*Accepted Paper*

**Inserted:** 8 apr 2017

**Last Updated:** 11 dec 2017

**Journal:** J. Math. Pures Appl.

**Year:** 2017

**Abstract:**

We prove uniqueness of blow up solutions of the mean field equation as $\rho_n\rightarrow 8\pi m$, $m\in\mathbb{N}$. 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 (C.C. Chen - C.S. Lin) and a rather involved analysis of suitably defined Pohozaev-type identities as recently developed in (C.S. Lin - S. Yan) 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 (C.C. Chen - C.S. Lin) vanishes.

**Keywords:**
uniqueness results, blow-up analysis, Geometric PDEs, Mean field equation, Pohozaev identities

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