Local and nonlocal singular Liouville equations in Euclidean spaces

Abstract

We study metrics of constant $Q$-curvature in the Euclidean space with a prescribed singularity at the origin, namely solutions to the equation $$(-\Delta {)}^{\frac{n}{2}}w=e^{nw}-c{\delta }_0\text{ on }{\mathbb{R}}^n,$$ under a finite volume condition. We analyze the asymptotic behaviour at infinity and the existence of solutions for every $n\ge 3$ also in a supercritical regime. Finally, we state some open problems.

26 pages