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 $16\pi$).
Categories: math.AP