A multiplicity result for quasilinear problems with convex and concave nonlinearities and nonlinear boundary conditions in unbounded domains

Abstract

We study the following quasilinear problem with nonlinear boundary conditions -∆pu = λα(x)|u|p-2 u + k(x)|u|q-2 u - h(x)|u|s-2 u, in Ω, |∇u|p-2 ∇u ∙ η + b(x)|u|p-2u = 0 on ∂Ω, where Ω is an unbounded domain in ℝN with a noncompact and smooth boundary ∂Ω, η denotes the unit outward normal vector on ∂Ω, ∆pu = div(|∇u|p-2∇u) is the p-Laplacian, α, k, h and b are nonnegative essentially bounded functions, q < p < s and p* < s. The properties of the first eigenvalue λ1 and the associated eigenvectors of the related eigenvalue problem are examined. Then it is shown that if λ < λ1, the original problem admits an infinite number of solutions one of which is nonnegative, which if λ = λ1 it admits at least one nonnegative solution. Our approach is variational in character.