Pérez García, Victor M.. 

We study the generation of vortices in rotating Bose-Einstein condensates, a situation that has been realized in a recent experiment [K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Phys. Rev. Lett. 84 806 (2000)]. By combining a linear stability analysis with the global optimization of the nonlinear free-energy functional, we study the regimes that can be reached in current experiments. We find a hysteresis phenomenon in the vortex nucleation due to the metastabilization of the vortexless condensate. We also prove that for a fast enough rotating trap, the ground state of the condensate hosts one or more bent vortex lines.