L^{p(\cdot)}}$. \end{itemize} } \end{remark} In \cite{new4444, new34}, the subspace $S=\{ u\in W^{2,p(\cdot)}(\Omega): u|_{\partial \Omega} \equiv constant\}$ is analyzed and can be considered as $\{ S= u+b : u\in W^{2,p(x)}(\Omega)\cap W_{0}^{1,p(x)}(\Omega), b\in \mathbb{R}\}$. Moreover, $(S, \parallel u \parallel_{W^{2,p(x)}})$ is a separable and reflexive Banach space and the norms $\parallel u \parallel_{W^{2,p(\cdot)}(\Omega)}$, $\parallel u \parallel_a$, and $\parallel \Delta u \parallel_{L^{p(x)}}$ are equivalent. Throughout the remainder of this paper, for convenience we use $\parallel u \parallel$ instead of $\parallel u \parallel_{W^{2,p(\cdot)}(\Omega)}$ on $X$. Next, we present some concepts and results needed to prove our main theorems in this paper. \begin{proposition} {\rm (\cite{new9000, rep})} If $u\in L^{p(\cdot)}(\Omega)$, then the following relations hold: \begin{itemize} \item[{(i)}] $\parallel u \parallel^{p^{-}}\leq \int_{\Omega}\mid \Delta u \mid^{p(x)}dx \leq \| u \|^{p^{+}}$, \quad if $\| u \| \geq 1$. \item[{(ii)}] $\parallel u \parallel^{p^{+}}\leq \int_{\Omega}\mid \Delta u \mid^{p(x)}dx \leq \| u \|^{p^{-}}$, \quad if $\| u \| \leq 1$. \end{itemize} \end{proposition} Here we are studying the existence of weak solutions to problem \eqref{e1.1}, so we next define what that means. \begin{definition} \label{def2.1} We say that $u\in S$ is a weak solution to \eqref{e1.1} if for all $v\in S$, \[ \int_{\Omega}\mid \Delta u\mid^{p(x)-2}\Delta u \Delta v dx -\lambda\int_{\Omega}t(x,u(x))v(x)dx-\mu\int_{\Omega}k(x,u(x))v(x)dx=0. \] \end{definition} To be able to find weak solutions to \eqref{e1.1}, we consider the functional $I : S \to \mathbb{R}$ defined by \[ I(u)= I_{1} + \mu I_{2}, \] where \[ I_{1}=\int_{\Omega}\frac{1}{p(x)}\mid \Delta u \mid^{p(x)}dx - \lambda \int_{\Omega}T(x,u(x))dx, \quad I_{2} = -\int_{\Omega}K(x, u(x))dx, \] \[ T(x,c)=\int^{c}_{0}t(x,s)ds \quad \mbox{and} \quad K(x,c)=\int^{c}_{0}k(x,s)ds. \] Thus, weak solutions to problem \eqref{e1.1} are exactly the critical points of the functional $I$. \section{Ricceri's variational principle} We will use a somewhat standard notation that $\rightharpoonup$ and $\to$ denote weak and strong convergence, respectively. \begin{definition} \label{defn3.1} Let $D$ be a bounded open subset of $X$. We say that $D$ is a Ricceri block of $I_{1}$ of type $\kappa$ if \begin{equation*} \begin{cases} I_{1}(x)< \kappa, &x\in D, \\ I_{1}(x) =\kappa, &x\in \overline{\partial D}=\overline{D}^{w}\setminus D, \end{cases} \end{equation*} where $D^w$ means in the weak topology on $D$. If $c0$ sufficiently small, we have $\inf_{\partial