  We study a system with competing short- and global-range interactions in the
framework of the Bose-Hubbard model. Using a mean-field approximation we obtain
the phase diagram of the system and observe four different phases: a
superfluid, a supersolid, a Mott insulator and a charge density wave, where the
transitions between the various phases can be either of first or second order.
We qualitatively support these results using Monte-Carlo simulations. An
analysis of the low-energy excitations shows that the second-order phase
transition from the charge density wave to the supersolid is associated with
the softening of particle- and hole-like excitations which give rise to a
gapless mode and an amplitude Higgs mode in the supersolid phase. This
amplitude Higgs mode is further transformed into a roton mode which softens at
the supersolid to superfluid phase transition.
