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

摘要

It is known from [17] that the solvability of the mean field equation Delta u + e(u) = 8n pi delta(0) with n is an element of N->= 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 is an element of 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 16 pi).