Under an ellipse-shaped potential, that is, the bottom of the trapping potential V(x) is an ellipse, we consider a minimization problem for the energy functional associated with a Gross-Pitaevskii equation in bounded domains \(\Omega \subset \mathbb R^2\) , which arises in the study of an attractive Bose-Einstein condensate. It has been shown that there exists a constant \(\beta ^{*}>0\) such that minimizers exist if and only if \(0<\beta <\beta ^{*}\) . In the present paper, on one hand, we prove that if the interior of \(\Omega \) contains the endpoints of the major axis of the ellipse-shaped bottom, the mass of minimizers must concentrate at an inner point of \(\Omega \) as \(\beta \nearrow \beta ^{*}\) . On the other hand, if all endpoints of major axis of the ellipse-shaped bottom locate at the boundary of \(\Omega \) , we prove that the mass of minimizers must concentrate near the boundary of \(\Omega \) as \(\beta \nearrow \beta ^{*}\) .
