  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.

Differential Geometry