  We consider radial solutions to the Schr\"odinger-Poisson system in three
dimensions with an external smooth potential with Coulomb-like decay. Such a
system can be viewed as a model for the interaction of dark matter with a
bright matter background in the non-relativistic limit. We find that there are
infinitely many critical points of the Hamiltonian, subject to fixed mass, and
that these bifurcate from solutions to the associated linear problem at zero
mass. As a result, each branch has a different topological character defined by
the number of zeros of the radial states. We construct numerical approximations
to these nonlinear states along the first several branches. The solution
branches can be continued, numerically, to large mass values, where they become
asymptotic, under a rescaling, to those of the Schr\"odinger-Poisson problem
with no external potential. Our numerical computations indicate that the ground
state is orbitally stable, while the excited states are linearly unstable for
sufficiently large mass.
