  We study a gas of repulsively interacting bosons in the first excited band of
an optical lattice. We explore this p-band physics both within the framework of
a standard mean-field theory as well as with the more accurate generalized
Gutzwiller ansatz. We find the phase diagrams for two- and three-dimensional
systems and characterize the first Mott-states in detail. Furthermore, we find
that even though the p-band model has strongly anisotropic kinetic energies and
inter-flavor interaction terms are missing in the lowest band theory, the
mean-field theory becomes useful quite rapidly once the transition from the
Mott-insulator to the superfluid is crossed.
