The Nonlinear Schrödinger Equation for Orthonormal Functions: Existence of Ground States

We study the nonlinear Schrodinger equation for systems of N orthonormal functions. We prove the existence of ground states for all N when the exponent p of the non linearity is not too large, and for an infinite sequence $$N_j$$ tending to infinity in the whole range of possible p’s, in dimensions $$d\ge 1$$ . This allows us to prove that translational symmetry is broken for a quantum crystal in the Kohn–Sham model with a large Dirac exchange constant.