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arXiv 1502.01839math.AP
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具有多阱势的吸引Gross-Pitaevskii方程的基态性质

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

Yujin Guo, Zhi-Qiang Wang, Xiaoyu Zeng, Huan-Song Zhou

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23 pages
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摘要：

我们对$\R^2$中的吸引Gross-Pitaevskii（GP）方程感兴趣，其中外势$V(x)$在$m$不相交的有界域$Ω_i\subset \R^2\ (i=1,2,\cdots,m)$和$V(x)\to\infty$上为零，为$|x|\to\infty$，即这些$Ω_i$的并集是势阱的底部.通过对GP方程的能量泛函进行精细的估计，证明了当相互作用强度$a$接近某个临界值$a^*$时，基态在某个最大内径的$Ω_j$的内圆中心集中并爆炸.在进一步的条件下，我们证明了GP方程的基态是唯一的，并且对几乎每一个$V(x)$至少是径向对称的.

英文摘要：

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 $Ω_i\subset \R^2\ (i=1,2,\cdots,m)$ and $V(x)\to\infty$ as $|x|\to\infty$, 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 $a \in (0, a^*)$.

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