Kirchhoff-Type Problem with Mixed Boundary Condition in a Variable Exponent Sobolev Space

Abstract

In this paper, we consider a mixed boundary value problem for the stationary Kirchhoff-type equation containing $p(·)$-Laplacian. More precisely, we are concerned with the problem with the Dirichlet condition on a part of the boundary and the Steklov boundary condition on an another part of the boundary. We show the existence of at least one, two or infinitely many nontrivial weak solutions according to hypotheses on given functions.