# Mean field equations on tori: existence and uniqueness of evenly symmetric blow-up solutions

## TL;DR

This paper investigates the existence and uniqueness of symmetric blow-up solutions to mean field equations on tori, demonstrating that symmetric solutions with degenerate blow-up sets are unique and originate from one-point blow-up solutions.

## Contribution

It proves the uniqueness of evenly symmetric solutions with degenerate blow-up sets on tori and constructs such solutions using Lyapunov-Schmidt reduction, linking them to solutions on a half torus.

## Key findings

- Symmetric solutions with degenerate blow-up sets are unique.
- Existence of symmetric blow-up solutions is established.
- All symmetric blow-up solutions derive from one-point solutions on a half torus.

## Abstract

We are concerned with the blow-up analysis of mean field equations. It has been proven in [6] that solutions blowing-up at the same non-degenerate blow-up set are unique. On the other hand, the authors in [18] show that solutions with a degenerate blow-up set are in general non-unique. In this paper we first prove that evenly symmetric solutions on a flat torus with a degenerate two-point blow-up set are unique. In the second part of the paper we complete the analysis by proving the existence of such blow-up solutions by using a Lyapunov-Schmidt reduction method. Moreover, we deduce that all evenly symmetric blow-up solutions come from one-point blow-up solutions of the mean field equation on a "half" torus.