On the existence of extremal functions for a weighted Sobolev embedding with critical exponent

We investigate the existence of ground state solutions to the Dirichlet problem -div(|x|α∇u) = |u|2*α-2u in Ω, u = 0 on ∂Ω, where α ε (0,2), 2*α = 2n/n-2+α and Ω is a domain in R^n. In particular we prove that a non negative ground state solution exists when the domain Ω is a cone, including the case Ω = R^n. Moroever, we study the case of arbitrary domains, showing how the geometry of the domain near the origin and at infinity affects the existence or non existence of ground state solutions.