Normalized solutions for 3-coupled nonlinear Schrödinger equations

Chuangye Liu; Rushun Tian

doi:10.3934/cpaa.2020229

Article Contents

2020, Volume 19, Issue 11: 5115-5130. Doi: 10.3934/cpaa.2020229

This issue Previous Article A new family of boundary-domain integral equations for the diffusion equation with variable coefficient in unbounded domains Next Article Uniform stabilization of the Klein-Gordon system

Normalized solutions for 3-coupled nonlinear Schrödinger equations

Chuangye Liu^1, and
Rushun Tian^2, ,

1.
School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China
2.
School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

^* Corresponding author
^* Corresponding author

Received: February 29, 2020

Revised: March 31, 2020

Early access: July 2020

Published: November 2020

The first author is supported by the Fundamental Research Funds for the Central Universities CCNU19QN079, NSFC 11971191. The second author is supported by KZ202010028048, NSFC 11771302, 11601353

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Abstract

In this paper, we study the existence of $ L^2 $-normalized solutions for the following 3-coupled nonlinear Schrödinger equations in $ [H_r^1( \mathbb{R}^N)]^3 $,

$ \begin{equation*} \begin{cases} -\Delta u_i = \lambda_i u_i+\mu_i|u_i|^{p_i-2}u_i+\beta r_i|u_i|^{r_i-2}\big(\sum\limits_{j\neq i}|u_j|^{r_j}\big)u_i,\\ |u_i|_2^2 = a_i, \quad i, j = 1,2,3, \end{cases} \end{equation*} $

where $ \mu_i, \beta $ and $ a_i $ are given positive constants, $ \lambda_i $ appear as unknown parameters, and $ H_r^1( \mathbb{R}^N) $ denotes the radial subspace of Hilbert space $ H^1( \mathbb{R}^N) $. For $ p_i, r_i $ satisfying $ L^2 $-subcritical or $ L^2 $-supercritical conditions, we obtain positive solutions of this system using variational methods and perturbation methods.

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Mathematics Subject Classification: Primary: 35J50, 35Q55.

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