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 $J_{ρ,φ}\colon H^1(Σ)→ \mathbb{R}$ by\begin{equation*}J_{ρ,φ}(u)=\frac{1}{2 } \bigg(\int_{Σ}| abla_g u|^{2} d v_g+\int_{Σ}φ (u-\overline{u}) d v_g \bigg)-ρ\ln \int_{ Σ} h e^{u-\overline{u}} d v_g,\end{equation*}where $h\colon Σ\ra\mathbb{R}$ is a smooth positive function,$φ\colon \mathbb{R}\ra\mathbb{R}$ is a smooth one-variable function and$\overline{u}=\int_Σ ud v_g/|Σ|$.If $ρ\in (8kπ,8(k+1)π)$ ($k\in \mathbb{N}^{*}$), $φ$ 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 schemesof Ding, Jost, Li and Wang \cite{DJLW99}, Djadli \cite{Djadli}and Malchiodi \cite{Malchiodi-DCDS}.