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October  2019, 39(10): 6069-6102. doi: 10.3934/dcds.2019265

Infinitely many segregated solutions for coupled nonlinear Schrödinger systems

Lushun Wang , , Minbo Yang and  Yu Zheng

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

* Corresponding author: Lushun Wang

Received  February 2019 Revised  April 2019 Published  July 2019

Fund Project: The second author is partially supported by NSFC (11571317, 11671364) and ZJNSF (LD19A010001).

In this paper, we consider the following coupled nonlinear Schrödinger system
$ \left\{\begin{array}{ll} -\Delta u+(1+\delta a(x))u = \mu_1 u^3+\beta uv^2 &\mbox{in }\mathbb{R}^3,\\ -\Delta v+(1+\delta b(x))v = \mu_2 v^3+\beta u^2v &\mbox{in }\mathbb{R}^3,\\ u\to 0,\quad v\to 0, &\mbox{as } |x|\to\infty \end{array}\right. $
where
$ \mu_1>0 $
,
$ \mu_2>0 $
,
$ \beta\in\mathbb{R} $
,
$ \delta\in\mathbb{R} $
, and
$ a(x) $
 and
$ b(x) $
 are two
$ C^\alpha $
 potentials with
$ 0<\alpha<1 $
, satisfying some slow decay assumptions, but do not need to fulfill any symmetry property. Using the Lyapunov–Schmidt reduction method and some variational techniques, we show that there exist
$ 0<\delta_0<1 $
 and
$ 0<\beta_0<\min\{\mu_1,\mu_2\} $
 such that the above system has infinitely many positive segregated solutions for any
$ 0<\delta<\delta_0 $
 and
$ 0<\beta<\beta_0 $
.
Keywords: Schrödinger system, Lyapunov–Schmidt reduction, segregated solutions.
Mathematics Subject Classification: Primary: 35J60, 35J75; Secondary: 35J80.
Citation: Lushun Wang, Minbo Yang, Yu Zheng. Infinitely many segregated solutions for coupled nonlinear Schrödinger systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6069-6102. doi: 10.3934/dcds.2019265
References:
[1]

W. Ao and J. Wei, Infinitely many positive solutions for nonlinear equations with non-symmetric potentials, Calc. Var., 51 (2014), 761-798.  doi: 10.1007/s00526-013-0694-5.  Google Scholar

[2]

W. Ao, J. Wei and J. Zeng, An optimal bound on the number of interior peak solutions for the Lin-Ni-Takagi problem, J. Funct. Anal., 265 (2013), 1324-1356.  doi: 10.1016/j.jfa.2013.06.016.  Google Scholar

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A. Bahri and Y. Li, On a minimax produre for the existence of a positive solution for certain scaler field equation in $ {\Bbb R}^n$, Revista Mat. Iberoa., 6 (1990), 1-15.  doi: 10.4171/RMI/92.  Google Scholar

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T. Bartsch, N. Dancer and Z. Wang, A Liouville theorem, a priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var., 37 (2010), 345-361.  doi: 10.1007/s00526-009-0265-y.  Google Scholar

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T. Bartsch, Z. Wang and J. Wei, Bound states for a couple Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.  doi: 10.1007/s11784-007-0033-6.  Google Scholar

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G. Cerami, D. Passaseo and S. Solimini, Infinitely many positive solutions to some scalar field equations with nonsymmetric coefficients, Comm. Pure Appl. Math., 66 (2013), 372-413.  doi: 10.1002/cpa.21410.  Google Scholar

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E. N. Dancer, J. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953-969.  doi: 10.1016/j.anihpc.2010.01.009.  Google Scholar

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B. D. Esry, C. H. Greene, J. P. Burker Jr. and J. L. Bohn, Hartee-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3584-3597.   Google Scholar

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B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $ {\Bbb R}^n$, Advances in Math., Supplementary Studies, 7A (1981), 369-402.   Google Scholar

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N. Hirano, Multiple existence of nonradial positive solutions for a coupled nonlinear Schrödinger system, NoDEA Nonlinear Diff. Eq. Appl., 16 (2009), 159-188.  doi: 10.1007/s00030-008-7047-7.  Google Scholar

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F. Lin, W. Ni and J. Wei, On the number of interior peak solutions for singularly pertubered Neumann problem, Comm. Pure Appl. Math., 60 (2007), 252-281.  doi: 10.1002/cpa.20139.  Google Scholar

[12]

J. Liu, X. Liu and Z. Wang, Multiple mixed states of nodal solutions for nonlinear Schrödinger systems, Calc. Var., 52 (2015), 565-586.  doi: 10.1007/s00526-014-0724-y.  Google Scholar

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Z. Liu and Z. Wang, Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Stud., 10 (2010), 175-193.  doi: 10.1515/ans-2010-0109.  Google Scholar

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R. Mandel, Minimal energy solutions for cooperative nonlinear Schrödinger systems, NoDEA Nonlinear Diff. Eq. Appl., 22 (2015), 239-262.  doi: 10.1007/s00030-014-0281-2.  Google Scholar

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S. Peng and Z. Wang, Segregated and synchronized vector solutions for nonlinear Schrödinger systems, Arch. Rational Mech. Anal., 208 (2013), 305-339.  doi: 10.1007/s00205-012-0598-0.  Google Scholar

