# Uniqueness of bubbling solutions of mean field equations with non-quantized singularities

## TL;DR

This paper proves the uniqueness of bubbling solutions in singular mean field equations on compact Riemann surfaces, especially when blowup points coincide with bubbling sources and under specific non-degeneracy conditions.

## Contribution

It extends previous results by establishing uniqueness of bubbling solutions with non-quantized singularities and non-degenerate assumptions.

## Key findings

- Uniqueness of bubbling solutions when blowup points coincide with bubbling sources.
- Uniqueness under non-degeneracy when source strength is not multiple of 4π.
- Extension of prior work by Bartolucci et al.

## Abstract

For singular mean field equations defined on a compact Riemann surface, we prove the uniqueness of bubbling solutions if some blowup points coincide with bubbling sources. If the strength of the bubbling sources at blowup points are not multiple of $4\pi$ we prove that bubbling solutions are unique under non-degeneracy assumptions. This work extends a previous work of Bartolucci, et, al \cite{bart-4}.