A generalized mean field type flow on a closed Riemann surface

Let $(\mathrm{\Sigma },g)$ be a closed Riemann surface. Let ψ, h be two smooth functions on Σ with ${\int }_{\mathrm{\Sigma }}\psi d{v}_g\ne 0$ and $h\ge 0,h\not\equiv 0$. In this paper, using the method of flow due to Cast$\acute{\mathrm{e}}$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 exists under given conditions. This gives a new proof of the main results of Zhu (Nonlinear Anal. 169(2018), 38–58).