# Existence and multiplicity of solutions for a $p(x)$-Kirchhoff type equation

### Ghasem A. Afrouzi

University of Mazandaran, Babolsar, Iran### Maryam Mirzapour

University of Mazandaran, Babolsar, Iran### Nguyen Thanh Chung

Quang Binh University, Vietnam

## Abstract

This paper is concerned with the existence and multiplicity to $p(x)$-Kirchhoff type problem of the following form

-M \big{(} \int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx\big {)}\mathrm{div}(|\nabla u|^{p(x)-2}\nabla u) =f(x,u) \quad \text{in } \Omega,

$u=0 \quad \mathrm {on} \: \partial \Omega.$

By means of a direct variational approach and the theory of the variable exponent Sobolev spaces, we establish conditions ensuring the existence and multiplicity of solutions for the problem.

## Cite this article

Ghasem A. Afrouzi, Maryam Mirzapour, Nguyen Thanh Chung, Existence and multiplicity of solutions for a $p(x)$-Kirchhoff type equation. Rend. Sem. Mat. Univ. Padova 136 (2016), pp. 95–109

DOI 10.4171/RSMUP/136-8