A singular Kazdan–Warner problem on a compact Riemann surface

Abstract

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 $$\begin{aligned} \Delta v + 4\pi \sum _{i=1}^{\ell }\alpha _i\left( 1-\delta _{p_i}\right) = \rho \left( 1-\frac{he^v}{\int _Mhe^vd\mu }\right) , \end{aligned}$$ 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 $$\rho =\overline{\rho }:=8\pi (1+\min \{0,\min _{1\le i\le \ell }\alpha _i\})$$ . These results can be seen as partial generalizations of works of Chen and Li (J Geom Anal 1:359–372, 1991), Ding et al. (Asian J Math 1:230–248, 1997), Mancini (J Geom Anal 26:1202–1230, 2016), Yang and Zhu (Proc Amer Math Soc 145:3953–3959, 2017), Sun and 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.