Existence of an unbounded branch of the set of solutions for Neumann problems involving the p(x)-Laplacian

摘要

We are concerned with the following nonlinear problem: ? div ( w ( x ) | ? u | p ( x ) ? 2 ? u ) + | u | p ( x ) ? 2 u = μ g ( x ) | u | p ( x ) ? 2 u + f ( λ , x , u , ? u ) in Ω, ? u ? n = 0 on ?Ω, which is subject to a Neumann boundary condition, provided that μ is not an eigenvalue of the p ( x ) -Laplacian. The aim of this paper is to study the structure of the set of solutions for the degenerate p ( x ) -Laplacian Neumann problems by applying a bifurcation result for nonlinear operator equations. MSC: 35B32, 35D30, 35J70, 47J10, 47J15.