Constrained Moser-Trudinger-Onofri inequality and a uniqueness criterion for the mean field equation

TL;DR

This paper develops new constrained Moser-Trudinger-Onofri inequalities involving second order moments and uses these to establish a uniqueness criterion for solutions of a specific mean field equation on the sphere when a parameter is near a critical value.

Contribution

It introduces a novel inequality under second order moment constraints and applies it to derive a uniqueness criterion for the mean field equation on the sphere.

Findings

Established Moser-Trudinger-Onofri inequalities with second order moment constraints.

Identified a threshold for deviation that ensures uniqueness of solutions.

Linked the inequality to the behavior of solutions near a critical parameter value.

Abstract

We establish Moser-Trudinger-Onofri inequalities under constraint of a deviation of the second order moments from $0$, which serves as an intermediate one between Chang-Hang's inequalities under first and second order moments constraints. A threshold for the deviation is a uniqueness criterion for the mean field equation $$-a\Delta_{\mathbb{S}^2}u+1=e^{2u} \quad \mathrm{~~on~~} \quad \mathbb{S}^2$$ when the constant $a$ is close to $\frac{1}{2}$.