# Existence and multiplicity of solutions for a $p(x)$-Kirchhoff type problem via variational techniques

## A. Mokhtari, T. Moussaoui, and D. O’Regan

A. Mokhtari, T. Moussaoui, Laboratory of Fixed Point Theory and Applications, Department of Mathematics, E.N.S. Kouba, Algiers, Algeria
National University of Ireland, School of Mathematics, Statistics and Applied Mathematics, Galway, Ireland King Abdulaziz University, NAAM Research Group, Jeddah, Saudi Arabia

Abstract: This paper discusses the existence and multiplicity of solutions for a class of $p(x)$-Kirchhoff type problems with Dirichlet boundary data of the following form ${\left\rbrace \begin{array}{ll} -\Big (a+b\int _{\Omega }\frac{1}{p(x)}|\nabla u|^{p(x)}\; dx\Big )\textrm{div}\big (|\nabla u|^{p(x)-2 } \nabla u\big )= f(x,u)\,, & \mbox {in \quad \Omega } \\[6pt] u=0 & \mbox {on \quad \partial \Omega }\,, \end{array}\right.}$ where $\Omega$ is a smooth open subset of $\mathbb{R}^N$ and $p\in C(\overline{\Omega })$ with $N <p^-= \inf _{x\in \Omega } p(x)\le p^+= \sup _{x\in \Omega } p(x)<+\infty$, $a$, $b$ are positive constants and $f\colon \overline{\Omega }\times \mathbb{R}\rightarrow \mathbb{R}$ is a continuous function. The proof is based on critical point theory and variable exponent Sobolev space theory.

AMSclassification: primary 34B27; secondary 35J60, 35B05.

Keywords: existence results, genus theory, nonlocal problems Kirchhoff equation, critical point theory.