\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 97, 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 (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/97\hfil Existence of solutions for an elliptic equation] {Existence of solutions for an elliptic equation involving the $p(x)$-Laplace operator} \author[M.-M. Boureanu\hfil EJDE-2006/97\hfilneg] {Maria-Magdalena Boureanu} \address{Maria-Magdalena Boureanu \hfill\break Department of Mathematics, University of Craiova, 200585 Craiova, Romania} \email{mmboureanu@yahoo.com} \date{} \thanks{Submitted June 6, 2006. Published August 22, 2006.} \subjclass[2000]{35D05, 35J60, 35J70, 58E05, 76A02} \keywords{$p(x)$-Laplace operator; Sobolev space with variable exponent; \hfill\break\indent mountain pass theorem; weak solution} \begin{abstract} In this paper we study an elliptic equation involving the $p(x)$-Laplace operator on the whole space $\mathbb{R}^N$. For that equation we prove the existence of a nontrivial weak solution using as main argument the mountain pass theorem of Ambrosetti and Rabinowitz. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \section{Introduction} In this paper we discuss the existence of solutions for the problem $$\label{1} \begin{gathered} -\Delta_{p(x)} u(x)+b(x)|u(x)|^{p(x)-2}u=f(x,u), \quad \mbox{for } x\in\mathbb{R}^N \\ u\in W^{1,p(x)}_{0} (\mathbb{R}^N), \end{gathered}$$ where $N\geq 3$, $p:\mathbb{R}^N\to \mathbb{R}$ is Lipschitz continuous with 2\leq \mathop{\rm ess\,inf}_{\mathbb{R}^N}p(x)< \mathop{\rm ess\,sup}_{\mathbb{R}^N}p(x)1\;{\rm for}\;{\rm all}\;x\in\overline\Omega\}. $$For any h\in L_+^\infty(\Omega) we define$$ h^+=\mathop{\rm ess\,sup}_{x\in\Omega}h(x)\quad\mbox{and}\quad h^-= \mathop{\rm ess\,inf}_{x\in\Omega}h(x). For any p(x)\in L_+^\infty(\Omega), we define the variable exponent Lebesgue space \begin{align*} L^{p(x)}(\Omega)=\{&u: \mbox{is a measurable real-valued function such that}\\ &\int_\Omega|u(x)|^{p(x)}\,dx<\infty\}. \end{align*} We define a norm, the so-called {\it Luxemburg norm}, on this space by the formula |u|_{p(x)}=\inf\big\{\mu>0;\;\int_\Omega\big| \frac{u(x)}{\mu}\big|^{p(x)}\,dx\leq 1\big\}.$$Variable exponent Lebesgue spaces resemble classical Lebesgue spaces in many respects: they are Banach spaces \cite[Theorem 2.5]{KR}, the H\"older inequality holds \cite[Theorem 2.1]{KR}, they are reflexive if and only if 1 < p^-\leq p^+<\infty \cite[Corollary 2.7]{KR} and continuous functions are dense if p^+ <\infty \cite[Theorem 2.11]{KR}. The inclusion between Lebesgue spaces also generalizes naturally \cite[Theorem 2.8]{KR}: if 0 <|\Omega|<\infty and r_1, r_2 