  We numerically construct asymptotically Anti-de Sitter charged black hole
solutions of (4+1)-dimensional SU(2) Einstein-Yang-Mills theory that, for
sufficiently low temperature, develop vector hair. Via gauge-gravity duality,
these solutions describe a strongly-coupled conformal field theory at finite
temperature and density that undergoes a phase transition to a superfluid state
with spontaneously broken rotational symmetry (a p-wave superfluid state). The
bulk theory has a single free parameter, the ratio of the five-dimensional
gravitational constant to the Yang-Mills coupling, which we denote as alpha.
Previous analyses have shown that in the so-called probe limit, where alpha
goes to zero and hence the gauge fields are ignored in Einstein's equation, the
transition to the superfluid state is second order. We construct fully
back-reacted solutions, where alpha is finite and the gauge fields are included
in Einstein's equation, and find that for values of alpha above a critical
value alpha_c = 0.365 +- 0.001, the transition becomes first order.
