On some nonlinear Neumann problem with weight and critical Sobolev trace maps
Abstract
We consider the problem - div(p(x)del u) = lambda u + alpha vertical bar u vertical bar(r-1)u in Q, partial derivative u/partial derivative nu = Q(x)vertical bar u(q-2)u on partial derivative ohm, where ohm is a bounded smooth domain in R-N, N >= 3, q = 2(N - 1)/(N - 2) and 2 < r < q. Under some conditions on partial derivative ohm, p, Q, lambda, alpha and the mean curvature at some point x(0), we prove the existence of solutions of the above problem. We use variational arguments, namely the concentration-compactness principle, min-max principle and the mountain-pass theorem.