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

TL;DR

This paper investigates the existence and multiplicity of solutions for a new class of nonlocal $p(x)$-Kirchhoff problems involving variable exponent spaces, introducing a novel nonlocal term and addressing associated analytical challenges.

Contribution

It introduces a new nonlocal term in the $p(x)$-Kirchhoff problem and develops an analytical framework to handle the complexities arising from variable exponents and nonlocality.

Findings

Established existence of solutions for various parameter ranges.

Proved multiplicity results under certain conditions.

Developed new analytical techniques for nonlocal variable exponent problems.

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_\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.