Critical points of a mean field type functional on a closed Riemann surface

Abstract

Let $(\Sigma,g)$ be a closed Riemann surface and $H^1(\Sigma)$ be the usual Sobolev space. For any real number $\rho$, we define a generalized mean field type functional $J_{\rho,\phi}\colon H^1(\Sigma)\rightarrow \mathbb{R}$ by\begin{equation*}J_{\rho,\phi}(u)=\frac{1}{2 } \bigg(\int_{\Sigma}|\nabla_g u|^{2} d v_g+\int_{\Sigma}\phi (u-\overline{u}) d v_g \bigg)-\rho\ln \int_{ \Sigma} h e^{u-\overline{u}} d v_g,\end{equation*}where $h\colon \Sigma\to\mathbb{R}$ is a smooth positive function, $\phi\colon \mathbb{R}\to\mathbb{R}$ is a smooth one-variable function and $\overline{u}=\int_\Sigma ud v_g/|\Sigma|$. If $\rho\in (8k\pi,8(k+1)\pi)$ ($k\in \mathbb{N}^{*}$), $\phi$ satisfies $|\phi(t)|\leq C (|t|^p+1)\ (1< p< 2)$ and $|\phi^\prime(t)|\leq C (|t|^{p-1}+1)$ for some constant $C$, then we get critical points of $J_{\rho,\phi}$ by adapting min-max schemes of Ding, Jost, Li and Wang [13], Djadli [14] and Malchiodi [22].