Positive solutions of a Neumann problem with competing critical nonlinearities

Abstract

We present existence results for a Neumann problem involving
critical Sobolev nonlinearities both on the right hand side of
the equation and at the boundary condition. Positive
solutions are obtained through constrained minimization on the
Nehari manifold. Our approach is based on the concentration
compactness principle of P. L. Lions and M. Struwe.