\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 107, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2016 Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2016/107\hfil Infinitely many solutions] {Infinitely many solutions via variational-hemivariational inequalities under Neumann boundary conditions} \author[F. Fattahi, M. Alimohammady \hfil EJDE-2016/107\hfilneg] {Fariba Fattahi, Mohsen Alimohammady } \address{Fariba Fattahi \newline Department of Mathematics, University of Mazandaran, Babolsar, Iran} \email{F.Fattahi@stu.umz.ac.ir} \address{Mohsen Alimohammady \newline Department of Mathematics, University of Mazandaran, Babolsar, Iran} \email{Amohsen@umz.ac.ir} \thanks{Submitted November 21, 2015. Published April 26, 2016.} \subjclass{35J87, 49J40, 49J52, 49J53} \keywords{Nonsmooth critical point theory; infinitely many solutions; \hfill\break\indent variational-hemivariational inequality} \begin{abstract} In this article, we study the variational-hemivariational inequalities with Neumann boundary condition. Using a nonsmooth critical point theorem, we prove the existence of infinitely many solutions for boundary-value problems. Our technical approach is based on variational methods. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section {Introduction} In this article, we study following boundary-value problem, depending on the parameters $\lambda,\mu$ with nonsmooth Neumann boundary condition: \begin{equation}\label{e1} \begin{gathered} -\Delta_{p(x)}u +a(x)|u|^{p(x)-2}u=0 \quad\text{in }\Omega\\ -|\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu} \in -\lambda \theta(x)\partial F(u)-\mu \partial \vartheta(x)G(u) \quad\text{on } \partial \Omega, \end{gathered} \end{equation} where $\Omega\subset \mathbb{R}^{N} (N \geq 2)$ is a bounded smooth domain, $\frac{\partial u}{\partial \nu}$ is the outer unit normal derivative on $\partial\Omega$, $p :\bar{\Omega} \to \mathbb{R}$ is a continuous function satisfying $$1 < p^{-} = \min_{x\in \bar{\Omega}} p(x)\leq p(x) \leq p^{+} =\max _{x\in \bar{\Omega}} p(x) < +\infty.$$ Here $\lambda,\mu$ are real parameters, $\lambda\in ]0,\infty[,\mu\in [0,\infty[$ and $\theta,\vartheta\in L^{1}(\partial\Omega)$, where $\theta(x),\vartheta(x)\geq0$ for \textrm{a.e.} $x\in\partial\Omega$. $F,G : \mathbb{R} \to \mathbb{R}$ are locally Lipschitz functions given by $F(\omega)=\int_0^{\omega}f(t)dt$, $G(\omega)=\int_0^{\omega}g(t)dt$, $\omega\in\mathbb{R}$ such that $f,g:\mathbb{R}\to\mathbb{R}$ are locally essentially bounded functions. $\partial F (u),\partial G (u)$ denote the generalized Clarke gradient of $F (u),G (u)$. Let $X$ be real Banach space. We assume that it is also given a functional $\chi : X \to \mathbb{R}\cup \{+\infty\}$ which is convex, lower semicontinuous, proper whose effective domain $dom(\chi) = \{x \in X : \chi(x) < +\infty\}$ is a (nonempty, closed, convex) cone in $X$. Our aim is to study the following variational-hemivariational inequalities problem: Find $u\in \mathcal{B}$ which is called a \emph{weak solution} of problem \eqref{e1}, i.e; if for all $v\in \mathcal{B}$, \begin{equation}\label{t2} \begin{aligned} &\int_{\Omega} |\nabla u|^{p(x)-2}\nabla u \nabla (v-u) dx +\int_{\Omega}a(x)|u|^{p(x)-2} u(v-u) dx\\ &-\lambda\int_{\partial\Omega}\theta(x)F^{0}(u;u-v)d\sigma -\mu\int_{\partial\Omega}\vartheta(x)G^{0}(u;u-v)d\sigma\geq 0, \end{aligned} \end{equation} where $\mathcal{B}$ is a closed convex subset of $W^{1,p(\cdot)}_0(\Omega)$. For simplicity $\mathcal{B}=W^{1,p(\cdot)}_0(\Omega)$. Recently, many researchers have paid attention to impulsive differential equations by variational method. We refer the reader to \cite{Afrou,Chabrowski,Molica,Radu1,Rad2,Radu3,Rosiu} and references cited therein. The operator $\Delta_{p(x)}u=\operatorname{div} (|\nabla u|^{p(x)-2}\nabla u)$ is the so-called $p(x)$-Laplacian, which becomes $p$-Laplacian when $p(x)\equiv p$ is a constant. More recently, the study of $p(x)$-Laplacian problems has attracted more and more attention \cite{Allaoui,Zhou}. Variational-hemivariational inequalities have been extensively studied in recent years via variational methods: in \cite{Kri}, the author studied hemivariational inequalities on an unbounded strip-like domain; in \cite{mot5}, the authors studied variational-hemivariational inequalities for the existence of a whole sequence of solutions with non-smooth potential and non-zero Neumann boundary condition; in \cite{Bona2}, the authors studied variational-hemivariational inequalities involving the $p-$Laplace operator and a nonlinear Neumann boundary condition via abstract critical point result; in \cite{Ali}, the authors studied variational-hemivariational inequality on bounded domains by using the mountain pass theorem and the critical point theory for Motreanu-Panagiotopoulos type functionals. The aim of the present paper is find sufficient conditions to guarantee the existence of infinitely many weak solutions for a variational-hemivariational inequality depending on two parameters. Our approach is a variational method and the main tool is a general nonsmooth critical point theorem. \section{Preliminaries} In this section, we recall some definitions and results which are used further in this paper. The variable exponent Lebesgue space is defined by $$L^{p(\cdot)}(\Omega)=\{u:\Omega \to \mathbb{R}: \int_{\Omega}|u(x)|^{p(x)}dx<\infty \}$$ and is endowed with the Luxemburg norm $$\|u\|_{p(\cdot)}=\inf\:\{\:\lambda > 0 : \int_{\Omega}|\frac{u(x)}{\lambda}|^{p(x)}dx \leq 1 \}.$$ Note that, when $p$ is constant, the Luxemburg norm $\|\cdot\|_{p(\cdot)}$ coincides with the standard norm $\|\cdot\|_p$ of the Lebesgue space $L^{p}(\Omega)$. $(L^{p(\cdot)}(\Omega),\|\cdot\|_{p(\cdot)})$ is a Banach space. The generalized Lebesgue-Sobolev space $W^{L,p(\cdot)}(\Omega)$ for $L=1,2,\dots$ is defined by $$W^{L,p(\cdot)}(\Omega)=\{u\in L^{p(\cdot)}(\Omega): D^{\alpha}u\in L^{p(\cdot)}(\Omega),|\alpha| \leq L\},$$ where $D^{\alpha}u=\frac{\partial^{|\alpha|}}{\partial^{\alpha_1}x_1\dots \partial^{\alpha_n}x_n}$ with $\alpha=(\alpha_1,\alpha_2,\dots ,\alpha_{N})$ is a multi-index and $|\alpha|=\Sigma_{i=1}^{N}\alpha_{i}$. The space $W^{L,p(\cdot)}(\Omega)$ with the norm $$\|u\|_{W^{L,p(\cdot)}}(\Omega)=\sum_{|\alpha|\leq L}\|D^{\alpha}u\|_{p(\cdot)},$$ is a separable reflexive Banach space \cite{Die1}. $W^{L,p(\cdot)}_0(\Omega)$ denotes the closure in $W^{L,p(\cdot)}(\Omega)$ of the set of functions in $W^{L,p(\cdot)}(\Omega)$ with compact support. For every $u\in W^{L,p(\cdot)}_0(\Omega)$ the Poincar\'{e} inequality holds, where $C_p>0$ is a constant $$\|u\|_{L^{p(\cdot)}(\Omega)} \leq C_p\|\nabla u\|_{L^{p(\cdot)}(\Omega)}.