The Nonexistence of Vortices for Rotating Bose–Einstein Condensates with Attractive Interactions

This article is devoted to studying the model of two-dimensional attractive Bose–Einstein condensates in a trap V ( x ) rotating at the velocity Ω . This model can be described by the complex-valued Gross–Pitaevskii energy functional. It is shown that there exists a critical rotational velocity 0 < Ω ∗ : = Ω ∗ ( V ) ≤ ∞ , depending on the general trap V ( x ), such that for any rotational velocity 0 ≤ Ω < Ω ∗ , minimizers (i.e., ground states) exist if and only if a < a ∗ = ‖ w ‖ 2 2 , where a > 0 denotes the absolute product for the number of particles times the scattering length, and w > 0 is the unique positive solution of Δ w - w + w 3 = 0 in R 2 . If V ( x ) = | x | 2 and 0 < Ω < Ω ∗ ( = 2 ) is fixed, we prove that, up to a constant phase, all minimizers must be real-valued, unique and free of vortices as a ↗ a ∗ , by analyzing the refined limit behavior of minimizers and employing the non-degenerancy of w .