Communications in Analysis and Geometry

Volume 27 (2019)

Number 8

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

Pages: 1737 – 1755

DOI: https://dx.doi.org/10.4310/CAG.2019.v27.n8.a3

Authors

Zhijie Chen (Department of Mathematical Sciences, Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)

Ting-Jung Kuo (Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan)

Chang-Shou Lin (Taida Institute for Mathematical Sciences (TIMS), Center for Advanced Study in Theoretical Sciences (CASTS), National Taiwan University, Taipei, Taiwan)

Abstract

It is known from [17] that the solvability of the mean field equation $\Delta u + e^u = {8 n \pi \delta}_0$ with $n \in \mathbb{N}_{\geq 1}$ on a flat torus $E_{\tau}$ essentially depends on the geometry of $E_{\tau}$. A conjecture is the non-existence of solutions for this equation if $E_{\tau}$ is a rectangular torus, which was proved for $n = 1$ in [17]. For any $n \in \mathbb{N}_{\geq 2}$, 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 $16 \pi$).

Received 6 March 2017

Accepted 10 September 2017

Published 21 January 2020