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}=4παδ0 \quad\text{in }B,\quad \tilde u=0 \quad\text{on }\partial B, ] where $α>0$, $λ>0$ 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 $α>0$ and $λ\in(0,λα)$, the singular Liouville equation has exactly $\lceil α\rceil+2$ solutions (up to rotation): a unique minimal energy solution; a unique singular sequence blowing up at the origin; and for each $1\le m\le\lceil α\rceil$, 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.