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

Daniele Bartolucci; Changfeng Gui; Yeyao Hu; Aleks Jevnikar; Wen Yang

doi:10.3934/dcds.2020039

Article Contents

2020, Volume 40, Issue 6: 3093-3116. Doi: 10.3934/dcds.2020039 open access

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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: January 31, 2019

Revised: March 31, 2019

Published: March 2020

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

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Abstract

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.

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Mathematics Subject Classification: 35J61, 35Q35, 35Q82, 81T13.

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