Calculus of Variations and Geometric Measure Theory

D. Bartolucci - C. Gui - Y. Hu - A. Jevnikar - W. Yang

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

created by jevnikar on 19 Feb 2019
modified on 04 Jun 2019


Accepted Paper

Inserted: 19 feb 2019
Last Updated: 4 jun 2019

Journal: Discrete Contin. Dyn. Syst.
Year: 2019


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 a 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 by 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.

Keywords: uniqueness, Pohozaev identity, blow-up analysis, Mean field equation, Evenly symmetric solutions, Lyapunov-Schmidt reduction