A generalized mean field type flow on a closed Riemann surface

Abstract

Let false(normalΣ,gfalse)$(\Sigma ,g)$ be a closed Riemann surface. Let ψ, h be two smooth functions on Σ with ∫Σψdvg≠0$\int _\Sigma \psi dv_g\ne 0$ and h≥0,h≢0$h\ge 0, h\not\equiv 0$. In this paper, using the method of flow due to Castnormalé$\mathrm{\acute{e}}$ras (Pacific J. Math. 276(2015), no. 2, 321–345) and Sun–Zhu (Calc. Var. Partial Differential Equations 60(2021), no. 1, 26), we prove that the solution of the equation badbreak−Δgugoodbreak=8π()heu∫normalΣheudvg−ψ∫normalΣψdvg$$\begin{equation*} -\Delta _g u=8\pi {\left(\frac{h e^u}{\int _\Sigma h e^u dv_g}-\frac{\psi }{\int _\Sigma \psi dv_g}\right)} \end{equation*}$$exists under given conditions. This gives a new proof of the main results of Zhu (Nonlinear Anal. 169(2018), 38–58).