Gaussian curvature in the negative case

In this paper, we reinvestigate an old problem of prescribing Gaussian curvature in the negative case.

In 1974, Kazdan and Warner considered the equation

on any compact two dimensional manifold $M$with $\alpha < 0$. They showed that there exists a number $\alpha_o$, such that the equation is solvable for every $0 > \alpha > \alpha_o$ and it is not solvable for $\alpha < \alpha_o$.

Then one may naturally ask:

Is the equation solvable for $\alpha = \alpha_o$?

In this paper, we answer the question affirmatively. We show that there exists at least one solution for $\alpha = \alpha_o$.