doi: 10.3934/dcdsb.2021124

Multiple solutions for Schrödinger lattice systems with asymptotically linear terms and perturbed terms

1. 

School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, China

2. 

Department of Mathematics, University of California, Irvine, CA 92697-3875, USA

* Corresponding author: Guanwei Chen

Received  October 2020 Revised  March 2021 Published  April 2021

Fund Project: This work is supported by National Natural Science Foundation of China (No. 11771182)

In infinite $ m $-dimensional lattices, we obtain the existence of two nontrivial solutions for a class of non-periodic Schrödinger lattice systems with perturbed terms, where the potentials are coercive and the nonlinearities are asymptotically linear at infinity. In addition, examples are given to illustrate our results.

Citation: Guanwei Chen, Martin Schechter. Multiple solutions for Schrödinger lattice systems with asymptotically linear terms and perturbed terms. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021124
References:
[1]

G. Chen, S. Ma and Z-Q. Wang, Standing waves for discrete Schrödinger equations in infinite lattices with saturable nonlinearities, J. Differential Equations, 261 (2016) 3493–3518. doi: 10.1016/j.jde.2016.05.030.  Google Scholar

[2]

G. Chen and M. Schechter, Non-periodic discrete Schrödinger equations: Ground state solutions, Z. Angew. Math. Phys., 67 (2016), 72, 15 pp. doi: 10.1007/s00033-016-0665-8.  Google Scholar

[3]

G. Chen and S. Ma, Perturbed Schrödinger lattice systems: Existence of homoclinic solutions, Proc. Roy. Soc. Edinburgh Sect. A, 149 (2019), 1083-1096.  doi: 10.1017/prm.2018.106.  Google Scholar

[4]

G. Chen and M. Schechter, Non-periodic Schrödinger lattice systems with perturbed and asymptotically linear terms: Negative energy solutions, Appl. Math. Lett., 93 (2019), 34-39. doi: 10.1016/j.aml.2019.01.033.  Google Scholar

[5]

G. Chen and M. Schechter, Multiple solutions for non-periodic Schrödinger lattice systems with perturbation and super-linear terms, Z. Angew. Math. Phys., 70 (2019), 152, 9 pp. doi: 10.1007/s00033-019-1199-7.  Google Scholar

[6]

G. Chen and M. Schechter, Multiple homoclinic solutions for discrete Schrödinger equations with perturbed and sublinear terms, Z. Angew. Math. Phys., 72 (2021), 63. doi: 10.1007/s00033-021-01503-z.  Google Scholar

[7]

D. N. ChristodoulidesF. Lederer and Y. Silberberg, Discretizing light behaviour in linear and nonlinear waveguide lattices, Nature, 424 (2003), 817-823.  doi: 10.1038/nature01936.  Google Scholar

[8]

I. Ekeland, Non-convex minimization problems, Bull. Amer. Math. Soc., 1 (1979), 443-474.  doi: 10.1090/S0273-0979-1979-14595-6.  Google Scholar

[9]

I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-74331-3.  Google Scholar

[10]

L. ErbeB. Jia and Q. Zhang, Homoclinic solutions of discrete nonlinear systems via variational method, J. Appl. Anal. Comput., 9 (2019), 271-294.  doi: 10.11948/2019.271.  Google Scholar

[11]

G. KopidakisS. Aubry and G. P. Tsironis, Targeted energy transfer through discrete breathers in nonlinear systems, Phys. Rev. Lett., 87 (2001), 175-196.  doi: 10.1103/PhysRevLett.87.165501.  Google Scholar

[12]

G. Lin and Z. Zhou, Homoclinic solutions of discrete $\phi$-Laplacian equations with mixed nonlinearities, Comm. Pure Appl. Anal., 17 (2018), 1723-1747.  doi: 10.3934/cpaa.2018082.  Google Scholar

[13]

G. Lin and Z. Zhou, Homoclinic solutions in non-periodic discrete $\phi$-Laplacian equations with mixed nonlinearities, Appl. Math. Lett., 64 (2017), 15-20.  doi: 10.1016/j.aml.2016.08.001.  Google Scholar

