Singular Liouville Equations on $S^2$: Sharp Inequalities and Existence Results

TL;DR

This paper establishes a sharp inequality and analyzes the existence of solutions for a singular Liouville equation on the sphere, highlighting conditions for extremals and critical points.

Contribution

It provides a sharp Onofri-type inequality for singular potentials and explores existence and non-existence results for extremals and critical points.

Findings

Proved a sharp inequality for the Moser-Tudinger functional with singularities.

Demonstrated non-existence of extremals in certain singular cases.

Identified conditions under which critical points exist.

Abstract

We prove a sharp Onofri-type inequality and non-existence of extremals for a Moser-Tudinger functional on the sphere in the presence of potentials having positive order singularities. We also investigate the existence of critical points and give some sufficient conditions under symmetry or nondegeneracy assumptions.