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

TL;DR

This paper investigates the existence of multiple blow-up solutions to a Liouville equation with singular data, extending previous results to multiple singular sources and establishing conditions for solutions with several concentration points.

Contribution

It extends prior work by demonstrating the existence of solutions with multiple blow-up points in the presence of several singular sources under specific weight restrictions.

Findings

Existence of solutions with multiple blow-up points proven.

Extension of previous results to multiple singular sources.

Conditions on weights for solution existence established.

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 $\sum_{p\in Z}\max\{n\in\N\,|\, n<1+\alpha_p\}$.