Existence of infinitely many weak solutions for the [formula omitted]-Laplacian with nonlinear boundary conditions

Abstract

In this paper we deal with the existence of weak solutions for quasilinear elliptic problem involving a p -Laplacian of the form { − Δ p u + λ ( x ) | u | p − 2 u = f ( x , u ) in Ω , | ∇ u | p − 2 ∂ u ∂ ν = η | u | p − 2 u on ∂ Ω . We consider the above problem under several conditions on f . For f “superlinear” and subcritical with respect to u , we prove the existence of infinitely many solutions of the above problem by using the “fountain theorem” and the “dual fountain theorem” respectively. For the case where f is critical with a subcritical perturbation, namely f ( x , u ) = | u | p ∗ − 2 u + | u | r − 2 u , 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 the “mountain pass theorem” and the “concentration–compactness principle” respectively.