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B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $ {\Bbb R}^n$, Comm. Math. Phys., 271 (2007), 199-221.  doi: 10.1007/s00220-006-0179-x.  Google Scholar

[17]

N. Soave, On existence and phase separation of solitary waves for nonlinear Schrödinger systems modelling simultaneous cooperation and competition, Calc. Var., 53 (2015), 689-718.  doi: 10.1007/s00526-014-0764-3.  Google Scholar

[18]

Y. Sato and Z. Wang, On the multiple existence of semi-positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 1-22.  doi: 10.1016/j.anihpc.2012.05.002.  Google Scholar

[19]

R. Tian and Z. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223.   Google Scholar

[20]

J. Wei and M. Winter, Mathematical Aspects of Pattern Formation in Biological Systems, Applied mathematical Science, 189, Springer, London, 2014. doi: 10.1007/978-1-4471-5526-3.  Google Scholar

[21]

J. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Commun. Pure App. Ana., 11 (2012), 1003-1011.  doi: 10.3934/cpaa.2012.11.1003.  Google Scholar

show all references

References:
[1]

W. Ao and J. Wei, Infinitely many positive solutions for nonlinear equations with non-symmetric potentials, Calc. Var., 51 (2014), 761-798.  doi: 10.1007/s00526-013-0694-5.  Google Scholar

[2]

W. Ao, J. Wei and J. Zeng, An optimal bound on the number of interior peak solutions for the Lin-Ni-Takagi problem, J. Funct. Anal., 265 (2013), 1324-1356.  doi: 10.1016/j.jfa.2013.06.016.  Google Scholar

[3]

A. Bahri and Y. Li, On a minimax produre for the existence of a positive solution for certain scaler field equation in $ {\Bbb R}^n$, Revista Mat. Iberoa., 6 (1990), 1-15.  doi: 10.4171/RMI/92.  Google Scholar

[4]

T. Bartsch, N. Dancer and Z. Wang, A Liouville theorem, a priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var., 37 (2010), 345-361.  doi: 10.1007/s00526-009-0265-y.  Google Scholar

[5]

T. Bartsch, Z. Wang and J. Wei, Bound states for a couple Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.  doi: 10.1007/s11784-007-0033-6.  Google Scholar

[6]

G. Cerami, D. Passaseo and S. Solimini, Infinitely many positive solutions to some scalar field equations with nonsymmetric coefficients, Comm. Pure Appl. Math., 66 (2013), 372-413.  doi: 10.1002/cpa.21410.  Google Scholar

[7]

E. N. Dancer, J. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953-969.  doi: 10.1016/j.anihpc.2010.01.009.  Google Scholar

[8]

B. D. Esry, C. H. Greene, J. P. Burker Jr. and J. L. Bohn, Hartee-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3584-3597.   Google Scholar

[9]

B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $ {\Bbb R}^n$, Advances in Math., Supplementary Studies, 7A (1981), 369-402.   Google Scholar

[10]

N. Hirano, Multiple existence of nonradial positive solutions for a coupled nonlinear Schrödinger system, NoDEA Nonlinear Diff. Eq. Appl., 16 (2009), 159-188.  doi: 10.1007/s00030-008-7047-7.  Google Scholar

[11]

F. Lin, W. Ni and J. Wei, On the number of interior peak solutions for singularly pertubered Neumann problem, Comm. Pure Appl. Math., 60 (2007), 252-281.  doi: 10.1002/cpa.20139.  Google Scholar

[12]

J. Liu, X. Liu and Z. Wang, Multiple mixed states of nodal solutions for nonlinear Schrödinger systems, Calc. Var., 52 (2015), 565-586.  doi: 10.1007/s00526-014-0724-y.  Google Scholar

[13]

Z. Liu and Z. Wang, Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Stud., 10 (2010), 175-193.  doi: 10.1515/ans-2010-0109.  Google Scholar

[14]

R. Mandel, Minimal energy solutions for cooperative nonlinear Schrödinger systems, NoDEA Nonlinear Diff. Eq. Appl., 22 (2015), 239-262.  doi: 10.1007/s00030-014-0281-2.  Google Scholar

[15]

S. Peng and Z. Wang, Segregated and synchronized vector solutions for nonlinear Schrödinger systems, Arch. Rational Mech. Anal., 208 (2013), 305-339.  doi: 10.1007/s00205-012-0598-0.  Google Scholar

[16]

B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $ {\Bbb R}^n$, Comm. Math. Phys., 271 (2007), 199-221.  doi: 10.1007/s00220-006-0179-x.  Google Scholar

[17]

N. Soave, On existence and phase separation of solitary waves for nonlinear Schrödinger systems modelling simultaneous cooperation and competition, Calc. Var., 53 (2015), 689-718.  doi: 10.1007/s00526-014-0764-3.  Google Scholar

[18]

Y. Sato and Z. Wang, On the multiple existence of semi-positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 1-22.  doi: 10.1016/j.anihpc.2012.05.002.  Google Scholar

[19]

R. Tian and Z. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223.   Google Scholar

[20]

J. Wei and M. Winter, Mathematical Aspects of Pattern Formation in Biological Systems, Applied mathematical Science, 189, Springer, London, 2014. doi: 10.1007/978-1-4471-5526-3.  Google Scholar

[21]

J. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Commun. Pure App. Ana., 11 (2012), 1003-1011.  doi: 10.3934/cpaa.2012.11.1003.  Google Scholar

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