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 the functional $$J(u)=\frac{1}{2}{\int }_M\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{2}d{v}_g+8\pi {\int }_{M}ud{v}_g-8\pi \log {\int }_{M}h{e}^{u}d{v}_g$$ 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 (arXiv:2012.12840) showed Ding-Jost-Li-Wang condition is also sufficient for $J$ achieves its minimum when $h$ changes sign, which was reproved later by Wang and Yang (J. Funct. Anal. 282: Paper No. 109449, 2022) and Li and Xu (Calc. Var. 61: Paper No. 143, 2022) respectively using 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.