On the nonhomogeneous Neumann problem with weight and with critical nonlinearity in the boundary

We consider the problem: −div(p(x)∇u)=λu+f(x) in Ω, ∂u∂ν=Q(x)|u|2N−2u on ∂Ω, where Ω is a bounded smooth domain in RN,N≥3. Under some conditions on ∂Ω, p, Q, f, λ and the mean curvature at some point x0, we prove the existence of solutions of the above problem. We use variational arguments, namely Ekeland's variational principle, the min-max principle and the mountain pass theorem.