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<p>We establish Moser-Trudinger-Onofri inequalities under constraint of a deviation of the second order moments from 0, which serves as intermediate ones 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 <Equation ID="Equ52"> <EquationSource Format="TEX">\(\begin{aligned} -a\Delta _{\mathbb {S}^2}u+1=e^{2u} \quad \mathrm {~~on~~} \quad \mathbb {S}^2 \end{aligned}\)</EquationSource> </Equation>when the constant <i>a</i> is close to <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\frac{1}{2}\)</EquationSource> </InlineEquation>.</p>

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

  • 期刊论文

  • 发表日期：

    2025年10月10日

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  • Xuezhang Chen,
  • Shihong Zhang

摘要

We establish Moser-Trudinger-Onofri inequalities under constraint of a deviation of the second order moments from 0, which serves as intermediate ones 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 \(\begin{aligned} -a\Delta _{\mathbb {S}^2}u+1=e^{2u} \quad \mathrm {~~on~~} \quad \mathbb {S}^2 \end{aligned}\) when the constant a is close to \(\frac{1}{2}\) .