Infinitely many solutions of a class of elliptic equations with variable exponent
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 $Ω\subset\R^N$ is a smooth bounded domain, $1 p^+$ and $Q: \overlineΩ\to\R$ 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.
math.FA↗