Infinitely many weak solutions for a p-Laplacian equation with nonlinear boundary conditions

Abstract

We study the following quasilinear problem with nonlinear boundary conditions -Δpu + α(x)|u|p-2u = ƒ(x, u) in Ω, |∇u|p-2 ∂u/∂v = g(x, u) on ∂Ω, where Ω is a bounded domain in ℝN with smooth boundary and ∂/∂v is the outer normal derivative, Δpu = div(|∇u|p-2∇u) is the p-Laplacian with 1 < p < N. We consider the above problem under several conditions on ƒ and superlinear and subcritical with respect to u, then we prove the existence of infinitely many solutions of this problem by using "fountain theorem" and "dual fountain theorem" respectively. In the case, where g is superlinear but subcritical and ƒ is critical with a subcritical perturbation, namely ƒ(x, u) = |u|p*-2u + λ|u|r-2u, we show that there exists at least a nontrivial solution when p < r < p* and there exist infinitely many solutions when 1 < r < p, by using "mountain pass theorem" and concentration-compactness principle" respectively.