## Abstract

In this paper, we consider the mean field equation Δu + e^{u} = Σ_{3}_{i=0} 4πn_{i}δω_{i}/2 in E_{τ} , where ni ∈ ℤ_{≥}0, E_{τ} is the flat torus with periods ω_{1} = 1, ω_{2} = τ and Im τ > 0. Assuming N = Σ^{3}_{i}_{=0} n_{i} is odd, a non-critical case for the above PDE, we prove: (i) If among {ni|i = 0, 1, 2, 3} there is only one odd integer, then there is always an even solution. Furthermore, if n0 = 0 and n_{3} is odd, then up to SL_{2}(Z) action, except for finitely many E_{τ} , there are exactly n_{3}+1/2 even solutions. (ii) If there are exactly three odd integers in {ni|i = 0, 1, 2, 3}, then the equation has no even solutions for any flat torus E_{τ} . Our second result might suggest the symmetric solution of the above mean field equation does not hold in general.

Original language | English |
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Pages (from-to) | 1577-1590 |

Number of pages | 14 |

Journal | Proceedings of the American Mathematical Society |

Volume | 150 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2022 |

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics