Since the first experimental achievement of Bose Einstein condensates in 1995, and more recently the Nobel prizes in 2001 and this year, a lot of properties of these systems are studied experimentally and theoretically. A mathematical approach consists in studying the setady states which minimize and energy called the Gross-Pitaevskii energy. In this talk, I will present rigorous results obtained related to two different experiments: existence of vortex free solutions based on Pohozaev identity and constrained minimization and special vortex configurations exhibiting bending.