Non-existence of solutions for a mean field equation on flat tori at critical parameter 16π

Zhijie Chen, Ting Jung Kuo, Chang Shou Lin

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Abstract

It is known from [17] that the solvability of the mean field equation ∆u + eu = 8nπδ0 with n ∈ N 1 on a flat torus Eτ essentially depends on the geometry of Eτ. A conjecture is the non-existence of solutions for this equation if Eτ is a rectangular torus, which was proved for n = 1 in [17]. For any n ∈ N2, 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 languageEnglish
Pages (from-to)1737-1755
Number of pages19
JournalCommunications in Analysis and Geometry
Volume27
Issue number8
DOIs
Publication statusPublished - 2019 Jan 1

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ASJC Scopus subject areas

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

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