$$ (see \cite{Fan2}). Hence, an equivalent norm for the space $W^{L,p(\cdot)}_0(\Omega)$ is given by $$\|u\|_{W^{L,p(\cdot)}_0(\Omega)}=\sum_{|\alpha|= L}\|D^{\alpha}u\|_{p(\cdot)}.$$ Given $p(x)$, let $p^{\ast}_{L}$ denote the critical variable exponent related to $p$, defined for all $x \in\bar{\Omega}$ by the pointwise relation \begin{equation}\label{a3} p^{\ast}_{L}(x)=\begin{cases} \frac{Np(x)}{N-L p(x)} & L p(x)< N, \\ +\infty & L p(x)\geq N, \end{cases} \end{equation} is the critical exponent related to $p$. Let \begin{equation} \label{z5} \mathcal{K}=\sup_{u\in X \backslash \{0\}} \frac{\max_{x\in\bar{\Omega}}|u(x)|^{p}}{\|u\|^{p}},\quad \mathcal{M}=\inf_{u\in X \backslash \{0\}} \frac{\min_{x\in\bar{\Omega}}|u(x)|^{p}}{\|u\|^{p}}\,. \end{equation} Since $p>N$, $X$ are compactly embedded in $C^{0}(\bar{\Omega})$, it follows that $\mathcal{K},\mathcal{M}<\infty$. \begin{proposition} \label{prop1} For $\Phi(u)=\int_{\Omega}[|\nabla u|^{p(x)}+a(x)| u(x)|^{p(x)}]dx$, and $u,u_n\in X$, we have \begin{itemize} \item[(i)] $\|u\|<(=,>)1 \Leftrightarrow \Phi(u)<(=,>)1$, \item[(ii)] $\|u\|\leq 1 \Rightarrow \|u\|^{p^{+}}\leq\Phi(u)\leq \|u\|^{p^{-}}$, \item[(iii)] $\|u\|\geq 1 \Rightarrow \|u\|^{p^{-}}\leq\Phi(u)\leq \|u\|^{p^{+}}$, \item[(iv)] $\|u_n\|\to 0 \Leftrightarrow \Phi(u_n) \to 0$, \item[(v)] $\|u_n\|\to \infty \Leftrightarrow \Phi(u_n)\to \infty$. \end{itemize} \end{proposition} The proof of the above proposition is similar to that in \cite{Fan}. \begin{proposition}[\cite{Fan,Kov}] \label{prop2} For $p,q \in C_{+}(\overline{\Omega})$ in which $q(x) \leq p^{\ast}_{L}(x)$ for all $x \in \overline{\Omega}$, there is a continuous embedding $$W^{L,p(\cdot)}(\Omega)\hookrightarrow L^{q(\cdot)}(\Omega).$$ If we replace $\leq$ with $<$, the embedding is compact. \end{proposition} \begin{remark}\label{rmk1} \rm % s1 (i) By the proposition \ref{prop2} there is a continuous and compact embedding of $W^{1,p(\cdot)}_0(\Omega)$ into $L^{q(\cdot)}$ where $q(x) N$, we deduce that $W^{1,p^{-}}_0(\Omega)$ is compactly embedded in $C^{0}(\bar{\Omega})$, So, there exists a constant $c > 0$ such that \begin{equation} \label{s1} \|u\|_{\infty}\leq c \|u\|,\quad \forall u\in X, \end{equation} where $\|u\|_{\infty}:=\sup_{x\in\bar{\Omega}}|u(x)|$. (ii) Denote $$\|u\|=\inf\{\lambda>0:\int_{\Omega}[|\frac{\nabla u}{\lambda}|^{p(x)} +a(x) |\frac{ u}{\lambda}|^{p(x)} ]dx\leq 1 \},$$ which is a norm on $W^{1,p(\cdot)}_0(\Omega)$. \end{remark} Let $\eta : \partial \Omega \to \mathbb{R}$ be a measurable. Define the weighted variable exponent Lebesgue space by $$L^{p(x)}_{\eta(x)}(\partial\Omega) =\{u:\partial\Omega \to \mathbb{R}\textrm{ is measurable and } \int_{\partial\Omega} |\eta(x)||u|^{p(x)}d\sigma<\infty\},$$ with the norm $$|u|_{(p(x),\eta(x))}=\inf\{ \tau>0;\int_{\partial\Omega}|\eta(x)|\, |\frac{u}{\tau}|^{p(x)}d\sigma\leq1\},$$ where $d\sigma$ is the measure on the boundary. \begin{lemma}[\cite{Den}] \label{lem1} Let $\rho(x)=\int_{\partial\Omega} |\eta(x)||u|^{p(x)}d\sigma$ for $u\in L^{p(x)}_{\eta(x)}(\partial\Omega)$ we have \begin{gather*} |u|_{(p(x),\eta(x))}\geq 1\Rightarrow |u|_{(p(x),\eta(x))}^{p^{-}}\leq\rho(u)\leq |u|_{(p(x),\eta(x))}^{p^{+}},\\ |u|_{(p(x),\eta(x))}\leq 1\Rightarrow |u|_{(p(x),\eta(x))}^{p^{+}} \leq\rho(u)\leq |u|_{(p(x),\eta(x))}^{p^{-}}. \end{gather*} \end{lemma} For $A \subseteq \bar{\Omega}$ denote by $\inf_{x \in A} p(x)=p^{-},\:\sup_{x \in A} p(x)=p^{+}$. Define \begin{gather}\label{a3b} p^{\partial}(x)=(p(x))^{\partial} :=\begin{cases} \frac{(N-1)p(x)}{N- p(x)} \quad p(x)< N, \\ +\infty \quad p(x)\geq N, \end{cases} \\ p^{\partial}(x)_{r(x)}:=\frac{r(x)-1}{r(x)} p^{\partial}(x), \nonumber \end{gather} where $x\in\partial\Omega,r\in C(\partial\Omega,\mathbb{R})$ and $r(x)>1$. \begin{proposition}[\cite{Fan5,Kov}] \label{prop3} If $q \in C_{+}(\overline{\Omega})$ and $q(x) 0$ depending on $U$ such that $|h(y)-h(z)|\leq K \|y-z\|$ for all $y, z \in U$. For a locally Lipschitz function $h : X \to \mathbb{R}$ is defined by the generalized directional derivative of $h$ at $u \in X$ in the direction $\gamma \in X$ by $$h^{0}(u;\gamma)=\limsup_{w\to u,t\to 0^{+}}\frac{h(w+t\gamma)-h(w)}{t}.$$ The generalized gradient of $h$ at $u \in X$ is defined by $$\partial h(u)=\{x^{\star}\in X^{\star}: \langle x^{\star},\gamma\rangle _{X}\leq h^{0}(u;\gamma),\;\forall \gamma\in X\},$$ which is non-empty, convex and $w^{\star}-$compact subset of $X^{\star}$, where $<\cdot,\cdot>_{X}$ is the duality pairing between $X^{\star}$ and $X$. \begin{proposition}[\cite{Clar}] \label{prop4} Let $h,g:X\to\mathbb{R}$ be locally Lipschitz functions. Then: \begin{itemize} \item[(i)] $h^{0}(u;\cdot)$ is subadditive, positively homogeneous. \item[(ii)] $(-h)^{0}(u;v)=h^{0}(u;-v)$ for all $u,v\in X$. \item[(iii)] $h^{0}(u;v)=\max\{<\xi,v>:\xi\in\partial h(u)\}$ for all $u,v\in X$. \item[(iv)] $(h+g)^{0}(u;v)\leq h^{0}(u;v)+g^{0}(u;v)$ for all $u,v\in X$. \end{itemize} \end{proposition} \begin{definition}[\cite{pan2}] \label{def1} \rm Let $X$ be a Banach space, $\mathcal{I}:X\to (-\infty,+\infty]$ is called a Motreanu-Panagiotopoulos-type functional, if $\mathcal{I}=h+\chi$, where $h:X\to \mathbb{R}$ is locally Lipschitz and $\chi:X\to (-\infty,+\infty]$ is convex, proper and lower semicontinuous. \end{definition} \noindent \begin{definition}[\cite{Ian}] \label{x1} \rm An element $u\in X$ is said to be a critical point of $\mathcal{I}=h+\chi$ if $$h^{0}(u;v-u)+\chi(v)-\chi(u)\geq 0,\quad \forall v\in X.$$ \end{definition} Let $X$ is a reflexive real Banach space, $\phi: X \to \mathbb{R}$ is a sequentially weakly lower semicontinuous and coercive, $\Upsilon: X \to \mathbb{R}$ is a sequentially weakly upper semicontinuous, $\lambda$ is a positive real parameter, $\chi:X\to (-\infty,+\infty]$ is a convex, proper, lower semicontinuous functional and $D(\chi)$ is the effective domain of $\chi$. Assuming also that $\phi$ and $\Upsilon$ are locally Lipschitz continuous functionals. Set $\mathcal{E}:=\Upsilon -\chi, \quad \mathcal{L}_{\lambda}:=\phi-\lambda\mathcal{E} =(\phi-\lambda\Upsilon)+\lambda\chi.$ We assume that $$\phi^{-1}(]-\infty,r[) \cap D(\chi)\neq \emptyset,\quad \forall r>\inf_{X}\phi,$$ and define for every $r>\inf_{X}\phi$, $$\varphi(r)=\inf_{u\in \phi^{-1}(]-\infty,r[)} \frac{\Big(\sup_{v\in \phi^{-1}(]-\infty,r[)} \mathcal{E}(v)\Big)-\mathcal{E}(u)}{r-\phi(u)}$$ and $$\gamma:=\liminf_{r\to+\infty}\varphi(r),\quad \delta:=\liminf_{r\to (\inf_{X} \phi)^{+}}\varphi(r).