  We consider minimal setups arising from different truncations of N=8
five-dimensional SO(6) gauged supergravity to study phase transitions involving
spontaneous breaking of any of the U(1) symmetries in U(1)xU(1)xU(1)in SO(6).
These truncations only keep the three relevant vector fields, four complex
scalar fields carrying U(1) charges, plus two neutral scalar fields required by
consistency. By considering thermal ensembles with different fixed U(1) charge
densities and solving the complete equations including the full back-reaction,
in some cases we find instabilities towards the formation of hairy black holes,
which lead to second order transitions, resulting from a thermodynamical
competition between different sectors. We argue that these should be the
dominant thermodynamical instabilities in the full ten-dimensional type IIB
theory. In other cases we find unstable branches of hairy black holes that
extend to temperatures above a critical temperature (`retrograde
condensation'). The results can be used as a first step to understand new
aspects of the phase diagram of large N, N=4 SU(N) super Yang-Mills theory with
fixed charge densities.
