  We investigate the existence of ground state solutions of a Gross-Pitaevskii
equation modeling the dynamics of pumped Bose Einstein condensates (BEC). The
main interest in such BEC comes from its important nature as macroscopic
quantum system, constituting an excellent alternative to the classical
condensates which are hard to realize because of the very low temperature
required. Nevertheless, the Gross Pitaevskii equation governing the new
condensates presents some mathematical challenges due to the presence of the
pumping and damping terms. Following a self-contained approach, we prove the
existence of ground state solutions of this equation under suitable
assumptions: This is equivalent to say that condensation occurs in these
situations. We also solve the Cauchy problem of the nonlinear Schroedinger
equation and prove some corresponding laws.