[14]

G. LinZ. Zhou and J. Yu, Ground state solutions of discrete asymptotically linear Schrödinger equations with bounded and non-periodic potentials, J. Dyn. Differ. Equ., 32 (2020), 527-555.  doi: 10.1007/s10884-019-09743-4.  Google Scholar

[15]

R. LiviR. Franzosi and G.-L. Oppo, Self-localization of Bose-Einstein condensates in optical lattices via boundary dissipation, Phys. Rev. Lett., 97 (2006), 3633-3646.  doi: 10.1103/PhysRevLett.97.060401.  Google Scholar

[16]

A. Pankov, Gap solitons in periodic discrete nonlinear equations, Nonlinearity, 19 (2006), 27-40.  doi: 10.1088/0951-7715/19/1/002.  Google Scholar

[17]

A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations. Ⅱ. A generalized Nehari manifold approach, Discrete Contin. Dyn. Syst., 19 (2007), 419-430.  doi: 10.3934/dcds.2007.19.419.  Google Scholar

[18]

A. Pankov and V. Rothos, Periodic and decaying solutions in discrete nonlinear Schrödinger with saturable nonlinearity, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 464 (2008), 3219-3236.  doi: 10.1098/rspa.2008.0255.  Google Scholar

[19]

A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations with saturable nonlinearities, J. Math. Anal. Appl., 371 (2010), 254-265.  doi: 10.1016/j.jmaa.2010.05.041.  Google Scholar

[20]

A. Pankov and G. Zhang, Standing wave solutions for discrete nonlinear Schrödinger equations with unbounded potentials and saturable nonlinearity, J. Math. Sci., 177 (2011), 71-82.  doi: 10.1007/s10958-011-0448-x.  Google Scholar

[21]

A. Pankov, Standing waves for discrete nonlinear Schrödinger equations: Sign-changing nonlinearities, Appl. Anal., 92 (2013), 308-317.  doi: 10.1080/00036811.2011.609987.  Google Scholar

[22]

M. Schechter, A variation of the mountain pass lemma and applications, J. London Math. Soc., (2) 44 (1991), 491-502.  doi: 10.1112/jlms/s2-44.3.491.  Google Scholar

[23]

H. Shi and H. Zhang, Existence of gap solitons in periodic discrete nonlinear Schrödinger equations, J. Math. Anal. Appl., 361 (2010), 411-419.  doi: 10.1016/j.jmaa.2009.07.026.  Google Scholar

[24]

M. YangW. Chen and Y. Ding, Solutions for Discrete Periodic Schrödinger Equations with Spectrum 0, Acta Appl. Math., 110 (2010), 1475-1488.  doi: 10.1007/s10440-009-9521-6.  Google Scholar

[25]

G. Zhang and A. Pankov, Standing waves of the discrete nonlinear Schrödinger equations with growing potentials, Commun. Math. Anal., 5 (2008), 38-49.   Google Scholar

[26]

G. Zhang, Breather solutions of the discrete nonlinear Schrödinger equations with unbounded potentials, J. Math. Phys., 50 (2009), 013505. doi: 10.1063/1.3036182.  Google Scholar

[27]

G. Zhang and A. Pankov, Standing wave solutions of the disrete non-linear Schrödinger equatins with unbounded potentials, Ⅱ, Appl. Anal., 89 (2010), 1541-1557.  doi: 10.1080/00036810902942234.  Google Scholar

[28]

Z. ZhouJ. Yu and Y. Chen, On the existence of gap solitons in a periodic discrete nonlinear Schrödinger equation with saturable nonlinearity, Nonlinearity, 23 (2010), 1727-1740.  doi: 10.1088/0951-7715/23/7/011.  Google Scholar

[29]

Z. Zhou and J. Yu, On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems, J. Differential Equations, 249 (2010), 1199-1212.  doi: 10.1016/j.jde.2010.03.010.  Google Scholar

[30]

