Bol's type inequality for singular metrics and its application to prescribing $Q$-curvature problems

TL;DR

This paper extends Bol's inequality to singular metrics, providing conditions for solutions to a $Q$-curvature problem and establishing bounds on total $Q$-curvature using compactness arguments.

Contribution

It introduces a higher-order Bol's inequality for singular metrics and applies it to derive existence criteria and bounds for a singular $Q$-curvature problem.

Findings

Necessary and sufficient conditions for radial solutions

Uniform bounds on total $Q$-curvature

Extension of Bol's inequality to singular metrics

Abstract

In this article, we study higher-order Bol's inequality for radial normal solutions to a singular Liouville equation. By applying these inequalities along with compactness arguments, we derive necessary and sufficient conditions for the existence of radial normal solutions to a singular $Q$-curvature problem. Moreover, under suitable assumptions on the $Q$-curvature, we obtain uniform bounds on the total $Q$-curvature.