are variable exponents so that r_1(x) \leq r_2(x) almost everywhere in \Omega then there exists the continuous embedding L^{r_2(x)}(\Omega)\hookrightarrow L^{r_1(x)}(\Omega), whose norm does not exceed |\Omega|+1. We denote by L^{p'(x)}(\Omega) the conjugate space of L^{p(x)}(\Omega), where 1/p(x)+1/p'(x)=1. For any u\in L^{p(x)}(\Omega) and v\in L^{p'(x)}(\Omega) the H\"older type inequality $$\label{Hol} \big|\int_\Omega uv\,dx\big|\leq\big(\frac{1}{p^-}+ \frac{1}{{p'}^-}\big)|u|_{p(x)}|v|_{p'(x)}$$ holds true. An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the {\it modular} of the L^{p(x)}(\Omega) space, which is the mapping \rho_{p(x)}:L^{p(x)}(\Omega)\to \mathbb{R} defined by$$ \rho_{p(x)}(u)=\int_\Omega|u|^{p(x)}\,dx. If u\in L^{p(x)}(\Omega) and p^+<\infty then the following relations hold \begin{gather}\label{L4} |u|_{p(x)}>1\;\Rightarrow \; |u|_{p(x)}^{p^-}\leq\rho_{p(x)}(u) \leq|u|_{p(x)}^{p^+}; \\ \label{L5} |u|_{p(x)}<1\; \Rightarrow \; |u|_{p(x)}^{p^+}\leq \rho_{p(x)}(u)\leq|u|_{p(x)}^{p^-}; \\ \label{L6} |u_n-u|_{p(x)}\to 0\;\Leftrightarrow\;\rho_{p(x)} (u_n-u)\to 0. \end{gather} We also consider the weighted variable exponent Lebesgue spaces. Let b : \mathbb{R}^N\to \mathbb{R} be a measurable real function such that b(x)>0 a.e. x\in\Omega. We define \begin{align*} L_{b(x)}^{p(x)}(\Omega)=\{& u :u \mbox{ is a measurable real-valued function such that}\\ & \int_{\Omega} b(x)|u(x)|^{p(x)}dx < \infty\}. \end{align*} The space L_{b(x)}^{p(x)}(\Omega) endowed with the above norm is a Banach space which has similar properties with the usual variable exponent Lebesgue spaces. The modular of this space is \rho_{b(x);p(x)}:L_{b(x)}^{p(x)}(\Omega)\to \mathbb{R} defined by \rho_{b(x);p(x)}(u)=\int_{\Omega}b(x)|u|^{p(x)}dx. $$If u\in L_{b(x)}^{p(x)}(\Omega), then the following relations hold \begin{gather*} |u|_{({b(x)},{p(x)})}>1 \Rightarrow |u|_{({b(x)},{p(x)})}^{p^{-}} \leq \rho_{b(x);p(x)}(u)\leq |u|_{({b(x)},{p(x)})}^{p^{+}}, \\ |u|_{({b(x)},{p(x)})}<1 \Rightarrow |u|_{({b(x)},{p(x)})}^{p^{+}} \leq \rho_{b(x);p(x)}(u)\leq |u|_{({b(x)},{p(x)})}^{p^{-}}. \end{gather*} We define also the variable Sobolev space$$ W^{1,p(x)}(\Omega)=\{u\in L^{p(x)}(\Omega):\;|\nabla u|\in L^{p(x)} (\Omega) \}. $$On W^{1,p(x)}(\Omega) we may consider one of the following equivalent norms$$ \|u\|_{p(x)}=|u|_{p(x)}+|\nabla u|_{p(x)} $$or$$ |\|u\||=\inf\big\{\mu>0;\;\int_\Omega\big(\big| \frac{\nabla u(x)}{\mu}\big|^{p(x)}+\big| \frac{u(x)}{\mu}\big|^{p(x)}\big)\,dx\leq 1\big\}. $$We define also W_0^{1,p(x)}(\Omega) as the closure of C_0^\infty(\Omega) in W^{1,p(x)}(\Omega). Assuming p^->1 the