  In this work, we study pancake-shaped Bose-Einstein condensates confined by
both a cylindrically symmetric harmonic potential and an optical lattice with
equal periodicity in two orthogonal directions. We first identify the spectrum
of the underlying two-dimensional linear problem through multiple-scale
techniques. Then, we use the results obtained in the linear limit as a starting
point for a nonlinear existence and stability analysis of the lowest energy
states, emanating from the linear ones, in the nonlinear problem. Two-parameter
continuations of these states are performed for increasing nonlinearity and
optical lattice strengths, and their instabilities and temporal evolution are
investigated. It is found that the ground state as well as one of the excited
states are either stable or weakly unstable for both attractive and repulsive
interatomic interactions.
