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

Afrouzi, Ghasem; Mirzapour, Maryam; Chung, Nguyen Thanh

Rendiconti del Seminario Matematico della Università di Padova, Tome 136 (2016), pp. 95-109.

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

$$-M\left(\right({\int }_{\Omega }\frac{1}{p\left(x\right)}{\left|\nabla u\right|}^{p\left(x\right)}{dx\left)\mathrm{div}\right(\left|\nabla u\right|}^{p\left(x\right)-2}\nabla u)=f(x,u)\text{in}\Omega ,$$

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

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.