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.