Z. ZhouJ. Yu and Y. Chen, Homoclinic solutions in periodic difference equations with saturable nonlinearity, Sci. China Math., 54 (2011), 83-93.  doi: 10.1007/s11425-010-4101-9.  Google Scholar

[31]

Z. Zhou and D. Ma, Multiplicity results of breathers for the discrete nonlinear Schrödinger equations with unbounded potentials, Sci. China Math., 58 (2015), 781-790.  doi: 10.1007/s11425-014-4883-2.  Google Scholar

show all references

References:
[1]

G. Chen, S. Ma and Z-Q. Wang, Standing waves for discrete Schrödinger equations in infinite lattices with saturable nonlinearities, J. Differential Equations, 261 (2016) 3493–3518. doi: 10.1016/j.jde.2016.05.030.  Google Scholar

[2]

G. Chen and M. Schechter, Non-periodic discrete Schrödinger equations: Ground state solutions, Z. Angew. Math. Phys., 67 (2016), 72, 15 pp. doi: 10.1007/s00033-016-0665-8.  Google Scholar

[3]

G. Chen and S. Ma, Perturbed Schrödinger lattice systems: Existence of homoclinic solutions, Proc. Roy. Soc. Edinburgh Sect. A, 149 (2019), 1083-1096.  doi: 10.1017/prm.2018.106.  Google Scholar

[4]

G. Chen and M. Schechter, Non-periodic Schrödinger lattice systems with perturbed and asymptotically linear terms: Negative energy solutions, Appl. Math. Lett., 93 (2019), 34-39. doi: 10.1016/j.aml.2019.01.033.  Google Scholar

[5]

G. Chen and M. Schechter, Multiple solutions for non-periodic Schrödinger lattice systems with perturbation and super-linear terms, Z. Angew. Math. Phys., 70 (2019), 152, 9 pp. doi: 10.1007/s00033-019-1199-7.  Google Scholar

[6]

G. Chen and M. Schechter, Multiple homoclinic solutions for discrete Schrödinger equations with perturbed and sublinear terms, Z. Angew. Math. Phys., 72 (2021), 63. doi: 10.1007/s00033-021-01503-z.  Google Scholar

[7]

D. N. ChristodoulidesF. Lederer and Y. Silberberg, Discretizing light behaviour in linear and nonlinear waveguide lattices, Nature, 424 (2003), 817-823.  doi: 10.1038/nature01936.  Google Scholar

[8]

I. Ekeland, Non-convex minimization problems, Bull. Amer. Math. Soc., 1 (1979), 443-474.  doi: 10.1090/S0273-0979-1979-14595-6.  Google Scholar

[9]

I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-74331-3.  Google Scholar

[10]

L. ErbeB. Jia and Q. Zhang, Homoclinic solutions of discrete nonlinear systems via variational method, J. Appl. Anal. Comput., 9 (2019), 271-294.  doi: 10.11948/2019.271.  Google Scholar

[11]

G. KopidakisS. Aubry and G. P. Tsironis, Targeted energy transfer through discrete breathers in nonlinear systems, Phys. Rev. Lett., 87 (2001), 175-196.  doi: 10.1103/PhysRevLett.87.165501.  Google Scholar

[12]

G. Lin and Z. Zhou, Homoclinic solutions of discrete $\phi$-Laplacian equations with mixed nonlinearities, Comm. Pure Appl. Anal., 17 (2018), 1723-1747.  doi: 10.3934/cpaa.2018082.  Google Scholar

[13]

G. Lin and Z. Zhou, Homoclinic solutions in non-periodic discrete $\phi$-Laplacian equations with mixed nonlinearities, Appl. Math. Lett., 64 (2017), 15-20.  doi: 10.1016/j.aml.2016.08.001.  Google Scholar

[14]

G. LinZ. Zhou and J. Yu, Ground state solutions of discrete asymptotically linear Schrödinger equations with bounded and non-periodic potentials, J. Dyn. Differ. Equ., 32 (2020), 527-555.  doi: 10.1007/s10884-019-09743-4.  Google Scholar

