  We use a mix of analytic and numerical methods to exhaustively study a class
of asymptotically global AdS solitons and hairy black hole solutions in
negative cosmological constant Einstein Maxwell gravity coupled to a charged
massless scalar field. Our results depend sensitively on the charge 'e' of the
scalar field. The solitonic branch of solutions we study hit the Chandrashekhar
limit at finite mass at small 'e', but extends to arbitrarily large mass at
larger 'e'. At low values of 'e' no hairy black holes exist. At intermediate
values of 'e' hairy black holes exist above a critical charge. At large 'e'
hairy black holes exist at all values of the charge. The lowest mass hairy
black holes is a smooth zero entropy soliton at small charge, but a (probably)
singular nonzero entropy hairy black hole at larger charge. In a phase diagram
of solutions, the hairy black holes merge with the familiar
Reissner-Nordstrom-AdS black holes along a curve that is determined by the
onset of the superradiant instability in the latter family.
