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

Abstract

We study the blow-up behavior of solutions to the singular Liouville equation \[ Δ\tilde u+λe^{\tilde u}=4παδ_0 \quad\text{in }B,\quad \tilde u=0 \quad\text{on }\partial B, \] 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.

26 pages