Two-dimensional solutions of a mean field equation on flat tori

Abstract

We study the mean field equation on the flat torus Tσ:=C/(Z+Zσ)Δu+ρ(eu∫Tσeu−1|Tσ|)=0, where ρ is a real parameter. For a general flat torus, we obtain the existence of two-dimensional solutions bifurcating from the trivial solution at each eigenvalue (up to a multiplicative constant |Tσ|) of Laplace operator on the torus in the space of even symmetric functions. We further characterize the subset of all eigenvalues through which only one bifurcating curve passes. Finally local convexity near bifurcating points of the solution curves are obtained.