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arXiv · 1704.02354

Uniqueness of bubbling solutions of mean field equations

Daniele Bartolucci·Aleks Jevnikar·Youngae Lee·Wen Yang
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Abstract

We prove uniqueness of blow up solutions of the mean field equation as $ρ_n \rightarrow 8πm$, $m\in\mathbb{N}$. If $u_{n,1}$ and $u_{n,2}$ are two sequences of bubbling solutions with the same $ρ_n$ 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 $ρ_n-8πm$ in case the first order evaluated in [24] vanishes.

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Daniele Bartolucci, Aleks Jevnikar, Youngae Lee, Wen Yang. 2018-01-07. Uniqueness of bubbling solutions of mean field equations. https://arxiv.org/abs/1704.02354

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