February  2021, 20(2): 867-884. doi: 10.3934/cpaa.2020294

Multiple positive solutions for coupled Schrödinger equations with perturbations

1. 

Center for Applied Mathematics, Tianjin University, Tianjin 300072, China

2. 

College of Mathematics and Informatics, Fujian Normal University, Fuzhou 350117, China

3. 

Department of Mathematics and Statistics, Utah State University, Logan, Utah 84322, USA

* Corresponding author

Dedicated to Jiaquan Liu with admiration on the occasion of his 75th birthday

Received  September 2020 Revised  October 2020 Published  February 2021 Early access  December 2020

Fund Project: This research is supported by CNSF(11771324, 11831009, 11811540026)

For coupled Schrödinger equations with nonhomogeneous perturbations we give several results on the existence of multiple positive solutions. In particular in one case we consider perturbations of the permutation symmetry.

Citation: Haoyu Li, Zhi-Qiang Wang. Multiple positive solutions for coupled Schrödinger equations with perturbations. Communications on Pure & Applied Analysis, 2021, 20 (2) : 867-884. doi: 10.3934/cpaa.2020294
References:
[1]

N. Ackermann and T. Bartsch, Superstable manifolds of semilinear parabolic problems, J. Dyn. Differ. Equ., 17 (2005), 115-173.  doi: 10.1007/s10884-005-3144-z.  Google Scholar

[2]

S. Adachi and K. Tanaka, Four positive solutions for the semilinear elliptic equation: $-\Delta u+u = a(x)u^p+f(x)$ in $\mathbb{R}^N$, Calc. Var. Partial Differ. Equ., 11 (2000), 63-95.  doi: 10.1007/s005260050003.  Google Scholar

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S. Alarcón, Multiple solutions for a critical nonhomogeneous elliptic problem in domains with small holes, Commun. Pure Appl. Anal., 8 (2009), 1269-1289.  doi: 10.3934/cpaa.2009.8.1269.  Google Scholar

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A. Bahri and H. Berestycki, Existence of forced oscillations for some nonlinear differential equations, Commun. Pure Appl. Math., 37 (1984), 403-442.  doi: 10.1002/cpa.3160370402.  Google Scholar

[5]

A. Bahri and P. L. Lions, Morse index of some min-max critical points. I. Application to multiplicity results, Commun. Pure Appl. Math., 41 (1988), 1027-1037.  doi: 10.1002/cpa.3160410803.  Google Scholar

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N. P. Bhatia and G. P. Szegö, Dynamical Systems: Stability Theory and Applications, Springer, Berlin, Heidelberg, 1967. doi: 10.1007/BFb0080630.  Google Scholar

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D. M. Cao and H. S. Zhou, Multiple positive solutions of nonhomogeneous semilinear elliptic equations in $\mathbb{R}^N$, Commun. Pure Appl. Math., 41 (1988), 1027-1037.  doi: 10.1017/S0308210500022836.  Google Scholar

[8]

T. Cazenave and P.L. Lions, Solutions globales d'equations de la chaleur semi lineaire, Commun. Partial Differ. Equ., 9 (1984), 955-978.  doi: 10.1080/03605308408820353.  Google Scholar

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K. C. Chang, Infinite-dimensional Morse Theory and Multiple Solution Problems, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0385-8.  Google Scholar

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K. C. Chang, Heat method in nonlinear elliptic equations, in Topological Methods, Variational Methods and Their Applications, World Sci., (2003), 65–76.  Google Scholar

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K. C. ChangZ. Q. Wang and T. Zhang, On a new index theory and non semi-trivial solutions for elliptic systems, Discrete Contin. Dyn. Syst., 28 (2010), 809-826.  doi: 10.3934/dcds.2010.28.809.  Google Scholar

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M. ContiL. Merizzi and S. Terracini, Radial Solutions of Superlinear Equations on $\mathbb{R}^{N}$. Part I: Global Variational Approach, Arch. Ration. Mech. Anal., 153 (2000), 291-316.  doi: 10.1007/s002050050015.  Google Scholar

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D. Daners and P. K. Medina, Abstract Evolution Equations, Periodic Problems, and Applications, Longman Scientific and Technical, 1992. doi: 978-0582096356.  Google Scholar

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

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D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 2008. doi: 10.1007/BFb0089649.  Google Scholar

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N. Hirano and W. Zou, A perturbation method for multiple sign-changing solutions, Calc. Var. Partial Differ. Equ., 37 (2010), 87-98.  doi: 10.1007/s00526-009-0253-2.  Google Scholar

