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 and 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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[10]

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

[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.

[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.

[13]

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

[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.

[15]

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

[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.

[17]

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

[18]

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

[19]

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

[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.

[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.

[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.

[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.

[24]

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

[25]

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

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[10]

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

[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.

[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.

[13]

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

[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.

[15]

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

[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.

[17]

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

[18]

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

[19]

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

[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.

[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.

[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.

[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.

[24]

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

[25]

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

[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.

[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.

[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.

[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.

[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.

[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.

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