$$ We recall the following nonsmooth version of a critical point result. \begin{theorem}[\cite{Mara}]\label{thm1} Under the above assumptions on $X$, $\phi$ and $\mathcal{E}$, we have \begin{itemize} \item[(a)] For every $r > \inf_{X} \phi$, and every $\lambda\in (0,\frac{1}{\varphi(r)})$, the restriction of the functional $$\mathcal{L}_{\lambda}=\phi-\lambda\mathcal{E}$$ to $\phi^{-1}(-\infty,r)$ admits a global minimum, which is a critical point (local minimum) of $\mathcal{L}_{\lambda}$ in $X$. \item[(b)] If $\gamma < +\infty$, then for each $\lambda\in (0,1/\gamma)$, the following alternative holds: either (b1) $\mathcal{L}_{\lambda}$ possesses a global minimum, or (b2) there is a sequence $\{u_n\}$ of critical points (local minima) of $\mathcal{L}_{\lambda}$ such that $$\lim_{n\to+\infty} \phi(u_n)=+\infty.$$ \item[(c)] If $\delta < +\infty$, then for each $\lambda\in (0,\frac{1}{\delta})$, the following alternative holds: either (c1) there is a global minimum of $\phi$ which is a local minimum of $\mathcal{L}_{\lambda}$, or (c2) there is a sequence $\{u_n\}$ of pairwise distinct critical points (local minima) of $\mathcal{L}_{\lambda}$ that converges weakly to a global minimum of $\phi$. \end{itemize} \end{theorem} Consider $\phi,\mathcal{F},\mathcal{G}:X\to \mathbb{R}$, as follows \begin{gather*} \phi(u)=\int_{\Omega}\frac{1}{p(x)}[|\nabla u|^{p(x)}+a(x)|u|^{p(x)}]dx, \quad u\in W^{1,p(\cdot)}_0(\Omega),\\ \mathcal{F}(u)=\int_{\partial\Omega}F(u(x))d\sigma,\quad u\in W^{1,p(\cdot)}_0(\Omega),\\ \mathcal{G}(u)=\int_{\partial\Omega}G(u(x))d\sigma,\quad u\in W^{1,p(\cdot)}_0(\Omega). \end{gather*} The next lemma characterizes some properties of $\phi$ \cite{Abd}. \begin{lemma}\label{lem2} Let $\phi(u)=\int_{\Omega}\frac{1}{p(x)}[|\nabla u|^{p(x)}+a(x)|u|^{p(x)}]dx.$ Then \begin{itemize} \item[(i)] $\phi : X \to \mathbb{R}$ is sequentially weakly lower semicontinuous; \item[(ii)] $\phi'$ is of $(S_{+})$ type; \item[(iii)] $\phi'$ is a homeomorphism. \end{itemize} \end{lemma} \begin{proposition}[\cite{Kri}] \label{prop5} Let $F,G:\mathbb{R}\to\mathbb{R}$ be locally Lipschitz functions. Then $\mathcal{F}$ and $\mathcal{G}$ are well-defined and \begin{gather*} \mathcal{F}^{0}(u;v)\leq\int _{\partial\Omega}F^{0}(u(x);v(x))d\sigma, \quad \forall u,v\in W^{1,p(\cdot)}_0(\Omega),\\ \mathcal{G}^{0}(u;v)\leq\int _{\partial\Omega}G^{0}(u(x);v(x))d\sigma,\quad \forall u,v\in W^{1,p(\cdot)}_0(\Omega). \end{gather*} \end{proposition} \section{Main results} Let $f:\mathbb{R}\to\mathbb{R}$ be a locally essentially bounded function whose potential $F(t)=\int_0^{t}f(\omega)d\omega$ for all $t\in\mathbb{R}$. Set $\alpha:=\liminf_{\omega\to+\infty} \frac{\max_{|t|\leq \omega}F(t)}{|\omega|^{p^{-}}},\quad \beta:=\limsup_{\omega\to+\infty}\frac{F(\omega)}{|\omega|^{p^{+}}}.