B(x_{*},\theta)}I_{1} > I_{1}(x_{*})$. Moreover, the open ball $B(x_{*},\theta) = \{ x\in X : \|x-x_{*}\| \leq \theta \}$ is a Ricceri box of $I_{1}$. \end{remark} \begin{remark} {\rm (\cite{new35})} For $I_{1}$ and $I_{2}$, the following assertions hold: \begin{itemize} \item[(i)] If $D$ is a Ricceri box of $I_{1}$ of type $(c,b)$, then for each $\kappa\in (c,b]$, $I_{1}^{-1}(-\infty, \kappa)\bigcap D$ is a Ricceri block of $I_{1}$ of type $\kappa$. \item[(ii)] $I_{1}$ and $I_{2}$ are sequentially weakly lower semi continuous, that is, for any $x\in X$ and any subsequence $x_{n} \subset X$ such that $x_{n}\rightharpoonup x$ weakly, we have $I_{i}(x) \leq \liminf_{n\to\infty}I_{i}(x_{n})$ for $i = 1,2$. \item[(iii)] The mapping $I'_{2}$ is weakly-strongly continuous, i.e., if $x_n\rightharpoonup x$, then $I'_{2}(x_n)\to I'_{2}(x)$. \item[(iv)] The sum of a type $(S_{+})$ mapping and a weakly-strongly continuous mappings is also of the type $(S_{+})$. \end{itemize} \end{remark} \begin{theorem} \label{r-th} {\rm (\cite[Theorem 3.2]{new35})} %{\rm (\cite{new 25}, \cite[Theorem 3.2]{new 35})} Assume that $\mu_{*}:=\sup_{x\in D}\frac{\kappa - I_{1}(x)}{I_{2}(x)-\inf_{\overline{D}^{w}}I_{2}(x)}$, where $D$ is a Ricceri block of $I_{1}$ of type $\kappa$. Then: \begin{itemize} \item[$(i)$] For each $\mu\in (0, \mu_{*})$, the restriction of $I_{1} + \mu I_{2}$ to $\overline{D}^{w}$ attains its infimum at some $x_{*}\in D$, so that $x_{*}$ is a local minimizer of $I_{1} + \mu I_{2}$. \item[$(ii)$] $D$ is a Ricceri box of $I_{1} + \mu I_{2}$. \end{itemize} \end{theorem} \begin{proposition} {\rm (\cite{new25})} \label{j9} Assume that: \begin{itemize} \item[{$(i)$}] For $r>0$ and $x_{1}\in B(x_{0},r)$, we have $I_{1}(x_{0})= \inf_{B(x_{0}, r)}I_{1}=c_{0}$ and $\inf_{\partial B(x_{0}, r)}I_{1} = b > c_{0}$; \item[{$(ii)$}] $x_{1}$ is a strictly local minimizer of $I_{1}$ and $I_{1}(x_{1})= c_{1}> c_{0}$. \end{itemize} Then for $\theta > 0$ sufficient small, $\kappa_{1}> c_{1}$, $\kappa_{0}\in (c_{0},\min\{b,c_{1}\})$, and for all $\mu \in (0, \mu_{*})$, $I_{1} +\mu I_{2}$ has at least two local minima $x^{*}_{0}$ and $x^{*}_{1} \in B(x_{0}, r)$, where $x^{*}_{0}\in I^{-1}_{1}((-\infty,\kappa_{0}))\bigcap B(x_{0}, r)$, $x^{*}_{0} \not\in\overline{B(x_{1}, \theta)}$, and $x^{*}_{1}\in I^{-1}_{1}((-\infty, \kappa_{1}))\bigcap B(x_{1}, \theta)$. \end{proposition} \begin{theorem} {\rm (\cite{new25})} \label{th3.7} Let $Y$ be a reflexive Banach space and assume that: \begin{itemize} \item[$(i)$] For $\phi\in C^{1}(Y,\mathbb{R})$, the mapping $\phi':Y\to Y'$ is of type $(S_{+})$. \item[$(ii)$] $(D_{0},D)$ is a valley box of $\phi$ with $D_{0}$ and $D$ being connected and $0\in D_{0}$. \item[$(iii)$] There exist $e\in D_{0}$ and $r>0$ such that $\| e \| >r$ and $\inf_{\partial B(0,r)}\phi > \{\max {\phi(0), \phi(e)}\}$. \end{itemize} Then, the functional $\phi$ has at least one critical point $x_{0}\!