A complete classification of ground-states for a coupled nonlinear Schrödinger system

Chuangye Liu; Zhi-Qiang Wang

doi:10.3934/cpaa.2017005

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2017, Volume 16, Issue 1: 115-130. Doi: 10.3934/cpaa.2017005

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A complete classification of ground-states for a coupled nonlinear Schrödinger system

Chuangye Liu^1, and
Zhi-Qiang Wang^2,3, ,

1.
School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, P.R. China
2.
Center for Applied Mathematics, Tianjin University, Tianjin 300072, P.R.China
3.
Department of Mathematics and Statistics, Utah State University, Logan UT 84322, USA

Author Bio: E-mail address: chuangyeliu1130@126.com
E-mail address: zhi-qiang.wang@usu.edu
E-mail address: zhi-qiang.wang@usu.edu

Received: May 31, 2016

Revised: July 31, 2016

Published: January 2018

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Abstract

In this paper, we establish the existence of nontrivial ground-state solutions for a coupled nonlinear Schrödinger system

$-\Delta u_j+ u_j=\sum\limits_{i=1}^mb_{ij}u_i^2u_j, \quad\text{in}\ \mathbb{R}^n,\\ u_j(x)\to 0\ \text{as}\ |x|\ \to \infty, \quad j=1,2,\cdots, m,$

where $n=1, 2, 3, m\geq 2$ and $b_{ij}$ are positive constants satisfying $b_{ij}=b_{ji}.$ By nontrivial we mean a solution that has all components non-zero. Due to possible systems collapsing it is important to classify ground state solutions. For $m=3$, we get a complete picture that describes whether nontrivial ground-state solutions exist or not for all possible cases according to some algebraic conditions of the matrix $B = (b_{ij})$. In particular, there is a nontrivial ground-state solution provided that all coupling constants $b_{ij}, i\neq j$ are sufficiently large as opposed to cases in which any ground-state solution has at least a zero component when $b_{ij}, i\neq j$ are all sufficiently small. Moreover, we prove that any ground-state solution is synchronized when matrix $B=(b_{ij})$ is positive semi-definite.

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Mathematics Subject Classification: Primary: 35J20, 35J47; Secondary: 35J50.

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Table 1. The number of non-zero components of ground-state solutions

case	condition 1	condition 2	type
1	$det(B) > 0$	$\beta_1 > \sqrt{\mu_2\mu_3}, \beta_2 > \sqrt{\mu_1\mu_3}, \beta_3 > \sqrt{\mu_1\mu_2}$	$ p= 3 $
2	$det(B) > 0$	$\beta_1 < \sqrt{\mu_2\mu_3}, \beta_2 < \sqrt{\mu_1\mu_3}, \beta_3 < \sqrt{\mu_1\mu_2}$	$p=1$
3	$det(B) < 0$		$p=1, 2$
4	$rank(B)=1$		$p=1, 2, 3$
5	$rank(B)=2$	$\beta_1 > \sqrt{\mu_2\mu_3}, \beta_2 > \sqrt{\mu_1\mu_3}, \beta_3=\sqrt{\mu_1\mu_2}$	$p=2, 3$
6	$rank(B)=2$	$\beta_1 < \sqrt{\mu_2\mu_3}, \beta_2 < \sqrt{\mu_1\mu_3}, \beta_3=\sqrt{\mu_1\mu_2}$	$p=1, 2$
7	$rank(B)=2$	$\beta_1 > \sqrt{\mu_2\mu_3}, \beta_2 > \sqrt{\mu_1\mu_3}, \beta_3 > \sqrt{\mu_1\mu_2}$	$p=1, 2, 3$
8	$rank(B)=2$	$\beta_1 < \sqrt{\mu_2\mu_3}, \beta_2 < \sqrt{\mu_1\mu_3}, \beta_3 < \sqrt{\mu_1\mu_2}$	$p=1$

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