  We consider probe Dp-branes, p=3,5,7, in global AdS_5 x S^5, rotating along
an internal direction in the S^5. These are dual to strongly interacting N=4
SYM on S^3 with massless fundamental flavours, in the presence of an R-symmetry
chemical potential for flavour fermions. For massless, "AdS-filling" Dp-brane
embeddings at zero temperature, we find an infinite set of threshold values of
the chemical potential at which instabilities are triggered. The onset of
instability is always preceded by metastability of the zero density state. From
the onset values of the chemical potential, we infer that unstable directions
favour a homogeneous condensate of a bilinear made from fermion harmonics, or
Cooper pairing. We confirm this picture both numerically and analytically. The
linearized analysis showing the appearance of instabilities involves a charged
scalar in global AdS space coupled to a (large) constant background gauge
potential. The resulting frequency space correlator of the fermion bilinear at
strong coupling displays poles in the upper half plane. In contrast, the
correlator at zero coupling exhibits Pauli blocking due to occupation of states
below the Fermi level, but no instabilities. The end-point of the strong
coupling instability is not visible in our setup.
