Nonexistence, existence and multiplicity of positive solutions to the p(x)p(x)-Laplacian nonlinear Neumann boundary value problem

We study the nonexistence, existence and multiplicity of positive solutions for the nonlinear Neumann boundary value problem involving the p(x)p(x)-Laplacian of the form {−Δp(x)u+λ|u|p(x)−2u=f(x,u)in Ω|∇u|p(x)−2∂u∂η=g(x,u)on ∂Ω, where ΩΩ is a bounded smooth domain in RN, p∈C1(Ω¯) and p(x)>1p(x)>1 for x∈Ω¯. Using the sub–supersolution method and the variational principles, under appropriate assumptions on ff and gg, we prove that there exists λ∗>0λ∗>0 such that the problem has at least two positive solutions if λ>λ∗λ>λ∗, has at least one positive solution if λ=λ∗λ=λ∗ and has no positive solution if λ<λ∗λ<λ∗.