Properties of ground states of attractive Gross–Pitaevskii equations with multi-well potentials

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 making some delicate estimates on the 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 which has the largest inradius. Moreover, under some further conditions on V (x) we show that the ground states of GP equations are unique and radially symmetric at leat for almost every a ∈ (0, a * ).

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

“…On the other hand, the local uniqueness of positive minimizers for e(a) as a.e. a ր a * was also proved [11] by the ODE argument, for the case where V (r) = V (|x|) is radially symmetric and satisfies V ′ (r) ≥ 0, see Corollary 1.1 in [11] for details. Here the locality of uniqueness means that a is near a * .…”

“…In this paper positive minimizers of e(a) are called ground states of attractive BEC. Applying energy estimates and blow-up analysis, the spike profiles of positive minimizers for e(a) as a ր a * were recently discussed in [10,11,12] under different types of potentials V (x), see our Proposition 2.1 for some related results. In spite of these facts, it remains open to discuss the refined spike profiles of positive minimizers.…”

“…In Section 4 we shall extend the refined spike behavior of Theorem 1.2 to more general potentials V (x) = g(x)h(x), where h(−x) = h(x) is homogeneous and satisfies (1.14) and 0 ≤ C ≤ g(x) ≤ 1 C holds in R 2 , see Theorem 4.4 for details. To establish Theorem 1.2 and Theorem 4.4, our Proposition 2.1 shows that the arguments of [10,11,12] give the leading expansion terms of the minimizer u a and the associated Lagrange multiplier µ a satisfying (1.6) as well. In order to get (1.17) for the rest terms of u a , the difficulty is to obtain the more precise estimate of µ a , which is overcome by the very delicate analysis of the associated equation (1.6), together with the constraint condition of u a .…”

“…As addressed recently in [10,11], ground states of attractive BEC in R 2 can be described by the constraint minimizers of the GP energy e(a) := inf {u∈H, u 2 2 =1}…”

“…The minimization problem e(a) was analyzed recently in [2,10,11,12,26] and references therein. Existing results show that e(a) is an L 2 −critical constraint variational problem.…”