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

TL;DR

This paper proves the existence of infinitely many small and large solutions for a variable exponent p(x)-Laplacian equation with nonlinearities, using variational methods like Clark's theorem and the symmetric mountain pass lemma.

Contribution

It establishes the existence of infinitely many solutions for a class of p(x)-Laplacian equations, extending previous results to variable exponent settings.

Findings

Infinitely many small solutions exist.

Infinitely many large solutions exist.

Solutions are obtained via variational methods.

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.