  Weakly interacting atomic or molecular bosons in quantum degenerate regime
and trapped in harmonically confined optical lattices, exhibit a wedding cake
structure consisting of insulating (Mott) shells. It is shown that superfluid
regions emerge between Mott shells as a result of fluctuations due to finite
hopping. It is found that the order parameter equation in the superfluid
regions is not of the Gross-Pitaeviskii type except near the insulator to
superfluid boundaries. The excitation spectra in the Mott and superfluid
regions are obtained, and it is shown that the superfluid shells posses low
energy sound modes with spatially dependent sound velocity described by a local
index of refraction directly related to the local superfluid density. Lastly,
the Berezinskii-Kosterlitz-Thouless transition and vortex-antivortex pairs are
discussed in thin (wide) superfluid shells (rings) limited by three (two)
dimensional Mott regions.
