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Trivial solution and symmetries of nontrivial solutions to a mean field equation

Volume 125 / 2020

Jiaming Jin, Chuanxi Zhu Annales Polonici Mathematici 125 (2020), 229-254 MSC: 35J15, 25J60, 35J93. DOI: 10.4064/ap191126-30-6 Published online: 9 December 2020

Abstract

We consider the mean field equation $$ \frac {\alpha }{2}\varDelta _g u+e^u-1=0\quad \text {on } \mathbb {S}^2. $$ We show that under some technical conditions, $u$ has to be constantly zero for $ {1}/{3}\leq \alpha \lt 1$. In particular, this is the case if $u(x)=-u(-x)$ and $u$ is odd symmetric about a plane. In the cases $u(x)=-u(-x)$ with $ {1}/{3}\leq \alpha \lt 1$ and $u(x)=u(-x)$ with $ {1}/{4}\leq \alpha \lt 1$, we analyze the additional symmetries of the nontrivial solution in detail.

Authors

  • Jiaming JinSchool of Mathematics
    Hunan University
    Changsha, Hunan 410082, P.R. China
    e-mail
  • Chuanxi ZhuDepartment of Mathematics
    Nanchang University
    Nanchang, 330031, P.R. China
    e-mail

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