Blow-up phenomena for the Liouville equation with a singular source of integer multiplicity

TL;DR

This paper investigates conditions under which solutions to a Liouville equation with a singular source blow up at a point as a parameter approaches zero, revealing new insights into the solution behavior near singularities.

Contribution

It establishes existence criteria for blowing-up solutions to the Liouville equation with a singular source of integer multiplicity, detailing the asymptotic behavior as the parameter tends to zero.

Findings

Existence of solutions blowing up at the singularity under certain conditions.

Asymptotic behavior of solutions as the parameter approaches zero.

Quantitative relation between the integral of the exponential term and the integer multiplicity.

Abstract

We are concerned with the existence of blowing-up solutions to the following boundary value problem $$-\Delta u= \la a(x) e^u-4\pi N \delta_0\;\hbox{ in } \Omega,\quad u=0 \;\hbox{ on }\partial \Omega,$$ where $\Omega$ is a smooth and bounded domain in $\R^2$ such that $0\in\Omega$, $a(x)$ is a positive smooth function, $N$ is a positive integer and $\la>0$ is a small parameter. Here $\delta_0$ defines the Dirac measure with pole at $0$. We find conditions on the function $a$ and on the domain $\Omega$ under which there exists a solution $u_\la$ blowing up at $0$ and satisfying $\la\into a(x)e^{u_\la} \to 8\pi(N+1)$ as $\la\to 0^+$.