spaces W^{1,p(x)}(\Omega) and W_0^{1,p(x)}(\Omega) are separable and reflexive Banach spaces. Set$$ I_{p(x)}(u)=\int_\Omega\big(|\nabla u|^{p(x)}+|u|^{p(x)}\big)\,dx. $$For all u\in W_0^{1,p(x)}(\Omega) the following relations hold \begin{gather}\label{M4} |\|u\||>1\;\Rightarrow\;|\|u\||^{p^-}\leq I_{p(x)}(u) \leq|\|u\||^{p^+}; \\ \label{M5} |\|u\||<1\;\Rightarrow\;|\|u\||^{p^+}\leq I_{p(x)}(u) \leq|\|u\||^{p^-}. \end{gather} Finally, we remember some embedding results regarding variable exponent Lebesgue-Sobolev spaces. For the continuous embedding between variable exponent Lebesgue-Sobolev spaces we refer to \cite[Theorem 1.1]{FSZ}: if p:\Omega\to \mathbb{R} is Lipschitz continuous and p^+N, then for any q(x)\in L_+^\infty(\Omega) with \mathop{\rm ess\,inf}_{{x\in\overline\Omega}} \big(\frac{Np(x)}{N-p(x)}-q(x)\big)>0 there is a compact embedding W^{1,p(x)}(\Omega)\hookrightarrow L^{q(x)}(\Omega). \section{Main result} In this paper we assume that b and f satisfy the hypotheses: \begin{itemize} \item[(B1)] b\in L^{\infty}_{\rm loc}(\mathbb{R}^N) and there exists b_{0}>0 such that b(x)\geq b_{0}, for any x\in \mathbb{R}^N;\\ \smallskip \item[(F1)] f\in C^{1}(\mathbb{R}^N\times\mathbb{R}), with f=f(x,z), f(x,0)=0 and \lim_{z\to 0}\frac{f_{z}(x,z)}{|z|^{p^{+}-2}}=0, for all x\in\mathbb{R}^N; \item[(F2)] p^+<\frac{Np^-}{N-p^-} and there exist a_1,a_2>0 and s\in(p^{+}-1,Np^{-}/(N-p^{-})-1) such that$$ |f_{z}(x,z)|\leq a_1|z|^{p^{+}-2}+a_2|z|^{s-1},\quad \forall\; x\in\mathbb{R}^N, \forall\; z\in\mathbb{R}; $$\item[(F3)] there exists \mu >p^+ such that$$ 0<\mu F(x,z):=\mu \int_{0}^{z}f(x,t)dt \leq z f(x,z),\quad \forall\; x\in\mathbb{R}^N,\forall\; z\in\mathbb{R}\setminus\{0\}. $$\end{itemize} Let E be the space defined as the completion of C_{0}^{\infty}(\mathbb{R}^N) with respect to the norm$$\|u\|_1=|\nabla u|_{p(x)}+|u|_{({b(x)},{p(x)})}.$$\begin{remark} \label{rmk1} Condition (B1) implies that E\subset W_0^{1,p(x)}(\mathbb{R}^N). \end{remark} A simple calculation shows that the above norm is equivalent to$$ \|u\|=\inf\big\{\mu>0;\;\int_\Omega\big(\big| \frac{\nabla u(x)}{\mu}\big|^{p(x)}+b(x)\big| \frac{u(x)}{\mu}\big|^{p(x)}\big)\,dx\leq 1\big\}. $$Set$$ J(u):=\int_{\mathbb{R}^N}\left(|\nabla u|^{p(x)}+b(x)|u|^{p(x)}\right)dx. $$Then, for all u \in E the following relations hold: $$\label{2} \begin{gathered} \|u\|>1 \Rightarrow \|u\|^{p^{-}}\leq J(u)\leq \|u\|^{p^{+}}, \\ \|u\|<1 \Rightarrow \|u\|^{p^{+}}\leq J(u)\leq \|u\|^{p^{-}}. \end{gathered}$$ We say that u\in E is a {\it weak solution} of \eqref{1} if$$ \int_{\mathbb{R}^N}\big(|\nabla u|^{p(x)-2}\nabla u \nabla v +b(x)|u|^{p(x)-2}u