Critical points of a mean field type functional on a closed Riemann surface - Zhang, Mengjie; Yang, Yunyan (Topological Methods in Nonlinear Analysis, 2022)
Let (Σ,g) be a closed Riemann surface and H^1(Σ) be the usual Sobolev space. For any real number ρ, we define a generalized mean field type functional J_ρ,φH^1(Σ)→ R by equation* J_ρ,φ(u)=12 (_Σ| abla_g u|^2 d v_g+_Σφ (u-u) d v_g )-ρ_ Σ h e^u-u d v_g, equation* where hΣ aR is a smooth positive function, φR aR is a smooth one-variable function and u=_Σ ud v_g/|Σ|. If ρ(8kπ,8(k+1)π) (kN^*), φ satisfies |φ(t)|≤ C (|t|^p+1) (1< p< 2) and |φ^′(t)|≤ C (|t|^p-1+1) for some constant C, then we get critical points of J_ρ,φ by adapting min-max schemes of Ding, Jost, Li and Wang DJLW99, Djadli Djadli and Malchiodi Malchiodi-DCDS.