[17]

L. Jeanjean, Two positive solutions for a class of nonhomogeneous elliptic equations, Differ. Integral Equ., 10 (1997), 609-624.   Google Scholar

[18]

Y. LiZ. Liu and C. Zhao, Nodal solutions of a perturbed elliptic problem, Topol. Methods Nonlinear Anal., 32 (2008), 49-68.   Google Scholar

[19]

H. Li and Z. Q. Wang, Multiple nodal solutions having shared componentwise nodal numbers for coupled Schrödinger equations, arXiv: 2005.13860v1. Google Scholar

[20]

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

[21]

Z. Liu and Z. Q. Wang, Multiple bound states of nonlinear Schrödinger systems, Commun. Math. Phys., 282 (2008), 721-731.  doi: 10.1007/s00220-008-0546-x.  Google Scholar

[22]

Z. Liu and Z. Q. 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

[23]

W. Long and S. Peng, Positive vector solutions for a Schrödinger system with external source terms, Nonlinear Differ. Equ. Appl., 27 (2020), 36 pp. doi: 10.1007/s00030-019-0608-0.  Google Scholar

[24]

P. Quittner, Boundedness of trajectories of parabolic equations and stationary solutions via dynamical methods, Differ. and Integral Equ., 7 (1994), 1547-1556.   Google Scholar

[25]

P. H. Rabinowitz, Multiple critical points of perturbed symmetric functionals, T. Am. Math. Soc., 272 (1982), 753-769.  doi: 10.2307/1998726.  Google Scholar

[26]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, American Mathematical Society, Providence, 1986. doi: 10.1090/cbms/065.  Google Scholar

[27]

K. Tanaka, Morse indices at critical points related to the symmetric mountain pass theorem and applications, Commun. Partial Differ. Equ., 14 (1989), 99-128.  doi: 10.1080/03605308908820592.  Google Scholar

[28]

R. Tian and Z. Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topo. Methods Nonlinear Anal., 37 (2011), 203-223.  doi: 10.1016/j.matcom.2011.02.010.  Google Scholar

[29]

J. Wei and T. Weth, Radial Solutions and Phase Separation in a System of Two Coupled Schrödinger Equations, Arch. Ration. Mech. Anal., 190 (2008), 83-106.  doi: 10.1007/s00205-008-0121-9.  Google Scholar

[30]

J. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 11 (2012), 1003-1011.  doi: 10.3934/cpaa.2012.11.  Google Scholar

[31]

X. Yue and W. Zou, Infinitely many solutions for the perturbed Bose-Einstein condensates system, Nonlinear Anal., 94 (2014), 171-184.  doi: 10.1016/j.na.2013.08.012.  Google Scholar

show all references

References:
[1]

N. Ackermann and T. Bartsch, Superstable manifolds of semilinear parabolic problems, J. Dyn. Differ. Equ., 17 (2005), 115-173.  doi: 10.1007/s10884-005-3144-z.  Google Scholar

[2]

S. Adachi and K. Tanaka, Four positive solutions for the semilinear elliptic equation: $-\Delta u+u = a(x)u^p+f(x)$ in $\mathbb{R}^N$, Calc. Var. Partial Differ. Equ., 11 (2000), 63-95.  doi: 10.1007/s005260050003.  Google Scholar

[3]

S. Alarcón, Multiple solutions for a critical nonhomogeneous elliptic problem in domains with small holes, Commun. Pure Appl. Anal., 8 (2009), 1269-1289.  doi: 10.3934/cpaa.2009.8.1269.  Google Scholar

[4]

A. Bahri and H. Berestycki, Existence of forced oscillations for some nonlinear differential equations, Commun. Pure Appl. Math., 37 (1984), 403-442.  doi: 10.1002/cpa.3160370402.  Google Scholar

[5]

A. Bahri and P. L. Lions, Morse index of some min-max critical points. I. Application to multiplicity results, Commun. Pure Appl. Math., 41 (1988), 1027-1037.  doi: 10.1002/cpa.3160410803.  Google Scholar

[6]

N. P. Bhatia and G. P. Szegö, Dynamical Systems: Stability Theory and Applications, Springer, Berlin, Heidelberg, 1967. doi: 10.1007/BFb0080630.  Google Scholar

[7]

D. M. Cao and H. S. Zhou, Multiple positive solutions of nonhomogeneous semilinear elliptic equations in $\mathbb{R}^N$, Commun. Pure Appl. Math., 41 (1988), 1027-1037.  doi: 10.1017/S0308210500022836.  Google Scholar

