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} -Δ_{p(x)} u=Q(x)|u|^{r(x)-2}u, &\mbox{in}\ Ω,\\ u=0, &\mbox{on}\ \partial Ω, \end{array}\right. \end{equation} where $\Omega \subset {\mathbb{R}}^N$ is a smooth bounded domain, $1<p^{-}={\min }_{x\in \overline{\Omega }}p(x)\le p(x)\le {\max }_{x\in \overline{\Omega }}p(x)=p^{+}<N$, $1\le r(x)<p^{\ast }(x)=\frac{Np(x)}{N-p(x)}$, $r^{-}={\min }_{x\in \overline{\Omega }}r(x)<p^{-}$, $r^{+}={\max }_{x\in \overline{\Omega }}r(x)>p^{+}$ and $Q:\overline{\Omega }\to \mathbb{R}$ 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.