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We study the nonlinear Schr\"odinger equation for systems of $N$ orthonormalfunctions. We prove the existence of ground states for all $N$ when theexponent $p$ of the non linearity is not too large, and for an infinitesequence $N_j$ tending to infinity in the whole range of possible $p$'s, indimensions $d\geq1$. This allows us to prove that translational symmetry isbroken for a quantum crystal in the Kohn-Sham model with a large Dirac exchangeconstant.

The nonlinear Schr\"odinger equation for orthonormal functions: I. Existence of ground states