  We consider the nonlinear Gross-Pitaevskii equation at positive density, that
is, for a bounded solution not tending to 0 at infinity. We focus on infinite
ground states, which are by definition minimizers of the energy under local
perturbations. When the Fourier transform of the interaction potential takes
negative values we prove the existence of a phase transition at high density,
where the constant solution ceases to be a ground state. The analysis requires
mixing techniques from elliptic PDE theory and statistical mechanics, in order
to deal with a large class of interaction potentials.