\!\in\! \!\overline{D}$ with $\phi(x_{0})\!=\!d$, where $d\!=\!\inf_{\beta\in \Gamma}\sup_{s\in [0,1]}\phi(\beta(s))$ and $\Gamma = \{\beta\in C([0,1],D): \beta(0)=0, \ \beta(1)=e\}$. \end{theorem} \begin{proposition} {\rm (\cite[Corollary 2.1]{new40})} \label{j10} If $I_{2}':Y\to Y'$ is weakly-strongly continuous, and assumptions (i)--(iii) of Theorem \ref{th3.7} hold, then there exist $\mu\in (0,\mu_{*})$ such that $I_{1} + \mu I_{2}$ has a mountain pass type critical point $x_{2}\in \overline{D}$. \end{proposition} \section{Main results} The following assumptions will be needed in our main theorems: \begin{itemize} \item[(a)] $\lim_{\mid z \mid\to\infty}\frac{t(x,z)}{\mid t \mid^{p(x)-1}}=0$ uniformly for $x \in \Omega$. \item[{(b)}] $p^{+} < p^{*}(x)$ and there exist $b_{0}>0$ and $\delta>0$ such that \[ \mid T(x,z)\mid \leq b_{0}\mid z \mid^{q_{0}(x)} \quad \mbox{for all} \quad x\in \Omega \quad \mbox{and} \quad | z | < \delta. \] \item[{(c)}] Let $D$ be a ball with $\overline{D}\subset \Omega$. For any given $Carath\acute{e}odory$ function $t:\overline{\Omega}\times\mathbb{R}\to\mathbb{R}$, there exists $\gamma_{0}>0$ such that \[ \int_{D}T(x,\gamma_{0})dx>0. \] \item[{(d)}] There exist $\beta(x)\in C(\overline{\Omega})$ with $1\leq \beta(x)\leq \beta^{+}\leq p^{-}$, $b>0$, and $\gamma>0$ such that \[ K(x,z) \geq b \mid z \mid^{\beta^{+}} \quad \mbox{for all} \quad x\in \overline{\Omega} \quad \mbox{and} \quad | z | <\gamma. \] \item[{(e)}] $\lim_{\mid z\mid\to\infty}\frac{T(x,z)}{\mid z \mid^{p^{+}}} = +\infty$ uniformly for $x \in \Omega$. \item[{(f)}] $\limsup_{z\to 0}\frac{\inf_{x\in \Omega}K(x,z)}{\mid z \mid^{p^{-}}} = +\infty$. \item[{(k)}] There exists $\xi \in \mathbb{R}$ such that $T(x,\xi)> \int_{\Omega}\frac{\mid \xi \mid^{p(x)}+\mid \Delta \xi\mid^{p(x)}}{p^-}dx$. \end{itemize} Our first existence result in this section is contained in the following theorem. \begin{theorem}\label{t4} Assume that conditions (a)--(e) hold. Then there exists a constant $\lambda_{0}>0$ such that, for all $\lambda_{0}<\lambda$ and $\mu\in (0, \mu_{*})$, problem \eqref{e1.1} has at least three nontrivial weak solutions. \end{theorem} \begin{proof} The proof will be divided into five steps. \textbf{Step 1:} {\it $w_{0} = 0$ is a strictly local minimizer of $I_{1}$.