  We investigate the mean-field phase diagram of the Bose-Hubbard model with
infinite-range interactions in two dimensions. This model describes ultracold
bosonic atoms confined by a two-dimensional optical lattice and dispersively
coupled to a cavity mode with the same wavelength as the lattice. We determine
the ground-state phase diagram for a grand-canonical ensemble by means of
analytical and numerical methods. Our results mostly agree with the ones
reported in Dogra et al. [PRA 94, 023632 (2016)], and have a remarkable
qualitative agreement with the quantum Monte Carlo phase diagrams of Flottat et
al. [PRB 95, 144501 (2017)]. The salient differences concern the stability of
the supersolid phases, which we discuss in detail. Finally, we discuss
differences and analogies between the ground state properties of strong
long-range interacting bosons with the ones predicted for repulsively
interacting dipolar bosons in two dimensions.
