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

Abstract

It is known from \cite{LW} that the solvability of the mean field equation $\Delta u+e^u=8n\pi {\delta }_0$ with $n\in {\mathbb{N}}_{\ge 1}$ 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 \cite{LW}. For any $n\in {\mathbb{N}}_{\ge 2}$, 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 $16\pi$).

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