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We consider radial solutions to the Schr\"odinger-Poisson system in threedimensions with an external smooth potential with Coulomb-like decay. Such asystem can be viewed as a model for the interaction of dark matter with abright matter background in the non-relativistic limit. We find that there areinfinitely many critical points of the Hamiltonian, subject to fixed mass, andthat these bifurcate from solutions to the associated linear problem at zeromass. As a result, each branch has a different topological character defined bythe number of zeros of the radial states. We construct numerical approximationsto these nonlinear states along the first several branches. The solutionbranches can be continued, numerically, to large mass values, where they becomeasymptotic, under a rescaling, to those of the Schr\"odinger-Poisson problemwith no external potential. Our numerical computations indicate that the groundstate is orbitally stable, while the excited states are linearly unstable forsufficiently large mass.

Existence and Stability Properties of Radial Bound States for Schr\"odinger-Poisson with an External Coulomb Potential in Three Space Dimensions