A singular Kazdan-Warner problem on a compact Riemann surface

Let $(M,g)$ be a compact Riemann surface with unit area, $h\in C^{\infty}(M)$ a function which is positive somewhere, $\rho>0$, $p_i\in M$ and $\alpha_i\in(-1,+\infty)$ for $i=1,\cdots,\ell$, we consider the mean field equation \begin{align*} \Delta v + 4\pi\sum_{i=1}^{\ell}\alpha_i\left(1-\delta_{p_i}\right) = \rho\left(1-\frac{he^v}{\int_Mhe^vd\mu}\right), \end{align*} on $M$, where $\Delta$ and $d\mu$ are the Laplace-Beltrami operator and the area element of $(M,g)$ respectively. Using variational method and blowup analysis, we prove some existence results in the critical case $\rho=8\pi(1+\min\{0,\min_{1\leq i\leq\ell}\alpha_i\})$. These results can be seen as partial generalizations of works of Chen-Li (J. Geom. Anal. 1: 359--372, 1991), Ding-Jost-Li-Wang (Asian J. Math. 1: 230--248, 1997), Mancini (J. Geom. Anal. 26: 1202--1230, 2016), Yang-Zhu (Proc. Amer. Math. Soc. 145: 3953--3959, 2017), Sun-Zhu (arXiv:2012.12840) and Zhu (arXiv:2212.09943). Among other things, we prove that the blowup (if happens) must be at the point where the conical angle is the smallest one and $h$ is positive, this is the most important contribution of our paper.