Liouville type theorems for conformal Gaussian curvature equation

TL;DR

This paper establishes Liouville type theorems for the conformal Gaussian curvature equation in R^2, demonstrating non-existence of finite total curvature solutions under certain conditions on the function K(x).

Contribution

It provides new non-existence results for solutions of the mean field equation with sign-changing or monotone K(x) using moving plane and sphere methods.

Findings

Non-existence of finite total curvature solutions when K(x) is sign-changing.

Non-existence when K(x) is monotone non-decreasing along rays.

Application of moving plane and sphere methods to establish results.

Abstract

In this note, we study Liouville type theorem for conformal Gaussian curvature equation (also called the mean field equation) $$ -\Delta u=K(x)e^u, in R^2 $$ where $K(x)$ is a smooth function on $R^2$. When $K(x)=K(x_1)$ is a sign-changing smooth function in the real line $R$, we have a non-existence result for the finite total curvature solutions. When $K$ is monotone non-decreasing along every ray starting at origin, we can prove a non-existence result too. We use moving plane method and moving sphere method.