  In a recent letter, we presented numerical relativity simulations, solving
the full Einstein--Maxwell--Klein-Gordon equations, of superradiantly unstable
Reissner-Nordstr\"om black holes (BHs), enclosed in a cavity. Low frequency,
spherical perturbations of a charged scalar field, trigger this instability.
The system's evolution was followed into the non-linear regime, until it
relaxed into an equilibrium configuration, found to be a $\textit{hairy}$ BH: a
charged horizon in equilibrium with a scalar field condensate, whose phase is
oscillating at the (final) critical frequency. Here, we investigate the impact
of adding self-interactions to the scalar field. In particular, we find
sufficiently large self-interactions suppress the exponential growth phase,
known from linear theory, and promote a non-monotonic behaviour of the scalar
field energy. Furthermore, we discuss in detail the influence of the various
parameters in this model: the initial BH charge, the initial scalar
perturbation, the scalar field charge, mass, and the position of the cavity's
boundary (mirror). We also investigate the "explosive" non-linear regime
previously reported to be akin to a bosenova. A mode analysis shows that the
"explosions" can be interpreted as the decay into the BH of modes that exit the
superradiant regime.
