We consider probe Dp-branes, $p=3$, 5, 7, in global ${\mathrm{AdS}}_5\times S^5$, rotating along an internal direction in the $S^5$. These are dual to strongly interacting $\mathcal{N}=4$ super Yang-Mills on $S^3$ with massless fundamental flavors, in the presence of an $R$-symmetry chemical potential for flavor fermions. For massless, “anti de Sitter-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 favor 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 anti de Sitter 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.

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