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: March 2020

Revised: April 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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References

[1]	T. Bartsch and L. Jeanjean, Normalized solutions for nonlinear Schrödinger systems, P. Roy. Soc. Edinb. A, 148 (2018), 225-242. doi: 10.1017/S0308210517000087.
[2]	T. Bartsch, L. Jeanjean and N. Soave, Normalized solutions for a system of coupled cubic Schrödinger equations on $\mathbb{R}^3$, J. Math. Pure. Appl., 106 (2016), 583-614. doi: 10.1016/j.matpur.2016.03.004.
[3]	T. Bartsch and N. Soave, A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems, J. Funct. Anal., 272 (2017), 4998-5037. doi: 10.1016/j.jfa.2017.01.025.
[4]	T. Bartsch and N. Soave, Multiple normalized solutions for a competing system of Schrödinger equations, Calc. Var. Partial Differ. Equ., 58 (2019), 1-24. doi: 10.1007/s00526-018-1476-x.
[5]	H. Berestycki and P. L. Lions, Nonlinear scalar field equations, Ⅱ existence of infinitely many solutions, Arch. Ration. Mech. Anal., 82 (1983), 347-375. doi: 10.1007/BF00250556.
[6]	M. Du, L. Tian, J. Wang and F. Zhang, Existence of normalized solutions for nonlinear fractional Schrödinger equations with trapping potentials, P. Roy. Soc. Edinb. A, 149 (2018), 617-653. doi: 10.1017/prm.2018.41.
[7]	T. Gou and L. Jeanjean, Multiple positive normalized solutions for nonlinear Schrödinger systems, Nonlinearity, 31 (2018), 2319-2345. doi: 10.1088/1361-6544/aab0bf.
[8]	L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28 (1997), 1633-1659. doi: 10.1016/S0362-546X(96)00021-1.
[9]	L. Lu, $L^2$ normalized solutions for nonlinear Schrödinger systems in $\mathbb{R}^3$, Nonlinear Anal., 191 (2020), 1-19. doi: 10.1016/j.na.2019.111621.
[10]	N. V. Nguyen, On the orbital stability of solitary waves for the 2-coupled nonlinear Schrödinger system, Commun. Math. Sci., 9 (2011), 997-1012. doi: 10.4310/CMS.2011.v9.n4.a3.
[11]	N. V. Nguyen and Z. Q. Wang, Orbital stability of solitary waves for a nonlinear Schrödinger system, Adv. Differ. Equ., 16 (2011), 977-1000.
[12]	N. V. Nguyen and Z. Q. Wang, Orbital stability of solitary waves of a 3-coupled nonlinear Schrödinger system, Nonlinear Anal., 90 (2013), 1-26. doi: 10.1016/j.na.2013.05.027.
[13]	N. V. Nguyen and Z. Q. Wang, Existence and stability of a two-parameter family of solitary waves for a 2-coupled nonlinear Schrödinger system, Discrete Contin. Dyn. Syst., 36 (2015), 1005-1021. doi: 10.3934/dcds.2016.36.1005.
[14]	B. Noris, H. Tavares and G. Verzini, Stable solitary waves with prescribed $L^2$-mass for the cubic Schrödinger system with trapping potentials, Discrete Contin. Dyn. Syst., 35 (2015), 6085-6112. doi: 10.3934/dcds.2015.35.6085.
[15]	B. Noris, H. Tavares and G. Verzini, Normalized solutions for nonlinear Schrödinger systems on bounded domains, Nonlinearity, 32 (2019), 1044-1072. doi: 10.1088/1361-6544/aaf2e0.
[16]	M. Ohta, Stability of solitary waves for coupled nonlinear Schrödinger equations, Nonlinear Anal., 26 (1996), 933-939. doi: 10.1016/0362-546X(94)00340-8.
[17]	M. Willem, Minimax Theorems, Boston (1996). doi: 10.1007/978-1-4612-4146-1.