Let $(M,g)$ be a compact Riemannian surface without boundary, $W^{1,2}(M)$ be the usual Sobolev space,
$J: W^{1,2}(M)\ra \mathbb{R}$ be the functional defined by 
$$J(u)=\f{1}{2}\int_M|\nabla u|^2dv_g+8\pi \int_M udv_g-8\pi\log\int_M he^udv_g,$$
where $h$ is a positive smooth function on $M$. In an inspiring work (Asian J. Math., vol. 1, pp. 230-248, 1997),
Ding, Jost, Li and Wang obtained a sufficient condition under which $J$ achieves its minimum.
In this note, we prove that if the smooth function $h$ satisfies $h\geq 0$ and $h\not\equiv 0$, then
the above result still holds. Our method is to exclude blow-up points on the zero set of $h$. This is a joint work with Prof. Yunyan Yang.