Positive solutions of doubly coupled multicomponent nonlinear Schrödinger systems

Jiabao Su; Rushun Tian; Zhi-Qiang Wang

doi:10.3934/dcdss.2019138

Article Contents

2019, Volume 12, Issue 7: 2143-2161. Doi: 10.3934/dcdss.2019138

This issue Previous Article On periodic solutions in the Whitney's inverted pendulum problem Next Article Ground state homoclinic solutions for a second-order Hamiltonian system

Positive solutions of doubly coupled multicomponent nonlinear Schrödinger systems

1.
School of Mathematical Science, Capital Normal University, Beijing 10048, China
2.
Center for Applied Mathematics, Tianjin University, Tianjin 300072, China
3.
Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA

^* Corresponding author: Rushun Tian and Zhi-Qiang Wang
^* Corresponding author: Rushun Tian and Zhi-Qiang Wang

Received: October 31, 2017

Revised: March 31, 2018

Published: June 2019

This paper is supported by Beijing Natural Science Foundation (1174013), National Natural Science Foundation of China (11601353, 11771302, 11771324, 11671026, 11831009).

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Abstract

In this paper, we study the following doubly coupled multicomponent system

$ \begin{equation*} \left\{\begin{array}{ll} -\Delta u_j + \lambda_ju_j+ \sum_{k\neq j}\gamma_{jk}u_k = \mu_ju_j^3+ u_j\sum_{k\neq j}\beta_{jk}u_k^2,\\ u_j(x)\geq0\ \ \hbox{and}\ \ u_j\in H_0^1(\Omega), \end{array} \right. \end{equation*} $

where $ \Omega\subset \mathbb{R} ^N $ and $ N = 2,3 $; $ \lambda_j, \gamma_{jk} = \gamma_{kj}, \mu_j, \beta_{jk} = \beta_{kj} $ are constants, $ j, k = 1, 2, ..., n $, $ n\geq 2 $. We prove some existence and nonexistence results for positive solutions of this system. If the system is fully symmetric, i.e. $ \lambda_j\equiv\lambda, \gamma_{jk}\equiv\gamma, \mu_j\equiv\mu, \beta_{jk}\equiv\beta $, we study the multiplicity and bifurcation phenomena of positive solution.

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Mathematics Subject Classification: 35B05, 35J61, 58C40, 35J15, 58E07.

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