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

Abstract

We are concerned with the existence of blowing-up solutions to the following boundary value problem $$-Δu= \la a(x) e^u-4πN δ_0\;\hbox{ in } Ω,\quad u=0 \;\hbox{ on }\partial Ω,$$ where $\Omega$ is a smooth and bounded domain in ${\mathbb{R}}^2$ 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 $\la\into a(x)e^{u_\la} \to 8π(N+1)$ as $\la\to 0^+$.