  We study a discrete nonlinear Schr\"odinger lattice with a parabolic trapping
potential. The model, describing, e.g., an array of repulsive Bose-Einstein
condensate droplets confined in the wells of an optical lattice, is
analytically and numerically investigated. Starting from the linear limit of
the problem, we use global bifurcation theory to rigorously prove that - in the
discrete regime - all linear states lead to nonlinear generalizations thereof,
which assume the form of a chain of discrete dark solitons (as the density
increases). The stability of the ensuing nonlinear states is studied and it is
found that the ground state is stable, while the excited states feature a chain
of stability/instability bands. We illustrate the mechanisms under which
discreteness destabilizes the dark-soliton configurations, which become stable
only inside the continuum regime. Continuation from the anti-continuum limit is
also considered, and a rich bifurcation structure is revealed.
