November  2020, 19(11): 5115-5130. 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  November 2020 Early access  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 and Applied Analysis, 2020, 19 (11) : 5115-5130. 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.

[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.

[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. 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.

[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. 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.

[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.

[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.

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.

[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.

[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. 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.

[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. 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.

[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.

[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.

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