It is known from  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 . 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).
ASJC Scopus subject areas
- Statistics and Probability
- Geometry and Topology
- Statistics, Probability and Uncertainty