doi: 10.3934/cpaa.2020229

Normalized solutions for 3-coupled nonlinear Schrödinger equations

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

Received  March 2020 Revised  April 2020 Published  July 2020

Fund Project: 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

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.
Citation: Chuangye Liu, Rushun Tian. Normalized solutions for 3-coupled nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2020229
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.  Google Scholar

[2]

T. BartschL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

M. DuL. TianJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

B. NorisH. 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.  Google Scholar

[15]

B. NorisH. Tavares and G. Verzini, Normalized solutions for nonlinear Schrödinger systems on bounded domains, Nonlinearity, 32 (2019), 1044-1072.  doi: 10.1088/1361-6544/aaf2e0.  Google Scholar

[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.  Google Scholar

[17]

M. Willem, Minimax Theorems, Boston (1996). doi: 10.1007/978-1-4612-4146-1.  Google Scholar

show all references

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.  Google Scholar

[2]

T. BartschL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

M. DuL. TianJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

B. NorisH. 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.  Google Scholar

[15]

B. NorisH. Tavares and G. Verzini, Normalized solutions for nonlinear Schrödinger systems on bounded domains, Nonlinearity, 32 (2019), 1044-1072.  doi: 10.1088/1361-6544/aaf2e0.  Google Scholar

[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.  Google Scholar

[17]

M. Willem, Minimax Theorems, Boston (1996). doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[1]

Mohamad Darwich. On the $L^2$-critical nonlinear Schrödinger Equation with a nonlinear damping. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2377-2394. doi: 10.3934/cpaa.2014.13.2377

[2]

Yanfang Gao, Zhiyong Wang. Minimal mass non-scattering solutions of the focusing L2-critical Hartree equations with radial data. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1979-2007. doi: 10.3934/dcds.2017084

[3]

Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119

[4]

Ruoci Sun. Filtering the $ L^2- $critical focusing Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (10) : 5973-5990. doi: 10.3934/dcds.2020255

[5]

Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. Global well-posedness for the $L^2$ critical nonlinear Schrödinger equation in higher dimensions. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1023-1041. doi: 10.3934/cpaa.2007.6.1023

[6]

Myeongju Chae, Sunggeum Hong, Sanghyuk Lee. Mass concentration for the $L^2$-critical nonlinear Schrödinger equations of higher orders. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 909-928. doi: 10.3934/dcds.2011.29.909

[7]

Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa. Global well-posedness of critical nonlinear Schrödinger equations below $L^2$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1389-1405. doi: 10.3934/dcds.2013.33.1389

[8]

Aliang Xia, Jianfu Yang. Normalized solutions of higher-order Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 447-462. doi: 10.3934/dcds.2019018

[9]

Kun Cheng, Yinbin Deng. Nodal solutions for a generalized quasilinear Schrödinger equation with critical exponents. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 77-103. doi: 10.3934/dcds.2017004

[10]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309

[11]

Mingwen Fei, Huicheng Yin. Nodal solutions of 2-D critical nonlinear Schrödinger equations with potentials vanishing at infinity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2921-2948. doi: 10.3934/dcds.2015.35.2921

[12]

Abdelwahab Bensouilah, Van Duong Dinh, Mohamed Majdoub. Scattering in the weighted $ L^2 $-space for a 2D nonlinear Schrödinger equation with inhomogeneous exponential nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2735-2755. doi: 10.3934/cpaa.2019122

[13]

Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807

[14]

Youngwoo Koh, Ihyeok Seo. Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4877-4906. doi: 10.3934/dcds.2017210

[15]

D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563

[16]

Yuxia Guo, Zhongwei Tang. Multi-bump solutions for Schrödinger equation involving critical growth and potential wells. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3393-3415. doi: 10.3934/dcds.2015.35.3393

[17]

Yinbin Deng, Yi Li, Xiujuan Yan. Nodal solutions for a quasilinear Schrödinger equation with critical nonlinearity and non-square diffusion. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2487-2508. doi: 10.3934/cpaa.2015.14.2487

[18]

Van Duong Dinh. On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 689-708. doi: 10.3934/cpaa.2019034

[19]

Miaomiao Niu, Zhongwei Tang. Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3963-3987. doi: 10.3934/dcds.2017168

[20]

Santosh Bhattarai. Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1789-1811. doi: 10.3934/dcds.2016.36.1789

2019 Impact Factor: 1.105

Article outline

[Back to Top]