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 $0<\Omega^*:=\Omega^*(V)\leq \infty$, depending on the general trap $V(x)$, such that for any rotational velocity $0\leq \Omega <\Omega ^*$, 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 $\Delta w-w+w^3=0$ in $\mathbb{R}^2$. If $V(x)=|x|^2$ 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 $a \nearrow a^*$, by analyzing the refined limit behavior of minimizers and employing the non-degenerancy of $w$.