Singular elliptic problems with critical growth

We consider Dirichlet problems of the form $-|x|^\alpha\Delta u=|u|^{p-2}u$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is an arbitrary domain in $\mathbb{R}^N$, with $N\ge 3$, $\alpha\in(0,2)$, and $p={2(N-\alpha)\over N-2}$ is the corresponding critical exponent. A lack of compactness may occur when $0\in\overline\Omega$ or $\Omega$ is unbounded, because of concentration phenomena at the origin or vanishing, due to dilation invariance. We study the existence of positive solutions with respect to the geometry of the domain $\Omega$.