On the Classification of blow-up solutions of a singular Liouville equation on the disk

TL;DR

This paper classifies all blow-up solutions of a singular Liouville equation on the disk, identifying the exact number and structure of solutions, including their blow-up behavior and non-degeneracy.

Contribution

It provides a complete classification of blow-up solutions and determines their exact count for the singular Liouville equation on the disk.

Findings

Exactly eil; lpha solutions for fixed lpha, <<_lpha

Unique minimal energy solution and singular blow-up sequence identified

Existence of m-peak solutions with vertices forming a regular m-gon

Abstract

We study the blow-up behavior of solutions to the singular Liouville equation \[ \Delta \tilde u+\lambda e^{\tilde u}=4\pi\alpha\delta_0 \quad\text{in }B,\quad \tilde u=0 \quad\text{on }\partial B, \] where $\alpha>0$, $\lambda>0$ and $B\subset\mathbb R^2$ is the unit disk. Our main results give a complete classification of all blow-up solutions and determine the exact number of solutions to the above equation. More precisely, for fixed $\alpha>0$ and $\lambda\in(0,\lambda_\alpha)$, the singular Liouville equation has exactly $\lceil \alpha\rceil+2$ solutions (up to rotation): a unique minimal energy solution; a unique singular sequence blowing up at the origin; and for each $1\le m\le\lceil \alpha\rceil$, a unique $m$-peak sequence whose blow-up points are the vertices of a regular $m$-gon centered at the origin. This result answers the questions raised in Bartolucci-Montefusco \cite{Bartolucci-Montefusco06} and Bartolucci \cite{Bartolucci10}. We also prove the non-degeneracy of these solutions. Thus we provide a full description of the blow-up structure for the singular Liouville equation on the disk.