v \big)dx=\int_{\mathbb{R}^N}f(x,u)v dx, $$for any v\in C_0^\infty(\mathbb{R}^N). The main result of this paper is given by the following theorem. \begin{theorem}\label{t1} Assume conditions (B1) and (F1)-(F3) are fulfilled. Then problem \eqref{1} has a non-trivial weak solution. \end{theorem} We point out the fact that the result of Theorem \ref{t1} extends the results from \cite{R} and \cite{MR1} where similar equations are studied in the linear case. \section{Proof of Main Theorem} The energy functional corresponding to problem \eqref{1} is defined as I:E\to \mathbb{R},$$ I(u):=\int_{\mathbb{R}^N}\frac{1}{p(x)}\left(|\nabla u|^{p(x)}+b(x)|u|^{p(x)}\right)dx-\int_{\mathbb{R}^N}F(x,u)dx. $$Similar arguments as those used in \cite[Lemmas 3.1 and 3.2]{FH} assure that I\in C^1(E,\mathbb{R}) with$$ \langle I'(u),v\rangle=\int_{\mathbb{R}^N}\left(|\nabla u|^{p(x)-2}\nabla u \nabla v +b(x)|u|^{p(x)-2}u v \right)dx-\int_{\mathbb{R}^N}f(x,u)v dx, $$for any u, v\in E. Thus, we observe that the critical points of functional I are the weak solutions for equation \eqref{1}. Our idea is to prove Theorem \ref{t1} applying the mountain pass theorem (see e.g. \cite{AR}). With that end in view, we prove some auxiliary results which show that the functional I has a mountain pass geometry. \begin{lemma}\label{l1} If (B1) and (F1)-(F3) hold, then there exist \tau >0 and a>0 such that for all u\in E with \|u\|=\tau$$I(u)\geq a>0.$$\end{lemma} \begin{proof} Using (F1) and L'Hospital Theorem, we have$$ \lim_{z\to {0}}\frac{F(x,z)}{z^{p^{+}}} =\lim_{z\to {0}}\frac{f(x,z)}{p^{+}\cdot z^{p^{+}-1}} =\lim_{z\to {0}}\frac{f_{z}(x,z)}{p^{+}(p^{+}-1)\cdot z^{p^{+}-2}}=0, $$for all x\in \mathbb{R}^N. Thus, $$\label{3} \lim_{z\to {0}}\frac{F(x,z)}{z^{p^{+}}}=0.$$ Using (F2) we have$$ f_{z}(x,z)\leq|f_{z}(x,z)|\leq a_1|z|^{p^{+}-2}+a_2|z|^{s-1}. $$By integrating, we obtain$$ f(x,z)\leq \frac{a_1}{p^{+}-1}|z|^{p^{+}-1}+\frac{a_2}{s}|z|^{s}. $$We integrate again: \label{4} 00,\; \exists\; \delta_{1}>0 \mbox{ such that } \big|\frac{F(x,z)}{z^{p^{+}}}\big|<\varepsilon \mbox{ for all } z \mbox{ with } |z|<\delta_{1}; \\ \forall\; \varepsilon >0,\; \exists\; \delta_{2}>0 \mbox{ such that } \big|\frac{F(x,z)}{z^{Np^{-}/(N-p^{-})}}\big|<\varepsilon \mbox{ for all } z \mbox{ with } |z|>\delta_{2}. \end{gather*} Thus, for \varepsilon>0 there exist \delta_{1},\;\delta_{2}>0 such that$$ F(x,z)<\varepsilon\cdot|z|^{p^{+}},\;\;\; |z|<\delta_1 $$and$$ F(x,z)<\varepsilon\cdot|z|^{Np^{-}/(N-p^{-})},\quad |z|>\delta_2. $$Relation \eqref{4} implies that there exists a constant c>0 such that$$ F(x,z)\leq c\quad \mbox{for allzwith }|z|\in[\delta_1,\delta_2]. We conclude that for all \varepsilon>0 there exists c_{\varepsilon}>0 such that $$\label{6} F(x,z)\leq\varepsilon|z|^{p^{+}}+c_{\varepsilon}|z|^{Np^{-}/(N-p^{-})}.