Infinitely Many Solutions for a Class of Kirchhoff Problems Involving the $p(x)$-Laplacian Operator

Abstract

This article is devoted to studying a class of generalized $p(x)$-Laplacian Kirchhoff equations in the following form:

$$\begin{aligned} \, \begin{cases} -M\biggl(\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}\biggr)\operatorname{div} \biggl(|\nabla u|^{p(x)-2}\nabla u\biggr)=\lambda |u|^{r(x)-2}u +f(x,u) &\text{in }\Omega, \\ u=0 &\text{on }\partial\Omega, \end{cases} \end{aligned}$$

where $\Omega$ is a bounded domain of $\mathbb{R}^N (N\geq 2)$ with smooth boundary $\partial\Omega$, $\lambda>0$, and $p$ and $r$, are two continuous functions in $\overline{\Omega}$. Using variational methods combined with some properties of the generalized Sobolev spaces, under appropriate assumptions on $f$ and $M$, we obtain a number of results on the existence of solutions. In addition, we show the existence of infinitely many solutions in the case when $f$ satisfies the evenness condition.