On the construction of non-simple blow-up solutions for the singular Liouville equation with a potential

TL;DR

This paper investigates the existence of solutions with complex blow-up behavior for a singular Liouville equation involving a potential and a Dirac measure, revealing new non-simple blow-up profiles as a parameter approaches zero.

Contribution

It constructs non-simple blow-up solutions for the singular Liouville equation with a potential, under specific conditions on the potential's derivatives at zero.

Findings

Existence of non-simple blow-up solutions as  ^+

Identification of conditions on the potential V for blow-up

Analysis of blow-up profiles near the singularity

Abstract

We are concerned with the existence of blowing-up solutions to the following boundary value problem $$-\Delta u= \lambda V(x) e^u-4\pi N \delta_0\;\mbox{ in } B_1,\quad u=0 \;\mbox{ on }\partial B_1,$$ where $B_1$ is the unit ball in $\mathbb R^2$ centered at the origin, $V(x)$ is a positive smooth potential, $N$ is a positive integer ($N\geq 1$). Here $\delta_0$ defines the Dirac measure with pole at $0$, and $\lambda>0$ is a small parameter. We assume that $N=1$ and, under some suitable assumptions on the derivatives of the potential $V$ at $0$, we find a solution which exhibits a non-simple blow-up profile as $\lambda\to 0^+$.