$ \begin{theorem}\label{thm2} Let $\theta,\vartheta \in L^{1}(\partial \Omega)$ be non-negative and non-zero identically zero functions. Assume that \begin{equation} \label{ei} \alpha<\frac{p^{-}\mathcal{M}\theta^{\ast}\beta}{p^{+}\mathcal{K}} \end{equation} for each ${\lambda}\in (\lambda_1,\lambda_2)$, where $$\lambda_1=\frac{1}{p^{-} \mathcal{M}\theta^{\ast}\beta},\quad \lambda_2=\frac{1}{p^{+}\mathcal{K}\alpha},$$ and $\theta^{\ast}=\int_{\partial\Omega}\theta(x)d\sigma$. Also assume that for each locally essentially bounded function $g:\mathbb{R}\to\mathbb{R}$ with potential $G(t)=\int_0^{t}g(\omega)d\omega$, for all $t\in\mathbb{R}$, satisfies \begin{equation}\label{h1} G_{\infty}=\limsup_{\omega\to+\infty} \frac{\max_{|t|\leq\omega}G(t)}{|\omega|^{p^{-}}}<+\infty, \end{equation} for every $\mu \in [0, \mu_{G,\lambda})$, where $$\mu_{G,\lambda}=\frac{1}{p^{+}\mathcal{K}G_{\infty}}(1-p^{+} \mathcal{K}\lambda \alpha).$$ Then \eqref{e1} has a sequence of weak solutions for every $\mu \in [0, \mu_{G,\lambda})$ in $X$ such that $$\int_{\Omega}\frac{1}{p(x)}[|\nabla u_n|^{p(x)}+a(x)|u_n|^{p(x)}]dx\to+\infty.$$ \end{theorem} \begin{proof} Our strategy is to apply Theorem \ref{thm1} (b). \smallskip \noindent\textbf{Case 1.} Assume that $\|u\|\geq 1$. Let $\bar{\lambda}\in (\lambda_1,\lambda_2)$ and $G$ satisfy our assumptions. Since $\bar{\lambda}<\lambda_2$, it follows that $\mu_{G,\bar{\lambda}} =\frac{1}{p^{+}\mathcal{K}G_{\infty}} (1-p^{+} \mathcal{K}\bar{\lambda}\alpha).$ Fix $\bar{\mu}\in (0,\mu_{G,\bar{\lambda}})$ and define the functionals $\phi,\mathcal{E}:X\to\mathbb{R}$ for each $u \in X$ as follows: \begin{equation} \label{a3c} \begin{gathered} \phi(u)=\int_{\Omega}\frac{1}{p(x)}[|\nabla u|^{p(x)}+a(x)|u|^{p(x)}]dx, \\ \Upsilon(u)=\int_{\partial\Omega}\theta(x)[F(u(x))]d\sigma +\frac{\bar{\mu}}{\bar{\lambda}}\int_{\partial\Omega}\vartheta(x)[G(u(x))]d\sigma, \\ \chi (u)=\begin{cases} 0 & u\in\mathcal{B}, \\ +\infty & u \notin \mathcal{B}, \end{cases}\\ \mathcal{E}(u)=\Upsilon(u)-\chi(u)\,. \end{gathered} \end{equation} Then define the functional $$\mathcal{L}_{\bar{\lambda}}(u):=\phi(u)-\bar{\lambda}\mathcal{E}(u)$$ whose critical points are the weak solutions of \eqref{e1}. To apply Lemma \ref{lem2}, we assume that $\phi$ satisfies the regularity assumptions of Theorem \ref{thm1}. By standard argument, $\Upsilon$ is sequentially weakly continuous. First, we claim that $\bar{\lambda}<1/\gamma$. Note that $\phi(0) = \mathcal{E}(0) = 0$, then for every $n$ large enough, one has \begin{align*} \varphi(r) &=\inf_{u\in \phi^{-1}(]-\infty,r[)} \frac{\big(\sup_{v\in \phi^{-1}(]-\infty,r[)}\mathcal{E}(v)\big) -\mathcal{E}(u)}{r-\phi(u)}\\ &\leq \frac{\sup_{ v\in \phi^{-1}(]-\infty,r[)}\mathcal{E}(v)}{r}. \end{align*} Coercivity of $\phi$ implies that $\inf_{X}\phi=\phi(0)=0$. Since $\mathcal{B}$ contains constant functions, $0\in\mathcal{B}=D(\chi)$, thus $$0\in \phi^{-1}(]-\infty,r[) \cap D(\chi),\quad \forall r>\inf_{X}\phi.$$ For $v \in X$ with $\phi(v) < r$ and in view of $\eqref{z5}$, \begin{equation}\label{h5} \begin{aligned} \phi^{-1}(]-\infty,r[):&=\{v\in X :\phi(v)< r\} =\{v\in X :\frac{1}{p^{+}}\|v\|^{p^{-}}0$such that$\tau_n\to+\infty$and \begin{equation}\label{a2} \frac{1}{\bar{\lambda}}<\eta<\lim_{n\to +\infty}\frac{\mathcal{M} p^{-}\theta^{\ast}F(\tau_n)}{|\tau_n|^{p^{+}}}, \end{equation} for every$n \in \mathbb{N}$large enough. Let$\xi_n(x)=\tau_n$be a sequence in$X$for all$n \in \mathbb{N}$,$x\in\bar{\Omega}$. Fix$n \in \mathbb{N}$, by proposition \ref{prop1}, \begin{equation}\label{a3d} \phi(\xi_n)=\int_{\Omega}\frac{1}{p(x)}[|\nabla \xi_n|^{p(x)}+a(x)|\xi_n|^{p(x)}]dx \leq \frac{1}{p^{-}}\|\tau_n\|^{p^{+}} \leq \frac{1}{\mathcal{M}p^{-}}|\tau_n|^{p^{+}}. \end{equation} Since$G$is non-negative and from the definition of$\mathcal{E}\begin{equation}\label{a4} \begin{aligned} \mathcal{E}(\xi_n) &= {\int_{\partial\Omega}[\theta(x)F(\xi_n) +\frac{\bar{\mu}}{\bar{\lambda}}\vartheta(x)G(\xi_n)]d\sigma}-\chi(\xi_n)\\ &\geq \int_{\partial\Omega}\theta(x)F(\xi_n)d\sigma=\theta^{\ast}F(\tau_n). \end{aligned} \end{equation} According to \eqref{a2}, \eqref{a3} and \eqref{a4}, $$L_{\lambda}(\xi_n) \leq \frac{1}{p^{-}}\|\tau_n\|^{p^{+}}-\bar{\lambda} \int_{\partial\Omega}\theta(x)F(\tau_n)d\sigma <\frac{1}{\mathcal{M}p^{-}}|\tau_n|^{p^{+}} -\frac{1}{\mathcal{M}p^{-}}{\bar{\lambda}|\tau_n|^{p^{+}}\eta},$$ for every enough largen \in \mathbb{N}$. Since$\bar{\lambda}\eta>1$and$\lim_{n\to+\infty}\tau_n=+\infty$, it results that $$\lim_{n\to+\infty}\mathcal{L}_{\bar{\lambda}}(\xi_n)=-\infty.$$ Hence, the functional$\mathcal{L}_{\bar{\lambda}}$is unbounded from below, and it follows that$\mathcal{L}_{\bar{\lambda}}$has no global minimum. Therefore, applying \ref{a2} we deduce that there is a sequence${u_n} \in X$of critical points of$\mathcal{L}_{\bar{\lambda}}$such that $$\int_{\Omega}\frac{1}{p(x)}[|\Delta u_n|^{p(x)}+a(x)|u_n|^{p(x)}]dx\to+\infty.$$ \noindent\textbf{Case 2.} If$\|u\|\leq1$the proof is similar to the first case and the proof of theorem is complete. \end{proof} \begin{lemma}\label{lem3} Every critical point of the functional$\mathcal{L}_{\lambda}$is a solution of \eqref{e1}. \end{lemma} \begin{proof} By definition \ref{def1},$\mathcal{L}_{\lambda}=(\phi-\lambda\Upsilon)+\lambda\chi$is a Motreanu-Panagiotopoulos type functional. Let$\{u_n\}\subset X$be a critical sequence of$\mathcal{L}_{\lambda}=\phi-\lambda \mathcal{F}-\mu\mathcal{G}+\lambda\chi$then$u_n \in \mathcal{B}$, definition$\ref{x1}and proposition \ref{prop4} imply that $$(\phi-\lambda\Upsilon)^{0}(u_n;v-u_n)\geq 0,\quad \forall v\in \mathcal{B}.$$ Using proposition \ref{prop5}, \begin{equation}\label{t2b} \begin{aligned} &\int_{\Omega} |\nabla u_n|^{p(x)-2}\nabla u_n \nabla (v-u_n) dx +\int_{\Omega}a(x)|u_n|^{p(x)-2} u_n(v-u_n) dx \\ &-\lambda\int_{\partial\Omega}\theta(x)F^{0}(u_n;v-u_n)d\sigma -\mu\int_{\partial\Omega}\vartheta(x)G^{0}(u_n;v-u_n)d\sigma\geq 0. \end{aligned} \end{equation} for everyv\in \mathcal{B}$. This completes the proof. \end{proof} Now, we give a concrete application of Theorem \ref{thm2}. \begin{theorem} \label{thm3} Let$f:\mathbb{R}\to \mathbb{R}$be a non-negative, continuous function and set$F(\omega)=\int_0^{\omega}f(t)dt$for$\omega\in\mathbb{R}$. Assume that \begin{equation}\label{m2} \liminf_{\omega\to+\infty}\frac{F(\omega)}{\omega} < \frac{\mathcal{M}(\theta(1)+\theta(0))}{2\mathcal{K}} \limsup_{\omega\to+\infty}\frac{F(\omega)}{\omega^{2}}. \end{equation} Then, for each $\lambda\in\big]\frac{1}{\mu (\theta(1)+\theta(0)) \limsup_{\omega\to+\infty}\frac{F(\omega)}{\omega^{2}}}, \frac{1}{2\mathcal{K}\liminf_{\omega\to+\infty}\frac{F(\omega)}{\omega}}\big[\,,$ for each non-negative, continuous function$g:\mathbb{R}\to \mathbb{R}$, whose potential$G(\omega)=\int_0^{\omega}g(t)dt$satisfies $$\limsup_{\omega\to+\infty}\frac{G(\omega)}{\omega}<+\infty$$ and for every$ \mu\in [0,\mu_{G,\lambda}[$, where $$\mu_{G,\lambda}:=\frac{1}{2\mathcal{K}G_{\infty}}\big(1-2\mathcal{K} \lambda \liminf_{\xi\to+\infty}\frac{F(\omega)}{\omega}\big),$$ there is a sequence of pairwise distinct functions$\{u_n\}\subset W^{1,2-x}_0]0,1[$such that for all$n\in \mathbb{N}one has \begin{equation} \label{m3} \begin{gathered} -(|u'(x)|^{-x}u'(x))^{'}+|u(x)|^{-x}u(x)=0 \quad x\in]0,1[,\\ |u'_n(1)|^{-1}u'_n(1)=\bar{\lambda}\theta(1)f(u_n(1))+\bar{\mu}\vartheta(1)g(u_n(1)),\\ |u'_n(0)|^{-1}u'_n(0)=\bar{\lambda}\theta(0)f(u_n(0))+\bar{\mu}\vartheta(0)g(u_n(0)). \end{gathered} \end{equation} \end{theorem} \begin{proof} The first step is the inequality \begin{align*} &\int_0^{1}\theta(x)[F(u(x))]+\vartheta(x)[G(u(x))]d\sigma \\ &\leq (\theta(1)+\theta(0))\max_{|\omega|\leq\omega_n}F(\omega) +(\vartheta(1)+\vartheta(0))\max_{|\omega|\leq\omega_n}G(\omega). \end{align*} It results that $$\gamma\leq\liminf_{n\to+\infty}\varphi(r_n) \leq \mathcal{K}p^{+}\alpha(\theta(1)+\theta(0)) +\mathcal{K}p^{+}(\vartheta(1)+\vartheta(0))\frac{\bar{\mu}}{\bar{\lambda}} G_{\infty})<+\infty.$$ The second step is the inequality $$\int_0^{1}\vartheta(x)[G(\xi_n(x))]d\sigma =(\vartheta(1)+\vartheta(0))G(\tau_n) \geq (\vartheta(1)+\vartheta(0))\liminf_{\omega\to+\infty}G(\omega)\geq0,$$ which implies that\lim_{n\to+\infty}\mathcal{L}_{\bar{\lambda}}(\xi_n)=-\infty. The last one is \begin{align*} &\Big[\int_{\partial\Omega}\theta(x)F(u_n(x);v(x)-u_n(x))d\sigma +\int_{\partial\Omega}\vartheta(x)G(u_n(x);v(x)-u_n(x))d\sigma\Big]^{\circ}\\ &\leq \Big[\int_{\partial\Omega}\theta(x)F(u_n(x);v(x)-u_n(x))d\sigma\Big]^{\circ} +\Big[\int_{\partial\Omega}\vartheta(x)G(u_n(x);v(x)-u_n(x))d\sigma\Big]^{\circ} \\ &\leq \Big[\theta(1)F(u_n(1);v(1)-u_n(1))+\theta(0)F(u_n(0);v(0)-u_n(0))]^{\circ} \\ &\quad +\Big[\vartheta(1)G(u_n(1);v(1)-u_n(1))+\vartheta(0)G(u_n(0);v(0) -u_n(0))\Big]^{\circ} \\ &\leq \Big[\theta(1)F^{\circ}(u_n(1);v(1)-u_n(1)) +\theta(0)F^{\circ}(u_n(0);v(0)-u_n(0))\Big] \\ &\quad +\Big[\vartheta(1)G^{\circ}(u_n(1);v(1)-u_n(1)) +\vartheta(0)G^{\circ}(u_n(0);v(0)-u_n(0))\Big]. \end{align*} ChoosingX=W^{1,2-x}(]0,1[)$,$\Omega=]0,1[$,$p(x)=2-x$and$a(x)=1, then the conditions of Theorem \ref{thm2} hold. Hence, \begin{align*} &\int_0^{1}[|u_n'(x)|^{-x}u'_n(x)(v'-u'_n)+|u_n(x)|^{-x}u_n(x)(v-u_n)]dx\\ &-\bar{\lambda}[\theta(1)f(u_n(1))v(1)+\theta(0)f(u_n(0))v(0)]\\ &-\bar{\mu}[\vartheta(1)g(u_n(1))v(1)+\vartheta(0)g(u_n(0))v(0)]\geq 0. \end{align*} There exists an unbounded sequence\{u_n\}\subset W^{1,2-x}(]0,1[)such that \begin{align*} &\int_0^{1}[|u_n'(x)|^{-x}u'_n(x)v'(x)+|u_n(x)|^{-x}u_n(x)v(x)]dx\\ &-\Big(\bar{\lambda}\theta(1)f(u_n(1))+\bar{\mu}\vartheta(1)g(u_n(1))\\ &+\bar{\lambda}\theta(0)f(u_n(0))+\bar{\mu}\vartheta(0)g(u_n(0))\Big)\geq 0. \end{align*} Therefore\{u_n\}$is the unique solution of the problem \eqref{m3}. \end{proof} \subsection*{Acknowledgement} The authors are grateful to the anonymous referee for the careful reading and helpful comments. \begin{thebibliography}{00} \bibitem{Afrou} G. Afrouzi, M. Mirzapour, V. R\u{a}dulescu; \emph{Qualitative properties of anisotropic elliptic Schr\"{o}dinger equations}, Adv. 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