Uniqueness for bubbling solutions with collapsing singularities

TL;DR

This paper investigates the uniqueness of bubbling solutions with collapsing singularities in mean field equations, establishing that at most one such solution sequence exists under these conditions, despite the complexities introduced by kernel space considerations.

Contribution

It proves the uniqueness of bubbling solutions with collapsing singularities, addressing challenges posed by the kernel space of the linearized operator.

Findings

At most one bubbling solution sequence exists with collapsing singularities.

The limit after re-scaling is orthogonal to the kernel space, ensuring uniqueness.

The main technical challenge is handling the kernel space in the analysis.

Abstract

The seminal work \cite{bm} by Brezis and Merle showed that the bubbling solutions of the mean field equation have the property of mass concentration. Recently, Lin and Tarantello in \cite{lt} found that the "bubbling implies mass concentration" phenomena might not hold if there is a collapse of singularities. Furthermore, a sharp estimate \cite{llty} for the bubbling solutions has been obtained. In this paper, we prove that there exists at most one sequence of bubbling solutions if the collapsing singularity occurs. The main difficulty comes from that after re-scaling, the difference of two solutions locally converges to an element in the kernel space of the linearized operator. It is well-known that the kernel space is three dimensional. So the main technical ingredient of the proof is to show that the limit after re-scaling is orthogonal to the kernel space.