[15]

R. LiviR. Franzosi and G.-L. Oppo, Self-localization of Bose-Einstein condensates in optical lattices via boundary dissipation, Phys. Rev. Lett., 97 (2006), 3633-3646.  doi: 10.1103/PhysRevLett.97.060401.  Google Scholar

[16]

A. Pankov, Gap solitons in periodic discrete nonlinear equations, Nonlinearity, 19 (2006), 27-40.  doi: 10.1088/0951-7715/19/1/002.  Google Scholar

[17]

A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations. Ⅱ. A generalized Nehari manifold approach, Discrete Contin. Dyn. Syst., 19 (2007), 419-430.  doi: 10.3934/dcds.2007.19.419.  Google Scholar

[18]

A. Pankov and V. Rothos, Periodic and decaying solutions in discrete nonlinear Schrödinger with saturable nonlinearity, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 464 (2008), 3219-3236.  doi: 10.1098/rspa.2008.0255.  Google Scholar

[19]

A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations with saturable nonlinearities, J. Math. Anal. Appl., 371 (2010), 254-265.  doi: 10.1016/j.jmaa.2010.05.041.  Google Scholar

[20]

A. Pankov and G. Zhang, Standing wave solutions for discrete nonlinear Schrödinger equations with unbounded potentials and saturable nonlinearity, J. Math. Sci., 177 (2011), 71-82.  doi: 10.1007/s10958-011-0448-x.  Google Scholar

[21]

A. Pankov, Standing waves for discrete nonlinear Schrödinger equations: Sign-changing nonlinearities, Appl. Anal., 92 (2013), 308-317.  doi: 10.1080/00036811.2011.609987.  Google Scholar

[22]

M. Schechter, A variation of the mountain pass lemma and applications, J. London Math. Soc., (2) 44 (1991), 491-502.  doi: 10.1112/jlms/s2-44.3.491.  Google Scholar

[23]

H. Shi and H. Zhang, Existence of gap solitons in periodic discrete nonlinear Schrödinger equations, J. Math. Anal. Appl., 361 (2010), 411-419.  doi: 10.1016/j.jmaa.2009.07.026.  Google Scholar

[24]

M. YangW. Chen and Y. Ding, Solutions for Discrete Periodic Schrödinger Equations with Spectrum 0, Acta Appl. Math., 110 (2010), 1475-1488.  doi: 10.1007/s10440-009-9521-6.  Google Scholar

[25]

G. Zhang and A. Pankov, Standing waves of the discrete nonlinear Schrödinger equations with growing potentials, Commun. Math. Anal., 5 (2008), 38-49.   Google Scholar

[26]

G. Zhang, Breather solutions of the discrete nonlinear Schrödinger equations with unbounded potentials, J. Math. Phys., 50 (2009), 013505. doi: 10.1063/1.3036182.  Google Scholar

[27]

G. Zhang and A. Pankov, Standing wave solutions of the disrete non-linear Schrödinger equatins with unbounded potentials, Ⅱ, Appl. Anal., 89 (2010), 1541-1557.  doi: 10.1080/00036810902942234.  Google Scholar

[28]

Z. ZhouJ. Yu and Y. Chen, On the existence of gap solitons in a periodic discrete nonlinear Schrödinger equation with saturable nonlinearity, Nonlinearity, 23 (2010), 1727-1740.  doi: 10.1088/0951-7715/23/7/011.  Google Scholar

[29]

Z. Zhou and J. Yu, On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems, J. Differential Equations, 249 (2010), 1199-1212.  doi: 10.1016/j.jde.2010.03.010.  Google Scholar

[30]

Z. ZhouJ. Yu and Y. Chen, Homoclinic solutions in periodic difference equations with saturable nonlinearity, Sci. China Math., 54 (2011), 83-93.  doi: 10.1007/s11425-010-4101-9.  Google Scholar

[31]

