Nontrivial solutions for Kirchhoff-type problems involving the $p(x)$-Laplace operator

Abstract

In this article, we study the existence of nontrivial solutions for the following $p(x)$ Kirchhoff-type problem \begin{equation} \begin{cases}\smash {\!-M\big (\textstyle \int _{\Omega }A(x,\nabla u)\,dx\big ){div}(a(x,\nabla u))} =\lambda h(x)\frac{\partial F}{\partial u} (x,u), \quad \mbox {in } \Omega \\ u=0, \quad \mbox {on } \partial \Omega , \end{cases} \end{equation} where $\Omega \subset \mathbb {R}^{n}$, $n\geq 3$, is a smooth bounded domain, $\lambda >0$, $h\in C(\Omega )$, $F:\overline {\Omega }\times \mathbb {R}\rightarrow \mathbb {R}$ is continuously differentiable and $a, A:\Omega \times \mathbb {R}^{n}\rightarrow \mathbb {R}^{n}$ are continuous. The proof is based on variational arguments and the theory of variable exponent Sobolev spaces.