[8]

T. Cazenave and P.L. Lions, Solutions globales d'equations de la chaleur semi lineaire, Commun. Partial Differ. Equ., 9 (1984), 955-978.  doi: 10.1080/03605308408820353.  Google Scholar

[9]

K. C. Chang, Infinite-dimensional Morse Theory and Multiple Solution Problems, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0385-8.  Google Scholar

[10]

K. C. Chang, Heat method in nonlinear elliptic equations, in Topological Methods, Variational Methods and Their Applications, World Sci., (2003), 65–76.  Google Scholar

[11]

K. C. ChangZ. Q. Wang and T. Zhang, On a new index theory and non semi-trivial solutions for elliptic systems, Discrete Contin. Dyn. Syst., 28 (2010), 809-826.  doi: 10.3934/dcds.2010.28.809.  Google Scholar

[12]

M. ContiL. Merizzi and S. Terracini, Radial Solutions of Superlinear Equations on $\mathbb{R}^{N}$. Part I: Global Variational Approach, Arch. Ration. Mech. Anal., 153 (2000), 291-316.  doi: 10.1007/s002050050015.  Google Scholar

[13]

D. Daners and P. K. Medina, Abstract Evolution Equations, Periodic Problems, and Applications, Longman Scientific and Technical, 1992. doi: 978-0582096356.  Google Scholar

[14]

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

[15]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 2008. doi: 10.1007/BFb0089649.  Google Scholar

[16]

N. Hirano and W. Zou, A perturbation method for multiple sign-changing solutions, Calc. Var. Partial Differ. Equ., 37 (2010), 87-98.  doi: 10.1007/s00526-009-0253-2.  Google Scholar

[17]

L. Jeanjean, Two positive solutions for a class of nonhomogeneous elliptic equations, Differ. Integral Equ., 10 (1997), 609-624.   Google Scholar

[18]

Y. LiZ. Liu and C. Zhao, Nodal solutions of a perturbed elliptic problem, Topol. Methods Nonlinear Anal., 32 (2008), 49-68.   Google Scholar

[19]

H. Li and Z. Q. Wang, Multiple nodal solutions having shared componentwise nodal numbers for coupled Schrödinger equations, arXiv: 2005.13860v1. Google Scholar

[20]

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

[21]

Z. Liu and Z. Q. Wang, Multiple bound states of nonlinear Schrödinger systems, Commun. Math. Phys., 282 (2008), 721-731.  doi: 10.1007/s00220-008-0546-x.  Google Scholar

[22]

Z. Liu and Z. Q. 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

[23]

W. Long and S. Peng, Positive vector solutions for a Schrödinger system with external source terms, Nonlinear Differ. Equ. Appl., 27 (2020), 36 pp. doi: 10.1007/s00030-019-0608-0.  Google Scholar

[24]

P. Quittner, Boundedness of trajectories of parabolic equations and stationary solutions via dynamical methods, Differ. and Integral Equ., 7 (1994), 1547-1556.   Google Scholar

[25]

P. H. Rabinowitz, Multiple critical points of perturbed symmetric functionals, T. Am. Math. Soc., 272 (1982), 753-769.  doi: 10.2307/1998726.  Google Scholar

[26]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, American Mathematical Society, Providence, 1986. doi: 10.1090/cbms/065.  Google Scholar

[27]

K. Tanaka, Morse indices at critical points related to the symmetric mountain pass theorem and applications, Commun. Partial Differ. Equ., 14 (1989), 99-128.  doi: 10.1080/03605308908820592.  Google Scholar

[28]

R. Tian and Z. Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topo. Methods Nonlinear Anal., 37 (2011), 203-223.  doi: 10.1016/j.matcom.2011.02.010.  Google Scholar

[29]

J. Wei and T. Weth, Radial Solutions and Phase Separation in a System of Two Coupled Schrödinger Equations, Arch. Ration. Mech. Anal., 190 (2008), 83-106.  doi: 10.1007/s00205-008-0121-9.  Google Scholar

[30]

J. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 11 (2012), 1003-1011.  doi: 10.3934/cpaa.2012.11.  Google Scholar

[31]

X. Yue and W. Zou, Infinitely many solutions for the perturbed Bose-Einstein condensates system, Nonlinear Anal., 94 (2014), 171-184.  doi: 10.1016/j.na.2013.08.012.  Google Scholar

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