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We use quantum Monte Carlo simulations to determine the ground state phase diagram of the Bose-Hubbard Hamiltonian with long-range repulsive interactions. At half filling one finds superfluidity and insulating solid phases. Depending on the relative sizes of near-neighbor and next-near-neighbor interactions, this solid follows either a checkerboard or a striped pattern. In neither case is there a coexistence with superfluidity. However, at incommensurate densities ``supersolid'' phases appear with simultaneous diagonal and off-diagonal long-range order.
Batrouni et al. (Mon,) studied this question.