A Multiplicity Result for Quasilinear Problems with Convex and Concave Nonlinearities and Nonlinear Boundary Conditions in Unbounded Domains

We study the following quasilinear problem with nonlinear boundary conditions −∆pu = λa(x)|u|p−2u+ k(x)|u|q−2u− h(x)|u|s−2u, 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, a, 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, while if λ = λ1 it admits at least one nonnegative solution. Our approach is variational in character.