Solutions for a [formula omitted]-Kirchhoff type equation with Neumann boundary data

Abstract

This paper is concerned with the existence and multiplicity of solutions to a class of p ( x ) -Kirchhoff type problem with Neumann boundary data of the following form { − M ( ∫ Ω 1 p ( x ) ( | ∇ u | p ( x ) + | u | p ( x ) ) d x ) ( div ( | ∇ u | p ( x ) − 2 ∇ u ) − | u | p ( x ) − 2 u ) = f ( x , u ) in Ω , ∂ u ∂ υ = 0 on ∂ Ω . By means of a direct variational approach and the theory of the variable exponent Sobolev spaces, under appropriate assumptions on f and M , we obtain a number of results on the existence and multiplicity of solutions for the problem. In particular, we also obtain some results which can be considered as extensions of the classical result named “combined effects of concave and convex nonlinearities”. Moreover, the positive solutions and the regularity of weak solutions of the problem are considered.