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 $$-Δu=\e^2 e^u-4π\sum_{p\in Z}α_p δ_{p}\;\hbox{in} Ω,\quad u=0 \;\hbox{on}\partial Ω,$$ where $\Omega$ is a smooth and bounded domain in ${\mathbb{R}}^2$, ${\alpha }_p$'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 ${\sum }_{p\in Z}\max \{n\in \mathbb{N}|n<1+{\alpha }_p\}$.