Blow-up analysis of a nonlocal Liouville-type equation

TL;DR

This paper investigates the blow-up behavior and quantization phenomena of solutions to a nonlocal Liouville-type equation involving the fractional Laplacian on the circle, relating it to a similar equation on the real line and interpreting it as a prescribed curvature problem.

Contribution

It provides a detailed blow-up and quantization analysis for a nonlocal Liouville equation on the circle and establishes a connection with an analogous equation on the real line.

Findings

Characterization of blow-up behavior of solutions.

Quantization results for the prescribed curvature equation.

Relation between equations on $S^1$ and $ $.

Abstract

In this paper we perform a blow-up and quantization analysis of the following nonlocal Liouville-type equation \begin{equation}(-\Delta)^\frac12 u= \kappa e^u-1~\mbox{in $S^1$,} \end{equation} where $(-\Delta)^\frac{1}{2}$ stands for the fractional Laplacian and $\kappa$ is a bounded function. We interpret the above equation as the prescribed curvature equation to a curve in conformal parametrization. We also establish a relation between this equation and the analogous equation in $\mathbb{R}$ \begin{equation} (-\Delta)^\frac{1}{2} u =Ke^u \quad \text{in }\mathbb{R}, \end{equation} with $K$ bounded on $\mathbb{R}$.