  We investigate the ground-state properties of monolayer MoS$_2$ incorporating
the Coulomb interaction together with a short-range intervalley interaction
between charged particles between two valleys within the Hartree-Fock
approximation. We consider four variables as independent parameters, namely
homogeneous charge (electron or hole) density, averaged dielectric constant,
spin degree of freedom and finally the Hubbard repulsion coefficient which
originates mostly from $4d$ orbits of Mo atoms. We find the electronic charge
compressibility within the mean-field approximation and show that non-monotonic
behavior of the compressibility as a function of carrier density which is
rather different from those of the two-dimensional electron gas. We also
explore a paramagnetic-to-ferromagnetic quantum phase transition for the wide
range of the electron density in the parameter space.
