Non-existence of solutions for a mean field equation on flat tori at critical parameter 16π
Journal
Communications in Analysis and Geometry
Journal Volume
27
Journal Issue
8
Pages
1737-1755
Date Issued
2019
Author(s)
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 ∈ 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).
Type
journal article