On a nonlinear problem with zero Dirichlet boundary condition

We are concerned with the following PDE: -u = K(x)u(n+2/n-2), u > 0 in Omega, u = 0, on partial derivative Omega, where Omega is a bounded domain of R-n, n >= 3, and K : (Omega) over bar -> R is a given function. We provide a complete description of the loss of compactness of the equation under the assumption that K is strictly decreasing in the outward normal direction on partial derivative Omega and flat near its critical points. As product, we prove an existence result through an Euler-Poincare characteristic argument.