Existence of Weak Solution for $p(x)$-Kirchhoff Type Problem Involving the $p(x)$-Laplacian-like Operator by Topological Degree

Abstract

In this paper, we study the existence of "weak solution" for a class of $p(x)$-Kirchhoff type problem involving the $p(x)$-Laplacian-like operator depending on two real parameters with Neumann boundary condition. Using a topological degree for a class of demicontinuous operator of generalized $(S_+)$ type and the theory of the variable exponent Sobolev space, we establish the existence of "weak solution" of this problem.