The $C^0$-convergence at the Neumann boundary for Liouville equations

TL;DR

This paper establishes $C^0$-convergence results for solutions to Liouville equations with exponential Neumann boundary conditions during boundary blow-up scenarios, extending previous interior case analyses.

Contribution

It introduces a new method to prove $C^0$-convergence at boundary blow-up points, differing from prior approaches based on the moving planes or classification techniques.

Findings

Proved $C^0$-convergence at boundary blow-up points.

Extended asymptotic estimates to boundary cases.

Developed a novel proof technique for boundary convergence.

Abstract

In this paper, we study the blow-up analysis for a sequence of solutions to the Liouville type equation with exponential Neumann boundary condition. For interior case, i.e. the blow-up point is an interior point, Li \cite{Li} gave a uniform asymptotic estimate. Later, Zhang \cite{Zhang} and Gluck \cite{Gluck} improved Li's estimate in the sense of $C^0$-convergence by using the method of moving planes or classification of solutions of the linearized version of Liouville equation. If the sequence blows up at a boundary point, Bao-Wang-Zhou \cite{Bao-Wang-Zhou} proved a similar asymptotic estimate of Li \cite{Li}. In this paper, we will prove a $C^0$-convergence result in this boundary blow-up process. Our method is different from \cite{Zhang,Gluck}.