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 $-\div(|x|^\alpha\nabla u)=|u|^{2^*_\alpha-2}u$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\alpha\in(0,2)$, $2^*_\alpha={2n\over n-2+\alpha}$ and $\Omega$ is a domain in $\mathbb{R}^n$. In particular we prove that a non negative ground state solution exists when the domain $\Omega$ is a cone, including the case $\Omega=\mathbb{R}^n$. Moreover, 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.