Solutions for a $p(x)$-Kirchhoff Type Problem with a Non-smooth Potential in $\mathbb{R}^N$

Abstract

This paper is concerned with a class of $p(x)$-Kirchhoff type problem in $\mathbb{R}^N$. By the theories of nonsmooth critical point and variable exponent Sobolev spaces, we establish the existence and multiplicity of solutions to the $p(x)$-Kirchhoff type problem under weaker hypotheses on the nonsmooth potential at zero (at infinity, respectively). Some recent results in the literature are generalized and improved.