A blow-up phenomenon for a non-local Liouville-type equation

TL;DR

This paper investigates a non-local Liouville equation related to prescribing geodesic curvature on a circle, demonstrating a blow-up phenomenon at specific critical points under certain conditions.

Contribution

It constructs a family of solutions that blow up at critical points of the harmonic extension of the prescribed curvature, revealing new insights into the equation's behavior.

Findings

Solutions blow up at critical points of the harmonic extension.

Blow-up occurs under generic assumptions.

Provides a method to construct blow-up solutions.

Abstract

We consider a non-local Liouville equation corresponding to the prescription of the geodesic curvature on the circle. We build a family of solutions which blow up at a critical point of the harmonic extension of the prescribed curvature function, provided some generic assumptions are satisfied.