Nonexistence, existence and multiplicity of positive solutions to the [formula omitted]-Laplacian nonlinear Neumann boundary value problem

Abstract

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