Note on super-critical charged boson stars

We study the transition of charged boson stars from sub- to super-criticality. This transition is defined as that choice of coupling constants for which the Coulomb repulsion of two individual bosons (that make up the star) exactly cancels their gravitational attraction. It was recently shown that without self-interaction super-critical boson stars are unstable to decay into their individual constituents. Here we show that this is no longer true for the self-interacting case and that boson stars can possess spatial oscillations in the scalar field. We also discuss the corresponding black hole solutions that carry charged scalar hair.