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Partly clustering solutions of nonlinear Schrödinger systems with mixed interactions
Journal of Functional Analysis ( IF 1.7 ) Pub Date : 2021-03-17 , DOI: 10.1016/j.jfa.2021.108987
Jaeyoung Byeon , Youngae Lee , Sang-Hyuck Moon

In this paper, we prove a partly clustering phenomenon for nonlinear Schrödinger systems with large mixed couplings of attractive and repulsive forces, which arise from the models in Bose-Einstein condensates and nonlinear optics. More precisely, we consider a system with three components where the interaction between the first two components and the third component is repulsive, and the interaction between the first two components is attractive. Recent studies [10], [11], [12], [13] in this case show that for large interaction forces, the first two components are localized in a region with a small energy and the third component is close to a solution of a single equation. Especially, the results in the works [12], [13] say that the region of localization for a (locally) least energy vector solution on a ball in the class of radially symmetric functions is the origin or the whole boundary depending on the space dimension 1n3. In this paper we construct a new type of solutions with a region of localization different from the origin or the whole boundary. In fact, we show that there exist radially symmetric positive vector solutions with clustering multi-bumps for the first two components near the maximum point of rn1U3, where U is the limit of the third component and the maximum point is the only critical point different from the origin and the boundary.



中文翻译:

具有混合相互作用的非线性Schrödinger系统的部分聚类解决方案

在本文中,我们证明了具有吸引力和排斥力的大型混合耦合的非线性Schrödinger系统的部分聚集现象,这是由Bose-Einstein凝聚物和非线性光学模型引起的。更准确地说,我们考虑一个具有三个组件的系统,其中前两个组件和第三个组件之间的交互是排斥的,而前两个组件之间的交互是有吸引力的。最近的研究[10],[11],[12],[13]在这种情况下表明,对于较大的相互作用力,前两个分量位于能量较小的区域,而第三个分量接近于一个方程式。尤其是作品的成果[12],1个ñ3。在本文中,我们构造了一种新型的解决方案,其局部区域不同于原点或整个边界。实际上,我们表明存在径向对称的正矢量解,并且对于前两个分量的最大点附近存在聚类的多峰聚类。[Rñ-1个ü3,其中U是第三个分量的极限,最大点是唯一不同于原点和边界的临界点。

更新日期:2021-03-22
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