  In this paper, we study the sign-changing Kazdan-Warner's problem on two
dimensional closed Riemannian manifold with negative Euler number $\chi(M)<0$.
We show that once, the direct method on convex sets is used to find a minimizer
of the corresponding functional, then there is another solution via a use of
the variational method of mountain pass. In conclusion, we show that there are
at least two solutions to the Kazdan-Warner's problem on two dimensional
Kazdan-Warner equation provided the prescribed function changes signs and with
this average negative.
