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.

Bartolucci, D., Gui, C., Hu, Y., Jevnikar, A., Yang, W. (2020). Mean field equations on tori: existence and uniqueness of evenly symmetric blow-up solutions. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, 40(6), 3093-3116 [10.3934/dcds.2020039].

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

Bartolucci, D
Membro del Collaboration Group
;
2020-01-01

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.
2020
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05 - ANALISI MATEMATICA
English
mean field equation; evenly symmetric solutions; uniqueness; blow-up analysis; Pohozaev identity
D. Bartolucci is partially supported by MIUR Excellence Department Project awarded to the Department of Mathematics, Univ. of Rome Tor Vergata, CUP E83C18000100006
Bartolucci, D., Gui, C., Hu, Y., Jevnikar, A., Yang, W. (2020). Mean field equations on tori: existence and uniqueness of evenly symmetric blow-up solutions. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, 40(6), 3093-3116 [10.3934/dcds.2020039].
Bartolucci, D; Gui, C; Hu, Y; Jevnikar, A; Yang, W
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/256323
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