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

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_\Omega\frac{1}{p(x)}| \nabla u| ^{p(x)}dx\right)div(|\nabla u| ^{p(x)-2}\nabla u)=\lambda |u| ^{p(x)-2}u+g(x,u) \mbox{ in } \Omega, \\ u=0,\mbox{ on } \partial\Omega, \end{cases} \end{equation} where \(a\geq b > 0\) 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.