Uniqueness of bubbling solutions of mean field equations

We prove the uniqueness of blow up solutions of the mean field equation as rn ! 8pm, m 2 N. If un,1 and un,2 are two sequences of bubbling solutions with the same rn and the same (non degenerate) blow up set, then un,1 = un,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 rn 8pm in case the first order evaluated in [22] vanishes.