Abstract. We study ground states of two-dimensional Bose–Einstein condensates with repulsive ([Formula: see text]) or attractive ([Formula: see text]) interactions in a trap [Formula: see text] rotating at velocity [Formula: see text]. It is known that there exist critical parameters [Formula: see text] and [Formula: see text] such that if [Formula: see text], then there is no ground state for any [Formula: see text]; if [Formula: see text], then ground states exist if and only if [Formula: see text]. As a completion of the existing results, in this paper, we focus on the critical case where [Formula: see text] to classify the existence and nonexistence of ground states for any [Formula: see text]. Moreover, for a suitable class of radially symmetric traps [Formula: see text], employing the inductive symmetry method, we prove that up to a constant phase, ground states must be real valued, unique, and free of vortices as [Formula: see text], no matter whether the interactions of the condensates are repulsive or not.
We investigate the formation of vortices in quasi-two-dimensional dipolar Bose-Einstein condensates (BECs) through the interplay between two-body contact and long-ranged dipole-dipole interactions (DDIs), as both interactions can be tuned from repulsive to attractive. By solving the associated Gross-Pitaevskii equation for a rotating system, our initial approach concentrates on stabilizing a collapsing condensate with attractive
s-wave two-body interactions by employing sufficiently large repulsive DDIs. Subsequently, the same procedure was applied after reversing the signs of both interactions to evaluate the sensitivity of vortex formation to such an interchange of interactions. As a reference to guide our investigation, valid for generic dipolar atomic species, we have assumed a condensate with the strong dipolar dysprosium isotope,
164Dy. The correlation of the results with other dipolar BEC systems was exemplified by considering rotating BECs with two other isotopes, namely
168Er and
52Cr. For a purely dipolar condensate (no contact interactions) under fixed rotation, we show how the number of visible vortices increases as the DDI becomes more repulsive, accomplished by tuning the orientation of the dipoles through a characteristic angle parameter.