Uniqueness of bubbling solutions of mean field equations

Abstract

We prove uniqueness of blow up solutions of the mean field equation as ${\rho }_n\to 8\pi m$, $m\in \mathbb{N}$. 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 [24] 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 [24] vanishes.

arXiv commentAccepted for JMPA

Cited by in corpus (1)

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