Constrained Moser-Trudinger-Onofri inequality and a uniqueness criterion for the mean field equation
- Resource Type
- Working Paper
- Authors
- Chen, Xuezhang; Zhang, Shihong
- Source
- Subject
- Mathematics - Analysis of PDEs
Mathematics - Differential Geometry
- Language
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 $\frac{1}{2}$.
Comment: 22 pages