Existence and multiplicity results for a new $p(x)$-Kirchhoff problem

Abstract

We study the existence and multiplicity results for the following nonlocal $p(x)$-Kirchhoff problem: \begin{equation} \label{10} \begin{cases} -\left(a-b\int_Ω\frac{1}{p(x)}| \nabla u| ^{p(x)}dx\right)div(|\nabla u| ^{p(x)-2}\nabla u)=λ|u| ^{p(x)-2}u+g(x,u) \mbox{ in } Ω, \\ u=0,\mbox{ on } \partialΩ, \end{cases} \end{equation} where $a\ge b>0$ are constants, $\Omega \subset {\mathbb{R}}^N$ is a bounded smooth domain, $p\in C(\overline{\Omega })$ with $N>p(x)>1$, $\lambda$ is a real parameter and $g$ is a continuous function. The analysis developed in this paper proposes an approach based on the idea of considering a new nonlocal term which presents interesting difficulties.