We consider the stationary Gross-Pitaevskii equation for an extended function not tending to 0 at infinity, which describes the condensate part of an infinite Bose gas. We will explain how to mix tools from PDEs and statistical mechanics to study the phase diagram of this model. In particular, we will prove that there is a unique freezing transition for interaction potentials that are stable but not positive definite. The conjecture is that the solutions in the solid phase are periodic. Those describe a supersolid, a state of matter that has been the subject of much recent experimental work. Work in collaboration with Phan Thành Nam (Munich).