Local Uniqueness and Refined Spike Profiles of Ground States for Two-Dimensional Attractive Bose-Einstein Condensates

We consider ground states of two-dimensional Bose-Einstein condensates in a trap with attractive interactions, which can be described equivalently by positive minimizers of the L−critical constraint Gross-Pitaevskii energy functional. It is known that ground states exist if and only if a < a∗ := ∥w∥2, where a denotes the interaction strength and w is the unique positive solution of ∆w − w + w = 0 in R. In this paper, we prove the local uniqueness and refined spike profiles of ground states as a ↗ a∗, provided that the trapping potential h(x) is homogeneous and H(y) = ∫ R2 h(x+ y)w (x)dx admits a unique and non-degenerate critical point.