} \ From (a), we can conclude that for every $\epsilon > 0$ there is a $\delta_{\epsilon}> 0$ such that \begin{equation*} \mid t(x,z)\mid \leq \epsilon \mid z \mid^{p(x)-1} \quad \mbox{for all} \quad | z | > \delta_{\epsilon} \quad \mbox{and} \quad x\in \Omega. \end{equation*} Then, from the continuity of $t$, there exists $c_{0}>0$ such that \begin{equation} \label{we} | t(x,z) | \leq c_{0}+ \epsilon |z |^{p(x)-1} \end{equation} for all $z\in \mathbb{R}$ and all $x\in \Omega$. From \eqref{we} and $(b)$, it follows that there exists $q_{1}\in C(\bar{\Omega})$ with $p^{+} p^{+}$, there exists $\theta >0$ such that \[ I_{1}(u)>0 \quad \mbox{for all} \quad u\in \overline{B(0,\theta)}\setminus 0. \] \textbf{Step 2:} {\it $I_{1}$ has a global minimizer $w_{1} \neq 0$.} \ By \eqref{we} and the definition of $I_{1}$, \begin{equation*} \label{web} I_{1}(u) \geq \frac{1}{p^{+}}\int_{\Omega}[\mid\Delta u \mid^{p(x)} - \lambda\epsilon\mid u \mid^{p(x)}]dx - \lambda c_{0}\parallel u \parallel_{L^{1}}. \end{equation*} Choosing $\epsilon<\frac{1}{\lambda}$, by Proposition \ref{j4}, for $\parallel u \parallel \geq 1$ and $1 0$ sufficiently small, we can take \[ \overline{B_{\theta}}:=\overline{\{x\in\Omega : {\rm dist}(x, B) \leq \theta\}}\subset\Omega. \] Define the function \begin{equation*} u_{\theta}(x):= \begin{cases} t_{0}, &x\in B,\\ 0, &x\in \Omega \backslash B_{\theta}. \end{cases} \end{equation*} Then, \[ I_{1}(w_{1})\leq \int_{\Omega} \frac{1}{p(x)}\mid \Delta u_{\theta} \mid^{p(x)}dx-\lambda \int_{B}T(x,t_{0})dx-\lambda\int_{B_{\theta}\setminus B}T(x,u_{\theta})dx. \] Hence, take $\theta_{0}$ sufficiently small so that there is a positive constant $\alpha_{0}$ such that \[ I_{1}(w_{1})\leq \int_{\Omega} \frac{1}{p(x)}\mid \Delta u_{\theta_{0}} \mid^{p(x)}dx -\lambda \alpha_{0}\int_{B}T(x,t_{0})dx, \] and set \[ \lambda_{0}:=\frac{\int_{\Omega} \frac{1}{p(x)}\mid \Delta u_{\theta_{0}} \mid^{p(x)}dx}{\alpha_{0}\int_{B}T(x,t_{0})}>0. \] Then, $I_{1}(w_{1}) < 0$ for all $\lambda_{0}<\lambda$. Therefore, $w_{1}\neq 0$. \textbf{Step 3:} {\it $I=I_{1} + \mu I_{2}$ has two local minima.} \ Since $I_{1}$ is coercive, we can find $r>0$ large enough such that $w_{0}$, $w_{1}\in B(0,r)$ and $\inf_{\partial B(0,r)}I_{1} > I_{1}(w_{0}) > I_{1}(w_{1})$. By Proposition \ref{j9}, given any $\theta>0$, $\kappa_{1}\in (I_{1}(w_{1}), 0)$, and $\kappa_{2}>0$, we see that for all $\mu\in (0, \mu_{*})$, $I$ has at least two local minima $u_{0}\in B(0,\theta)\bigcap I_{1}^{-1}((-\infty, \kappa_{2}))$ and $u_{1}\in I_{1}^{-1}((-\infty, \kappa_{1}))$ with $u_{1}\not\in \overline{B(0, \theta)}$. \textbf{Step 4:} {\it $I=I_{1}+\mu I_{2}$ has a mountain pass type critical point.