Infinitely many low- and high-energy solutions for a class of elliptic equations with variable exponent

This paper is concerned with the $p(x)$-Laplacian equation of the form$$left{begin{array}{ll}-Delta_{p(x)} u=Q(x)|u|^{r(x)-2}u, &mbox{in} Omega,u=0, &mbox{on} partial Omega,end{array}right. eqno{0.1}$$where $OmegasubsetR^N$ is a smooth bounded domain, $1p^+$ and $Q: overline{Omega}toR$ is a nonnegative continuous function. We prove that (0.1) has infinitely many small solutions and infinitely many large solutions by using the Clark''s theorem and the symmetric mountain pass lemma.