$$ Let us assume that \|u\|<1. Then, using relations \eqref{2} and \eqref{6}, we have \begin{align*} I(u)&\geq \frac{1}{p^{+}}J(u)-\int_{\mathbb{R}^N}F(x,u)dx\\ &\geq \frac{1}{p^{+}}\|u\|^{p{+}}-\int_{\mathbb{R}^N}F(x,u)dx\\ &\geq \frac{1}{p^{+}}\|u\|^{p{+}}-\epsilon\int_{\mathbb{R}^N}|u|^{p^{+}}dx -c_{\varepsilon}\int_{\mathbb{R}^N}|u|^{Np^{-}/(N-p^{-})}dx. \end{align*} For p(x)\leq q(x)\leq \frac{Np(x)}{N-p(x)} we have W^{1,p(x)}(\mathbb{R}^N)\hookrightarrow L^{q(x)}(\mathbb{R}^N) continuous, so E\hookrightarrow L^{q(x)}(\mathbb{R}^N) continuous, thus |u|_{L^{q(x)}}\leq c \|u\|_{E}. Choosing q(x)=p^{+} and then q(x)=\frac{Np^{-}}{N-p^{-}} we obtain \begin{gather*} |u|_{{p^{+}}}\leq c_{1} \|u\|\Leftrightarrow \Big(\int_{\mathbb{R}^N}|u|^{p^{+}}dx\Big)^\frac{1}{p^{+}}\leq c_{1} \|u\|; \\ |u|_{{Np^{-}/(N-p^{-})}}\leq c_{2} \|u\|\Leftrightarrow \Big(\int_{\mathbb{R}^N}|u|^{Np^{-}/(N-p^{-})}dx\Big)^\frac{N-p^{-}}{Np^{-}} \leq c_{2} \|u\|\,. \end{gather*} Therefore, \begin{align*} I(u)&\geq \frac{1}{p^{+}}\|u\|^{p{+}}-\varepsilon c_{1}\|u\|^{p{+}} -c_2\cdot c_{\varepsilon}\|u\|^{Np^{-}/(N-p^{-})}\\ &\geq \|u\|^{p{+}}\left[(\frac{1}{p^{+}}-\varepsilon c_{1}) -c_2\cdot c_{\varepsilon}\|u\|^{Np^{-}/(N-p^{-})-p^{+}}\right]\geq a>0, \end{align*} for some fixed \varepsilon\in(0,\frac{1}{c_{1}p^{+}}) and a,\|u\| sufficiently small. \end{proof} \begin{lemma}\label{l2} Assume conditions (B1), (F1)-(F3) hold. Then there exists e\in E with \|e\|>\tau (\tau given in Lemma \ref{l1}) such that I(e)<0. \end{lemma} \begin{proof} Denote h(t)=\frac{F(x,tz)}{t^\mu},\quad \forall\; t>0. $$Then using (F3) we get$$ h'(t)=\frac{1}{t^{\mu+1}}[tzf(x,tz)-\mu F(x,tz)]\geq 0,\quad \forall\;t>0. Thus, we deduce that for any t\geq 1, F(x,tz)\geq t^\mu F(x,z). Choosing u\in E with \|u\|>1 and \int_{\mathbb{R}^N}F(x,u)dx>0 fixed and t>1, we have \begin{align*} I(tu)&=\int_{\mathbb{R}^N}\frac{1}{p(x)}\left(|\nabla (tu)|^{p(x)}+b(x)|tu|^{p(x)}\right)dx-\int_{\mathbb{R}^N}F(x,tu)dx\\ &=\int_{\mathbb{R}^N}\frac{1}{p(x)}t^{p(x)}\left(|\nabla u|^{p(x)}+b(x)|u|^{p(x)}\right)dx-\int_{\mathbb{R}^N}F(x,tu)dx\\ &\leq \frac{t^{p^{+}}}{p^{-}}\int_{\mathbb{R}^N}\left(|\nabla u|^{p(x)}+b(x)|u|^{p(x)}\right)dx-\int_{\mathbb{R}^N}F(x,tu)dx\\ &\leq \frac{t^{p^{+}}}{p^{-}}\|u\|^{p^{+}}-t^\mu\int_{\mathbb{R}^N}F(x,u)dx. \end{align*} But \mu> p^{+}, therefore I(tu)\to -\infty when t approaches +\infty, which concludes our lemma. \end{proof} \begin{proof}[Proof