A criterion for the properness of the K-energy in a general Kahler class

In this paper, we give a criterion for the properness of the K-energy in a general Kahler class of a compact Kahler manifold by using Song-Weinkove's result. As applications, we give some Kahler classes on $\mathbb{C}\mathbb{P}^2\#3\overline {\mathbb{C}\mathbb{P}^2}$ and $\mathbb{C}\mathbb{P}^2\#8\overline {\mathbb{C}\mathbb{P}^2}$ in which the K-energy is proper. Finally, we prove Song-Weinkove's result on the existence of critical points of $\hat J$ functional by the continuity method.