  We present analytical solutions for the mean-field master equation of the
driven-dissipative Bose-Hubbard model for cavity photons, in the limit of both
weak pumping and weak dissipation. Instead of pure Mott insulator states, we
find statistical mixtures with the same second-order coherence as a Fock state
with n photons, but a mean photon number of n/2. These mixed states occur when
n pump photons have the same energy as n interacting photons inside the
nonlinear cavity and survive up to a critical tunneling coupling strength,
above which a crossover to classical coherent state takes place. We also
explain the origin of both antibunching and superbunching predicted by
P-representation mean-field theory at higher pumping and dissipation. In
particular, we show that the strongly correlated region of the associated phase
diagram cannot be described within the semiclassical Gross-Pitaevski approach.
