  We study static, spherically symmetric, self-gravitating systems minimally
coupled to a scalar field with U(1) gauge symmetry: charged boson stars. We
find numerical solutions to the EinsteinMaxwell equations coupled to the
relativistic Klein-Gordon equation. It is shown that bound stable
configurations exist only for values of the coupling constant less than or
equal to a certain critical value. The metric coefficients and the relevant
physical quantities such as the total mass and charge, turn out to be in
general bound functions of the radial coordinate, reaching their maximum values
at a critical value of the scalar field at the origin. We discuss the stability
problem both from the quantitative and qualitative point of view. Taking
properly into account the electromagnetic contribution to the total mass, the
stability issue is faced also following an indication of the binding energy per
particle, verifying then the existence of configurations having positive
binding energy: objects apparently bound can be unstable against small
perturbations, in full analogy with what observed in the mass-radius relation
of neutron stars.