of Theorem \ref{t1}] We set \Gamma:=\{\gamma\in C([0,1],E): \gamma(0)=0,\;\gamma(1)=e\}, $$where e\in E is determined by Lemma \ref{l2}, and$$ c:=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}I(\gamma(t)). $$According to Lemma \ref{l2} we know that \|e\|>\tau, so every path \gamma\in \Gamma intersects the sphere \|w\|=\tau. Then Lemma \ref{l1} implies $$\label{kk} c\geq\inf_{\|u\|=\tau}I(u)\geq a,$$ with the constant a>0 in Lemma \ref{l1}, thus c>0. By the mountain-pass theorem (see, e.g., \cite{AR}) we obtain a sequence (u_n)_n\subset E such that $$\label{7} I(u_n)\to c,\quad I'(u_n)\to 0.$$ We claim that (u_{n})_{n} is bounded in E. Arguing by contradiction and passing to a subsequence, we have \|u_{n}\|\to \infty. Using \eqref{7} it follows that for n large enough, we have $$\label{8} c+1+\|u_{n}\|\ \geq I(u_{n})-\frac{1}{\mu} \langle I'(u_{n}),u_n\rangle .$$ Since$$ I(u_{n})=\int_{\mathbb{R}^N}\frac{1}{p(x)}\left(|\nabla u_{n}|^{p(x)}+b(x)|u_{n}|^{p(x)}\right)dx-\int_{\mathbb{R}^N}F(x,u_{n})dx,  \langle I'(u_{n}),u_n\rangle= \int_{\mathbb{R}^N}\left(|\nabla u_{n}|^{p(x)}+b(x)|u_{n}|^{p(x)}\right)dx-\int_{\mathbb{R}^N}f(x,u_{n})u_{n}dx, $$using \eqref{8} we obtain$$ c+1+\|u_{n}\|\ \geq \big(\frac{1}{p^{+}}-\frac{1}{\mu}\big) J(u_{n})-\int_{\mathbb{R}^N}\left[F(x,u_{n})-\frac{1}{\mu}f(x,u_{n})u_{n}\right]dx. $$By (F3) we have$$ \int_{\mathbb{R}^N}\big[F(x,u_{n})-\frac{1}{\mu}f(x,u_{n})u_{n}\big]dx\leq 0 . $$The above inequalities combined with relation \eqref{2} yield$$ c+1+\|u_{n}\|\ \geq \big(\frac{1}{p^{+}}-\frac{1}{\mu}\big) J(u_{n})\geq \big(\frac{1}{p^{+}}-\frac{1}{\mu}\big) \|u_{n}\|^{p^{-}}. $$We obtain $$\label{9} c+1+\|u_{n}\|\ \geq \big(\frac{1}{p^{+}}-\frac{1}{\mu}\big) \|u_{n}\|^{p^{-}} .$$ Now dividing by \|u_{n}\| in \eqref{9} and passing to the limit as n\to \infty we obtain a contradiction. So, up to a subsequence, (u_{n})_{n} converges weakly in E to some u\in E. If \Omega is bounded then there exists a compact embedding E(\Omega)\hookrightarrow L^{ \frac{Np^{-}}{N-p^{-}}} (\Omega). Then (u_{n})_{n} converges strongly in L^{ \frac{Np^{-}}{N-p^{-}}} (\Omega), for all \Omega bounded domains in \mathbb{R}^N. If we prove that $$\label{10} \langle I'(u_{n}),\varphi\rangle\to \langle I'(u),\varphi\rangle, \quad \forall \varphi\in C_{0}^{\infty}(\mathbb{R}^N).