On a variational problem involving critical Sobolev growth in dimension three

In this paper we consider the following nonlinear problem: \({{-\Delta u=Ku^{5}}}\), u > 0 in \({{\Omega}}\), u =  0 on \({{\partial \Omega}}\), where K > 0 in \({{\Omega}}\), K =  0 on \({{\partial \Omega}}\) and \({{\Omega}}\) is a bounded domain of \({{\mathbb{R}^{3}}}\). We prove a version of a Morse lemma at infinity for this problem, which allows us to describe the critical points at infinity of the associated variational functional. Using a topological argument, we prove an existence result.