  In this paper, we continue to develop Aubry-Mather and weak KAM theories for
contact Hamiltonian systems $H(x,u,p)$ with certain dependence on the contact
variable $u$. For the Lipschitz dependence case, we obtain some properties of
the Ma\~{n}\'{e} set. For the non-decreasing case, we provide some information
on the Aubry set, such as the comparison property, graph property and a
partially ordered relation for the collection of all projected Aubry sets with
respect to backward weak KAM solutions. Moreover, we find a new flow-invariant
set $\tilde{\mathcal{S}}_s$ consists of strongly static orbits, which coincides
with the Aubry set $\tilde{\mathcal{A}}$ in classical Hamiltonian systems.
Nevertheless, a class of examples are constructed to show
$\tilde{\mathcal{S}}_s\subsetneqq\tilde{\mathcal{A}}$ in the contact case. As
their applications, we find some new phenomena appear even if the strictly
increasing dependence of $H$ on $u$ fails at only one point, and we show that
there is a difference for the vanishing discount problem from the negative
direction between the minimal viscosity solution and non-minimal ones.
