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We study the Bose–Einstein condensates with attractive interaction in a bounded domain Ω ⊂ R 2 , described by Gross–Pitaevskii energy functional. We prove that there exists a constant a ∗ > 0 , such that minimizers exist if and only if 0 < a < a ∗ . As a ↗ a ∗ , the limit behavior of minimizers is also analyzed, where the mass must concentrate at the global minimum points of the trapping potential V (x). In particular, if V (x) has the flattest global minimum points both at the interior and the boundary of Ω, we prove that the mass must concentrate at an inner minimum point of Ω, rather than the neighborhood of the boundary minimum points. [ABSTRACT FROM AUTHOR]

This paper is concerned with constraint minimizers of an L2-critical minimization problem (1) in RN (N ≥ 1) under an L2-subcritical perturbation. We prove that the problem admits minimizers with mass ρN/2 if and only 4 if 0 ≤ ρ < ρ* := ‖Q‖24/N for b ≥ 0 and 0 < ρ ≤ ρ* for b < 0, where the constant b comes from the coefficient of the perturbation term, and Q is the unique positive radically symmetric solution of Δu(x) - u(x) + u1+ 4/N (x) = 0 in RN . Furthermore, we analyze rigorously the concentration behavior of minimizers as ρ /' ρ* for the case where b > 0, which shows that the concentration rates are determined by the subcritical perturbation, instead of the local profiles of the potential V (x). [ABSTRACT FROM AUTHOR]