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 Omega. This model can be described by the complex-valued Gross-Pitaevskii energy functional. It is shown that there exists a critical rotational velocity 0<Omega*:=Omega*(V)<=infinity, depending on the general trap V(x), such that for any rotational velocity 0 <=Omega<Omega*, minimizers (i.e., ground states) exist if and only if a<a* =parallel to w parallel to|(2)(2) a, 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 Delta w-w+w(3)=0 in R-2. If V(x)=vertical bar x vertical bar(2) and 0<omega<omega*(=2) is fixed, we prove that, up to a constant phase, all minimizers must be real-valued, unique and free of vortices as a, NE arrow a* by analyzing the refined limit behavior of minimizers and employing the non-degenerancy of w.