Z. Zhou and D. Ma, Multiplicity results of breathers for the discrete nonlinear Schrödinger equations with unbounded potentials, Sci. China Math., 58 (2015), 781-790.  doi: 10.1007/s11425-014-4883-2.  Google Scholar

[1]

Addolorata Salvatore. Sign--changing solutions for an asymptotically linear Schrödinger equation. Conference Publications, 2009, 2009 (Special) : 669-677. doi: 10.3934/proc.2009.2009.669

[2]

Fengshuang Gao, Yuxia Guo. Multiple solutions for a nonlinear Schrödinger systems. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1181-1204. doi: 10.3934/cpaa.2020055

[3]

Xiaoyan Lin, Yubo He, Xianhua Tang. Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1547-1565. doi: 10.3934/cpaa.2019074

[4]

Xueqin Peng, Gao Jia. Existence and asymptotical behavior of positive solutions for the Schrödinger-Poisson system with double quasi-linear terms. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021134

[5]

Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many solutions for a perturbed Schrödinger equation. Conference Publications, 2015, 2015 (special) : 94-102. doi: 10.3934/proc.2015.0094

[6]

Weichung Wang, Tsung-Fang Wu, Chien-Hsiang Liu. On the multiple spike solutions for singularly perturbed elliptic systems. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 237-258. doi: 10.3934/dcdsb.2013.18.237

[7]

Claudianor O. Alves, Chao Ji. Multiple positive solutions for a Schrödinger logarithmic equation. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2671-2685. doi: 10.3934/dcds.2020145

[8]

Ahmed Y. Abdallah. Asymptotic behavior of the Klein-Gordon-Schrödinger lattice dynamical systems. Communications on Pure & Applied Analysis, 2006, 5 (1) : 55-69. doi: 10.3934/cpaa.2006.5.55

[9]

Edcarlos D. Silva, José Carlos de Albuquerque, Uberlandio Severo. On a class of linearly coupled systems on $ \mathbb{R}^N $ involving asymptotically linear terms. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3089-3101. doi: 10.3934/cpaa.2019138

[10]

Shiwang Ma. Nontrivial periodic solutions for asymptotically linear hamiltonian systems at resonance. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2361-2380. doi: 10.3934/cpaa.2013.12.2361

[11]

Paolo Gidoni, Alessandro Margheri. Lower bound on the number of periodic solutions for asymptotically linear planar Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 585-606. doi: 10.3934/dcds.2019024

[12]

Minbo Yang, Yanheng Ding. Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part. Communications on Pure & Applied Analysis, 2013, 12 (2) : 771-783. doi: 10.3934/cpaa.2013.12.771

[13]

Alireza Khatib, Liliane A. Maia. A positive bound state for an asymptotically linear or superlinear Schrödinger equation in exterior domains. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2789-2812. doi: 10.3934/cpaa.2018132

[14]

Zhitao Zhang, Haijun Luo. Symmetry and asymptotic behavior of ground state solutions for schrödinger systems with linear interaction. Communications on Pure & Applied Analysis, 2018, 17 (3) : 787-806. doi: 10.3934/cpaa.2018040

[15]

Wulong Liu, Guowei Dai. Multiple solutions for a fractional nonlinear Schrödinger equation with local potential. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2105-2123. doi: 10.3934/cpaa.2017104

[16]

Walter Dambrosio, Duccio Papini. Multiple homoclinic solutions for a one-dimensional Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1025-1038. doi: 10.3934/dcdss.2016040

[17]

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

[18]

Tai-Chia Lin, Tsung-Fang Wu. Multiple positive solutions of saturable nonlinear Schrödinger equations with intensity functions. Discrete & Continuous Dynamical Systems, 2020, 40 (4) : 2165-2187. doi: 10.3934/dcds.2020110

[19]

Yanfang Xue, Chunlei Tang. Ground state solutions for asymptotically periodic quasilinear Schrödinger equations with critical growth. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1121-1145. doi: 10.3934/cpaa.2018054

[20]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (6) : 1857-1870. doi: 10.3934/dcdss.2020461

2019 Impact Factor: 1.27

Article outline

[Back to Top]