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 ${\mathbb{R}}^2$, where the external potential $V(x)$ vanishes on $m$ disjoint bounded domains ${\Omega }_i\subset {\mathbb{R}}^2(i=1,2,\cdots ,m)$ and $V(x)\to \infty$ as $|x|\to \infty$, that is, the union of these ${\Omega }_i$ 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 $a^{\ast }$ the ground states concentrate and blow up at the center of the incircle of some ${\Omega }_j$ 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 $a\in (0,a^{\ast })$.

arXiv comment23 pages

References in corpus (3)

• Theory of Bose-Einstein condensation in trapped gases
• Collapse and Bose-Einstein condensation in a trapped Bose-gas with negative scattering length
• Symmetry-breaking bifurcation in the nonlinear Schrödinger equation with symmetric potentials

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• A constrained minimization problem related to two coupled pseudo-relativistic Hartree equations
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• Local Uniqueness and Refined Spike Profiles of Ground States for Two-Dimensional Attractive Bose-Einstein Condensates