$$ Then, by \eqref{7}, u is a weak solution of \eqref{1}, since C_0^\infty(\Omega) is dense in E. To do this, let \varphi\in C_{0}^{\infty}(\mathbb{R}^N) be fixed. We set \Omega=\mathop{\rm supp}(\varphi). To prove \eqref{10}, first we prove that$$ \lim_{n\to \infty}\int_{\mathbb{R}^N}f(x,u_{n})\varphi dx= \int_{\mathbb{R}^N}f(x,u)\varphi dx. A simple calculation implies \begin{align*} \big|\int_{\Omega}(f(x,u_{n})-f(x,u))\varphi(x) dx\big| &\leq \int_{\Omega}|f(x,u_{n})-f(x,u)|\cdot |\varphi(x) |dx\\ &\leq \|\varphi\|_{L^{\infty}(\Omega)} \int_{\Omega}|f(x,u_{n})-f(x,u)|dx\\ &= \|\varphi\|_{L^{\infty}(\Omega)} \int_{\Omega}\big|\frac{f(x,u_{n})-f(x,u)}{u_{n}-u}\big|\cdot |u_{n}-u|dx\\ &\leq \|\varphi\|_{L^{\infty}(\Omega)} \int_{\Omega}|f_{z}(x,v_{n})|\cdot |u_{n}-u|dx, \end{align*} where v_{n}\in [u_{n},u] (or [u,u_{n}]). Using (F2), we obtain \begin{align*} &\Big|\int_{\Omega}(f(x,u_{n})-f(x,u))\varphi(x) dx\Big| \\ &\leq \|\varphi\|_{L^{\infty}(\Omega)} \int_{\Omega}|a_1|v_{n}|^{p^{+}-2}+a_2|v_{n}|^{s-1}|\cdot|u_{n}-u|dx\\ &\leq \|\varphi\|_{L^{\infty}(\Omega)}\cdot \Big[a_{1}\int_{\Omega}|v_{n}|^{p^{+}-2}\cdot|u_{n}-u|dx +a_{2}\int_{\Omega}|v_{n}|^{s-1}\cdot|u_{n}-u|dx\Big]. \end{align*} We have \frac{1}{p^{+}-1}+\frac{p^{+}-2}{p^{+}-1}=1 and \frac{1}{s}+\frac{s-1}{s}=1. Using H\"{o}lder inequality, \begin{align*} &\Big|\int_{\Omega}(f(x,u_{n})-f(x,u))\varphi(x) dx\Big|\\ &\leq \|\varphi\|_{L^{\infty}(\Omega)} \cdot [a_{1}\|v_{n}\|^{p^{+}-2}_{L^{p^{+}-1}(\Omega)} \cdot \|u_{n}-u\|_{L^{p^{+}-1}(\Omega)} + a_{2}\|v_{n}\|^{s-1}_{L^{s} (\Omega)} \cdot \|u_{n}-u\|_{L^{s}(\Omega)}]. \end{align*} Taking into account that u_{n}\to u strongly in L^{i}(\Omega), for all i\in [p^{+}-1,\frac{Np^{-}}{N-p^{-}}] and remarking that for all x\in \Omega and for all n\geq 1 there exists \lambda_{n}(x)\in [0,1] such that v_{n}(x)=\lambda_{n}(x)u_{n}(x)+[1-\lambda_{n}(x)]u(x) we deduce \int_{\Omega}|v_{n}-u|^{s}dx=\int_{\Omega} |\lambda_{n}(x)|^{s}\cdot|u_{n}-u|^{s}dx \leq \int_{\Omega} |u_{n}-u|^{s}dx\to 0, \mbox{ as } n\to \infty. $$It results that$$ \int_{\Omega}|v_{n}|^{s}dx\to \int_{\Omega}|u|^{s}dx \mbox{ , as } n\to \infty. $$From the above considerations, we obtain$$ \Big|\int_{\Omega}(f(x,u_{n})-f(x,u))\varphi(x) dx\Big| \to 0 ,\mbox{ as } n\to \infty. $$Since C_0^\infty(\mathbb{R}^N) is dense in E, the above relation implies$$ \lim_{n\to \infty}\int_{\Omega}(f(x,u_{n})-f(x,u))(u_n-u) dx=0. $$Next, since (u_n)_n converges weakly to u in E, it follows that$$ \lim_{n\to \infty}\int_{\Omega}f(x,u)(u_n-u) dx=0. $$Thus, actually we find$$ \lim_{n\to \infty}\int_{\Omega}f(x,u_n)(u_n-u) dx=0. $$On the other hand, we have$$ \lim_{n\to \infty}\langle I'(u_n),u_n-u\rangle=0. $$Combining the last two relations we deduce that $$\label{16} \lim_{n\to \infty}\int_{\mathbb{R}^N}\left(|\nabla u_{n}|^{p(x)-2}\nabla u_{n} \nabla(u_n-u) +b(x)|u_{n}|^{p(x)-2}u_{n} (u_n-u) \right)dx=0.