Uniqueness of bubbling solutions of mean field equations

Abstract

We prove the uniqueness of blow up solutions of the mean field equation as rho(n) -> 8 pi m, m is an element of N. If u(n,1) and u(n,2) are two sequences of bubbling solutions with the same rho(n), and the same (non degenerate) blow up set, then u(n,1) = u(n,2) for sufficiently large n. The proof of the uniqueness requires a careful use of some sharp estimates for bubbling solutions of mean field equations [22] and a rather involved analysis of suitably defined Pohozaev-type identities as recently developed in [51] in the context of the Chern-Simons-Higgs equations. Moreover, motivated by the Onsager statistical description of two dimensional turbulence, we are bound to obtain a refined version of an estimate about rho(n) - 8 pi m in case the first order evaluated in [22] vanishes.