  We consider the static charged black hole bomb system, originally designed
for a (uncharged) rotating superradiant system by Press and Teukolsky. A
charged scalar field confined in a Minkowski cavity with a Maxwell gauge field
has a quantized spectrum of normal modes that can fit inside the box.
Back-reacting non-linearly these normal modes, we find the hairy solitons,
a.k.a boson stars (depending on the chosen $U(1)$ gauge), of the theory. The
scalar condensate is totally confined inside the box and, outside it, we have
the Reissner-Nordstr\"om solution. The Israel junction conditions at the box
surface layer determine the stress tensor that the box must have to confine the
scalar hair. Some of these horizonless hairy solutions exist for any value of
the scalar field charge and not only above the natural critical charges of the
theory (namely, the critical charges for the onset of the near-horizon and
superradiant instabilities of the Reissner-Nordstr\"om black hole). However,
the ground state solutions have a non-trivial intricate phase diagram with a
main and a secondary family of solitons (some with a Chandrasekhar mass limit
but others without) and there are a third and a fourth critical scalar field
charges where the soliton spectra changes radically. Most of these intricate
properties are not captured by a higher order perturbative analysis of the
problem where we simply back-react a normal mode of the system.
