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arXiv · 2309.04706

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

Xuezhang Chen·Shihong Zhang
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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 $\frac{1}{2}$.

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Xuezhang Chen, Shihong Zhang. 2023-09-09. Constrained Moser-Trudinger-Onofri inequality and a uniqueness criterion for the mean field equation. https://arxiv.org/abs/2309.04706

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