Multiple Solutions for a Class of (P1(x), P2(x))-Laplacian Problems With Neumann Boundary Conditions

Abstract

In this paper, we study the existence of solutions for a class of nonlinear Neumann problems with variable exponents of the form      −div (|∇u| p 1 (x)−2 + |∇u| p 2 (x)−2)∇u + |u| pmax(x)−2 u = λf (x, u) + µg(x, u) in Ω, ∂u ∂ν = 0 on ∂Ω, where Ω ⊂ R N , N ≥ 3 is a smooth bounded domain, ν is the outward unit normal to ∂Ω, λ, µ are positive parameters, p i ∈ C + (Ω), inf x∈Ω p max (x) > N , p max (x) = max{p 1 (x), p 2 (x)} for all x ∈ Ω, f, g : Ω × R → R are Carathéodory functions. Our proofs are essentially based on the three critical points theorem due to Ricceri [18].