# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2020039

## Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions

 1 Department of Mathematics, University of Rome "Tor Vergata", Via della ricerca scientifica n.1, 00133 Roma, Italy 2 Department of Mathematics, The University of Texas at San Antonio, San Antonio, TX 78249, USA 3 Scuola Normale Superiore, Piazza dei Cavalieri 3, 56126 Pisa, Italy 4 Division of Mathematics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Science, Wuhan, Hubei 430071, China

Received  February 2019 Revised  April 2019 Published  October 2019

Fund Project: D. Bartolucci is partially supported by MIUR Excellence Department Project awarded to the Department of Mathematics, Univ. of Rome Tor Vergata, CUP E83C18000100006; C.Gui and Y. Hu are partially supported by NSF grant DMS-1601885; W. Yang is partially supported by NSFC No.11801550

We are concerned with the blow-up analysis of mean field equations. It has been proven in [6] that solutions blowing-up at the same non-degenerate blow-up set are unique. On the other hand, the authors in [18] show that solutions with a degenerate blow-up set are in general non-unique. In this paper we first prove that evenly symmetric solutions on an arbitrary flat torus with a degenerate two-point blow-up set are unique. In the second part of the paper we complete the analysis by proving the existence of such blow-up solutions using a Lyapunov-Schmidt reduction method. Moreover, we deduce that all evenly symmetric blow-up solutions come from one-point blow-up solutions of the mean field equation on a "half" torus.

Citation: Daniele Bartolucci, Changfeng Gui, Yeyao Hu, Aleks Jevnikar, Wen Yang. Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020039
##### References:
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