$$ Since relation \eqref{16} holds true and (u_n)_n converges weakly to u in E, by \cite[Lemma 3.1]{FH}, we deduce that (u_n)_n converges strongly to u in E. Then since I\in C^1(E,\mathbb{R}) we conclude $$\label{17} I'(u_n)\to I'(u) ,$$ as n\to \infty. Relations \eqref{7} and \eqref{17} show that I'(u)=0 and thus u is a weak solution for \eqref{1}. Moreover, by relation \eqref{7} it follows that I(u)>0 and thus, u is a nontrivial weak solution for \eqref{1}. The proof of Theorem \ref{t1} is complete. \end{proof} \begin{thebibliography}{99} \bibitem{AS} C. O. Alves and M. A. S. Souto, Existence of solutions for a class of problems in \mathbb{R}^N involving the p(x)-Laplacian, {\it Progress in Nonlinear Differential Equations and Their Applications} {\bf 66} (2005), 17-32. \bibitem{AR} A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory, {\it J. Funct. Anal.} {\bf 14} (1973), 349-381. \bibitem{D} L. Diening, {\it Theoretical and Numerical Results for Electrorheological Fluids}, Ph.D. thesis, University of Frieburg, Germany, 2002. \bibitem{FH} X. Fan and X. 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More exactly, condition (F2) must be replaced by \begin{itemize} \item[(F2)] p^+<\frac{Np^-}{N-p^-} and there exist s\in(p^{+}-1,Np^{-}/(N-p^{-})-1), \theta\in (s,Np^{-}/(N-p^{-})) and g_1\in L^{\infty}(\mathbb{R}^N)\cap L^{\theta /(\theta -p^{+}+1)}(\mathbb{R}^N), g_2\in L^{\infty}(\mathbb{R}^N)\cap L^{\theta /(\theta -s)}(\mathbb{R}^N), with g_{1}(x),g_{2}(x)\geq 0 such that$$ |f_{z}(x,z)|\leq g_{1}(x)|z|^{p^{+}-2}+g_{2}(x)|z|^{s-1},\quad \forall\, x\in\mathbb{R}^N,\; \forall\, z\in\mathbb{R}. $$\end{itemize} This new condition was inspired from the paper by Fan and Han \cite{FH}, and implies the old condition. \noindent\textbf{Remark.} Condition (F2) implies that there exist a_1,a_2>0 and s\in(p^{+}-1,Np^{-}/(N-p^{-})-1) such that$$ |f_{z}(x,z)|\leq a_1|z|^{p^{+}-2}+a_2|z|^{s-1},\quad \forall\, x\in\mathbb{R}^N,\; \forall\, z\in\mathbb{R}. $$After this correction, the proof of Theorem \ref{t1} will change as well. At the end of the proof, from If \Omega is bounded \dots'' on page 8, line 5, to  \lim_{n\to \infty}\int_{\Omega}f(x,u_n)(u_n-u) dx=0. '', page 9, line 8, will be replaced by: \begin{quote} Next, since (u_n)_n converges weakly to u in E, it follows that$$ \lim_{n\to \infty}\int_{\mathbb{R}^N}f(x,u)(u_n-u) dx=0. $$Using condition (F2) and \cite[Lemma 3.2]{FH}, we find$$ \lim_{n\to \infty}\int_{\mathbb{R}^N}f(x,u_n)(u_n-u) dx=0. $$Combining the above two relations we obtain$$ \lim_{n\to \infty}\int_{\mathbb{R}^N}(f(x,u_{n})-f(x,u))(u_n-u) dx=0.\$ \end{quote} End of corrigendum. \end{document}