  We study the stability of charged solitons in 5-dimensional Anti-de Sitter
(AdS) space-time. We show that for appropriate choices of the parameters of the
model these solutions become unstable to form scalar hair. We find that the
existence of charged solitons with scalar hair depends crucially on the charge
and the mass of the scalar field. We investigate the dependence of the spectrum
of solutions on the mass of the scalar field in detail. For positive mass of
the scalar field the hairy solitons can be interpreted as charged boson stars.
We find that for sufficiently small value of the charge of the scalar field a
"forbidden band" of the boson star mass and charge exists, while all our
results indicate that - contrary to the asymptotically flat space-time case -
boson stars in asymptotically AdS can have arbitrarily large charge and mass.
