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We are concerned with the blow-up analysis of mean field equations. It hasbeen proven in [6] that solutions blowing-up at the same non-degenerate blow-upset are unique. On the other hand, the authors in [18] show that solutions witha degenerate blow-up set are in general non-unique. In this paper we firstprove that evenly symmetric solutions on a flat torus with a degeneratetwo-point blow-up set are unique. In the second part of the paper we completethe analysis by proving the existence of such blow-up solutions by using aLyapunov-Schmidt reduction method. Moreover, we deduce that all evenlysymmetric blow-up solutions come from one-point blow-up solutions of the meanfield equation on a "half" torus.

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