Infinitely many solutions of a class of elliptic equations with variable exponent

Abstract

This paper is concerned with the $p(x)$-Laplacian equation of the form \begin{equation}\label{eq0.1} \left\{\begin{array}{ll} -\Delta_{p(x)} u=Q(x)|u|^{r(x)-2}u, &\mbox{in}\ \Omega,\\ u=0, &\mbox{on}\ \partial \Omega, \end{array}\right. \end{equation} where $\Omega\subset\R^N$ is a smooth bounded domain, $1<p^-=\min_{x\in\overline{\Omega}}p(x)\leq p(x)\leq\max_{x\in\overline{\Omega}}p(x)=p^+<N$, $1\leq r(x)<p^{*}(x)=\frac{Np(x)}{N-p(x)}$, $r^-=\min_{x\in \overline{\Omega}}r(x)<p^-$, $r^+=\max_{x\in\overline{\Omega}}r(x)>p^+$ and $Q: \overline{\Omega}\to\R$ is a nonnegative continuous function. We prove that \eqref{eq0.1} has infinitely many small solutions and infinitely many large solutions by using the Clark's theorem and the symmetric mountain pass lemma.