Existence results for the mean field equation on a closed symmetric Riemann surface

Abstract

Let (S,g) be a closed Riemann surface, G be a finite isometric group acting on it and ♯G(x) be the number of all distinct points in the set G(x) for x∈S. If there exists some ℓ∈N⁎ satisfying ♯G(x)≡ℓ for all x∈S, then we show that for any ρ∈(8πkℓ,8π(k+1)ℓ) with k∈N⁎, the mean field equationΔgu=ρ(heu∫Sheudvg−1|S|) has a G-invariant solution, where h is a strictly positive and G-invariant smooth function. Our method is a modification of the min-max scheme due to Ding-Jost-Li-Wang (1999), Djadli-Malchiodi (2008) and Djadli (2008).