  In this paper, we generalize our apriori estimates on cscK(constant scalar
curvature K\"ahler) metric equation to more general scalar curvature type
equations (e.g., twisted cscK metric equation). As applications, under the
assumption that the automorphism group is discrete, we prove the celebrated
Donaldson's conjecture that the non-existence of cscK metric is equivalent to
the existence of a destabilized geodesic ray where the $K$-energy is
non-increasing. Moreover, we prove that the properness of $K$-energy in terms
of $L^1$ geodesic distance $d_1$ in the space of K\"ahler potentials implies
the existence of cscK metric. Finally, we prove that weak minimizers of the
$K$-energy in $(\mathcal{E}^1, d_1)$ are smooth.
