Properties of ground states of attractive Gross–Pitaevskii equations with multi-well potentials. (9th February 2018)

• Record Type: Journal Article
• Authors: Guo, Yujin; Wang, Zhi-Qiang; Zeng, Xiaoyu; Zhou, Huan-Song
• Is Part Of: Nonlinearity. Volume 31:Number 3(2018:Mar.)
• Page Start: 957
• Page End: 979
• Publication Date: 2018-02-09
• Subjects: ground states -- constrained variational problem -- Gross–Pitaevskii equation -- multi-well potential; 36J20 -- 35J60 -- 35Q40; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/aa99a8
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: We are interested in the attractive Gross–Pitaevskii (GP) equation in R 2, where the external potential V ( x ) vanishes on m disjoint bounded domains Ω i ⊂ R 2 ( i = 1, 2, ⋯, m ) and V ( x ) → ∞ as | x | → ∞, that is, the union of these Ω i is the bottom of the potential well. By establishing some delicate estimates on the associated energy functional of the GP equation, we prove that when the interaction strength a approaches some critical value a ∗, the ground states concentrate and blow up at the center of the incircle of some Ω j with the largest inradius. Moreover, under some further conditions on V ( x ), we show that the ground states of the GP equations are unique and radially symmetric at least for almost every a ∈ ( 0, a ∗ ) .