### Abstract

It is known from [17] that the solvability of the mean field equation ∆u + e^{u} = 8nπδ_{0} with n ∈ N_{≥} _{1} on a flat torus E_{τ} essentially depends on the geometry of E_{τ}. A conjecture is the non-existence of solutions for this equation if E_{τ} is a rectangular torus, which was proved for n = 1 in [17]. For any n ∈ N_{≥}2, this conjecture seems challenging from the viewpoint of PDE theory. In this paper, we prove this conjecture for n = 2 (i.e. at critical parameter 16π).

Original language | English |
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Pages (from-to) | 1737-1755 |

Number of pages | 19 |

Journal | Communications in Analysis and Geometry |

Volume | 27 |

Issue number | 8 |

DOIs | |

Publication status | Published - 2019 Jan 1 |

### ASJC Scopus subject areas

- Analysis
- Statistics and Probability
- Geometry and Topology
- Statistics, Probability and Uncertainty

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## Cite this

Chen, Z., Kuo, T. J., & Lin, C. S. (2019). Non-existence of solutions for a mean field equation on flat tori at critical parameter 16π.

*Communications in Analysis and Geometry*,*27*(8), 1737-1755. https://doi.org/10.4310/cag.2019.v27.n8.a3