Existence of the normalized solutions to the nonlocal elliptic system with partial confinement

Jun Wang; Qiuping Geng; Maochun Zhu

doi:10.3934/dcds.2019092

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2019, Volume 39, Issue 4: 2187-2201. Doi: 10.3934/dcds.2019092

This issue Previous Article Exponential stabilization of the stochastic Burgers equation by boundary proportional feedback Next Article NLS bifurcations on the bowtie combinatorial graph and the dumbbell metric graph

Existence of the normalized solutions to the nonlocal elliptic system with partial confinement

Institute of Applied System Analysis, Jiangsu University, Zhenjiang, Jiangsu 212013, China

^* Corresponding author: Jun Wang
^* Corresponding author: Jun Wang

Received: June 2018

Revised: September 2018

Published: January 2019

This work was supported by NSFC grants: 11571140, 11671077, 11601190, Fellowship of Outstanding Young Scholars of Jiangsu Province(BK20160063), the Six big talent peaks project in Jiangsu Province(XYDXX-015), and NSF of Jiangsu Province Grants: BK20150478, BK20160483, and Jiangsu University Foundation Grant 16JDG043

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Abstract

In present paper we study the existence and orbital stability of the standing waves to the nonlocal elliptic system with partial confinement. This type equations arises from the basic quantum chemistry model of small number of electrons interacting with static nucleii. On the one hand, we prove the existence of global minimizer of the associate energy functional subject to the $ L^2 $-constraint. On the other hand, we discuss the orbital stability of the global minimizers. Comparing to the local case, we need establish the new inequality related to the Steiner rearrangement.

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Mathematics Subject Classification: Primary: 35J61, 35J20, 35Q55; Secondary: 49J40.

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