  We theoretically investigate the Cooper-pair symmetry to be realized in
hole-doped monolayer MoS$_2$ by solving linearized BCS gap equations on the
three-orbital attractive Hubbard-like model in the presence of the atomic
spin-orbit coupling. In hole-doped monolayer MoS$_2$, both spin-orbit coupling
and the multi-orbital effects are more prominent than those of electron-doped
system. Near the valence band edge, the Fermi surfaces are composed of three
different types of hole pockets, namely, one mainly consisting of the almost
spin-degenerate $|d_{z^{2}}\rangle$ orbital near $\Gamma$ point, and the others
of the spin-split upper and lower bands near ${\rm K}$ and ${\rm K}'$ points
arising from the $|d_{x^{2}-y^{2}}\rangle$ and $|d_{xy}\rangle$ orbitals. The
number of relevant Fermi pockets increases with increase of the doping. At very
low doping, the upper split bands of $|d_{x^{2}-y^{2}}\rangle$ and
$|d_{xy}\rangle$ are concerned, yielding extremely low $T_{\rm c}$ due to small
density of states of the split bands. For further doping, the conventional
spin-singlet state (SS) appears in the $\Gamma$ pocket, which has a mixture of
the spin-triplet (orbital-singlet) (ST-OS) and spin-singlet (orbital-triplet)
(SS-OT) states in the K and K$'$ pockets. The ratio of the mixture depends on
the relative strength of the interactions, and the sign of the exchange
interactions. Moderately strong ferromagnetic exchange interactions even lead
to the pairing state with the dominant ST-OS state over the conventional SS
one. With these observations, we expect that the fascinating pairing with
relatively high $T_{\rm c}$ emerges at high doping that involves all the three
Fermi pockets.
