A constrained variational problem arising in attractive Bose-Einstein condensate with ellipse-shaped potential

We consider a minimization problem for the variational functional associated with a Gross-Pitaevskii equation arising in the study of an attractive Bose-Einstein condensate. Under an ellipse-shaped trapping potential, that is, the bottom of the trapping potential is an ellipse, we prove that any minimizer of the minimization problem blows up at one of the endpoints of the major axis of the ellipse if the parameter associated to the attractive interaction strength approaches a critical value.