  Let $(M,g)$ be a compact Riemann surface, $h$ be a positive smooth function
on $M$. It is well known the functional $$J(u)=\frac{1}{2}\int_M|\nabla
u|^2dv_g+8\pi\int_M udv_g-8\pi\log\int_Mhe^{u}dv_g$$ 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.
