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

We consider the problem: −div(p(x)∇u)=λu+f(x) in ΩΩ, ∂u∂ν=Q(x)|u|2N−2u on ∂Ω∂Ω, where ΩΩ is a bounded smooth domain in RN,N≥3. Under some conditions on ∂Ω∂Ω, pp, QQ, ff, λλ and the mean curvature at some point x0x0, 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.