Existence of infinitely many weak solutions for the pp-Laplacian with nonlinear boundary conditions

In this paper we deal with the existence of weak solutions for quasilinear elliptic problem involving a pp-Laplacian of the form {−Δpu+λ(x)|u|p−2u=f(x,u)in Ω,|∇u|p−2∂u∂ν=η|u|p−2uon ∂Ω. We consider the above problem under several conditions on ff. For ff “superlinear” and subcritical with respect to uu, 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 ff is critical with a subcritical perturbation, namely f(x,u)=|u|p∗−2u+|u|r−2uf(x,u)=|u|p∗−2u+|u|r−2u, we show that there exists at least a nontrivial solution when p