Another remark on a result of Ding-Jost-Li-Wang.

Let (M,g) be a compact Riemann surface, h be a positive smooth function on M. It is well known that the functional \begin{equation*} J(u)=\frac {1}{2}\int _M|\nabla u|^2dv_g+8\pi \int _M udv_g-8\pi \log \int _Mhe^{u}dv_g \end{equation*} achieves its minimum under Ding-Jost-Li-Wang condition. This result was generalized to nonnegative h by Yang and the author. Later, Sun and Zhu [ Existence of Kazdan-Warner equation with sign-changing prescribed function , arXiv: 2012.12840 , 2020] showed the Ding-Jost-Li-Wang condition is also sufficient when h changes sign, which was reproved later by Wang and Yang [J. Funct. Anal. 282 (2022), Paper No. 109449] and Li and Xu [Calc. Var. Partial Differential Equations 61 (2022), Paper No. 143] respectively using a flow approach. The aim of this note is to give a new proof of Sun and Zhu's result. Our proof is based on the variational method and the maximum principle. [ABSTRACT FROM AUTHOR]