Multiple blow-up solutions for the Liouville equation with singular data

Abstract

We study the existence of solutions with multiple concentration to the following boundary value problem $$-\Delta u=\e^2 e^u-4\pi \sum_{p\in Z}\alpha_p \delta_{p}\;\hbox{in} \Omega,\quad u=0 \;\hbox{on}\partial \Omega,$$ where $\Omega$ is a smooth and bounded domain in $\R^2$, $\alpha_{p}$'s are positive numbers, $Z\subset \Omega$ is a finite set, $\delta_p$ defines the Dirac mass at $p$, and $\e>0$ is a small parameter. In particular we extend the result of Del-Pino-Kowalczyk-Musso (\cite{delkomu}) to the case of several singular sources. More precisely we prove that, under suitable restrictions on the weights $\alpha_p$, a solution exists with a number of blow-up points $\xi_j\in \Omega\setminus Z$ up to $\sum_{p\in Z}\max\{n\in\N\,|\, n<1+\alpha_p\}$.