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

We study the existence of solutions with multiple concentration to the following boundary value problem $$-\mathrm{\Delta }u={\text{e}}^2{e}^u-4\pi \sum _{p\in Z}{\alpha }_p{\delta }_p\text{in}\mathrm{\Omega },u=0\text{on}\mathrm{\partial }\mathrm{\Omega },$$ where $\mathrm{\Omega }$ is a smooth and bounded domain in ${\mathbb{R}}^2$, ${\alpha }_p$'s are positive numbers, $Z\subset \mathrm{\Omega }$ is a finite set, ${\delta }_p$ defines the Dirac mass at $p$, and $\text{e}>0$ 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 ${\xi }_j\in \mathrm{\Omega }\setminus Z$ up to $\sum_{p\in Z}\max{n\in\N,|, n<1+\alpha_p}$.