} \ Take a ball $B(0, r_{1}) \subset S$ such that $B(0, r_{1}) \supset I_{1}^{-1}(-\infty, \kappa_{1}) \bigcup B(0,\theta)$. Since $I_{1}$ is coercive, we can find $r_{2} > r_{1}$ with \[ \inf_{\partial B(0, r_{2})}I_{1}> \sup_{B(0, r_{1})}I_{1}. \] Then, $(B(0, r_{1}), B(0, r_{2}))$ is a valley box of $I_{1}$. Since $I_{1}(w_{1})<0$, by {\bf Step 1}, we have that for some $\epsilon_{0}>0$ with $\epsilon_{0}< \parallel w_{1} \parallel$, $\inf_{\partial B(0, \epsilon_{0})}I_{1} > \max \{I_{1}(0),I_{1} (w_{1}) \} = 0$. From Proposition \ref{j10}, we can conclude that for all $\mu\in (0, \mu_{*})$, $I$ admits a mountain pass critical point $u_{2}$. \textbf{Step 5:} {\it $u_{0} \neq 0$.} \ From condition (e), for all $L>0$ there exist $c_{L}>0$ such that \[ T(x,z)\geq L z^{p^{+}}- c_{L} \quad \mbox{for all} \quad x\in \Omega \quad \mbox{and} \quad z\geq 0. \] Moreover, by (d), (e), and choosing $\alpha \in (0,1)$, we can immediately see that \begin{align*} I_{1}(z\alpha) &\leq \alpha^{p^{+}}\int_{\Omega}\mid \Delta z \mid^{p(x)}dx-\lambda L \alpha^{p^{+}}\int_{\Omega}\mid z \mid^{p(x)}dx + \lambda c_{L}\mid\Omega\mid \\ &= \alpha^{p^{+}}(\int_{\Omega}\mid \Delta z \mid^{p(x)}dx-\lambda L\int_{\Omega}\mid z \mid^{p(x)}dx)+\lambda c_{L}\mid\Omega\mid. \end{align*} Since $L$ is arbitrary and large enough, we can obtain that \[ I_{1}(z\alpha)<0, \] and hence, \[ z\alpha \in B(0,\theta)\bigcap I_{1}^{-1}((-\infty, \kappa_{2})). \] On the other hand, \[ I_{2}(z\alpha)\leq -b\int_{\Omega}\mid z\alpha \mid^{\beta^{+}}dx= -b \alpha^{\beta^{+}}\int_{\Omega}\mid z \mid^{\beta^{+}}dx < 0. \] Therefore, \[I_{1}(u_{0})+\mu I_{2}(u_{0})< I_{1}(z\alpha)+\mu I_{2}(z\alpha)< 0,\] i.e., $u_{0}\neq 0$, and thus we obtain at least three nontrivial solutions to problem \eqref{e1.1}, which proves the theorem. \end{proof} In our next theorem, we replace assumptions (d) and (e) with (f) and (k). \begin{theorem} \label{t4a} Assume that conditions (a)--(c), (f), and (k) hold. Then, there exists a constant $\lambda_0>0$ such that for $1<\lambda_0 < \lambda$ and $\mu\in (0, \mu_{*})$, problem \eqref {e1.1} admits at least three nontrivial weak solutions. \end{theorem} \begin{proof} The proof is similar to the proof of the previous theorem. First we show that $I_{1}(w_{1})< 0$, and by condition (f), taking $z_{n}\to 0$ in {\bf Step 5}, we can show that $u_{0}\neq 0$. We may assume that $\parallel \xi \parallel < 1$; then \[I_{1}(\xi)< \frac{1}{p^{-}}\int_{\Omega}\mid \Delta \xi \mid^{p(x)}dx-\frac{\lambda}{p^{-}}\int_{\Omega}\mid \xi \mid^{p(x)}dx \leq (\frac{1}{p^{-}}-\frac{\lambda}{p^{-}})\parallel \xi \parallel^{p^{-}}<0. \] Thus, \[ I_{1}(w_{1})