Critical mean field equations for equilibrium turbulence with sign-changing prescribed functions

Abstract: Let $(M,g)$ be a compact Riemann surface with unit area. We investigate the mean field equation for equilibrium turbulence: \begin{align} \begin{cases} -\Delta u = \rho_1\left(\frac{h_1e^{u}}{\int_Mh_1e^udv_g}-1\right) - \rho_2\left(\frac{h_2e^{-u}}{\int_Mh_2e^{-u}dv_g}-1\right), \ \int_Mudv_g=0, \end{cases} \end{align} where $\rho_1=8\pi$ and $\rho_2\in(0,8\pi]$ are parameters, and $h_1, h_2$ are smooth functions on $M$ that are positive somewhere. By employing a refined Brezis-Merle type analysis, we establish sufficient conditions of Ding-Jost-Li-Wang type for the existence of solutions to this equation in critical cases, particularly when $h_1$ and $h_2$ may change signs. Our results extend Zhou's existence theorems (Nonlinear Anal. 69 (2008), no.~8, 2541--2552) for the case $h_1=h_2\equiv 1$.