A complementary result on a singular mean field equation with a sign-changing potential function

TL;DR

This paper investigates a singular mean field equation on a Riemann surface with a sign-changing potential, establishing a priori estimates and existence results even when singular sources are on the zero-level curve of the potential.

Contribution

It provides new a priori estimates and existence results for the mean field equation with sign-changing potential, especially when singular sources lie on the zero-level curve.

Findings

A priori estimates are valid when singular sources are on the zero-level curve.

Existence and multiplicity results depend on the topology of the manifold.

The results extend previous work to sign-changing potentials.

Abstract

In this note, we study the singular mean field equation defined on a Riemann surface with a sign-changing potential function. We prove if some singular sources happen to be placed on the zero-level curve of the potential function, a priori estimate can still be obtained. As a consequence of this estimate, existence and multiplicity results can still be obtained based on the topology of the manifold.