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 $n\in\mathbb{N}_{\geq 1}$ 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 \cite{LW}. For any $n\in \mathbb{N}_{\geq2}$, 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 $16\pi$).