Local and nonlocal singular Liouville equations in Euclidean spaces

TL;DR

This paper investigates solutions to singular Liouville equations with constant Q-curvature in Euclidean spaces, analyzing their asymptotic behavior and existence across various regimes, including supercritical cases.

Contribution

It provides new existence results and asymptotic analysis for singular solutions to higher-order Liouville equations in Euclidean spaces.

Findings

Existence of solutions for all dimensions n ≥ 3.

Asymptotic behavior characterized at infinity.

Open problems proposed for further research.

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.