On the location of two blow up points on an annulus for the mean field equation

TL;DR

This paper investigates the behavior of solutions to the mean field equation on annular domains, establishing that blow-up points must be symmetric with respect to the origin.

Contribution

It proves that for solutions with two blow-up points, these points are necessarily symmetric about the origin in annular domains.

Findings

Blow-up points are symmetric: P = -Q.

Characterization of blow-up behavior on annular domains.

Insight into symmetry properties of solutions.

Abstract

We consider the mean field equation on two-dimensional annular domains, and prove that if $P$ and $Q$ are two blow up points of a blowing-up solution sequence of the equation, then we must have $P=-Q$.