EXISTENCE OF INFINITELY MANY HOMOCLINIC SOLUTIONS OF DISCRETE SCHRÖDINGER EQUATIONS WITH LOCAL SUBLINEAR TERMS

Genghong Lin; Zhan Zhou; Jianshe Yu

doi:10.11948/20220047

2022 Volume 12 Issue 3

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2022 > 12(3): 964-980

Next Article Previous Article

Genghong Lin, Zhan Zhou, Jianshe Yu. EXISTENCE OF INFINITELY MANY HOMOCLINIC SOLUTIONS OF DISCRETE SCHRÖDINGER EQUATIONS WITH LOCAL SUBLINEAR TERMS[J]. Journal of Applied Analysis & Computation, 2022, 12(3): 964-980. doi: 10.11948/20220047

Citation:

Genghong Lin, Zhan Zhou, Jianshe Yu. EXISTENCE OF INFINITELY MANY HOMOCLINIC SOLUTIONS OF DISCRETE SCHRÖDINGER EQUATIONS WITH LOCAL SUBLINEAR TERMS[J]. Journal of Applied Analysis & Computation, 2022, 12(3): 964-980. doi: 10.11948/20220047

EXISTENCE OF INFINITELY MANY HOMOCLINIC SOLUTIONS OF DISCRETE SCHRÖDINGER EQUATIONS WITH LOCAL SUBLINEAR TERMS

1.
School of Mathematics and Information Science, Guangzhou University, 510006 Guangzhou, China
2.
Guangzhou Center for Applied Mathematics, Guangzhou University, 510006 Guangzhou, China

Dedicated to Professor Jibin Li on the occasion of his 80th birthday.
Corresponding author: Email: jsyu@gzhu.edu.cn (J. Yu)
Fund Project: This work was partially supported by National Natural Science Foundation of China (Nos. 12001127, 11971126), Guangdong Basic and Applied Basic Research Foundation (No. 2022A1515010667), and Program for Changjiang Scholars and Innovative Research Team in University (No. IRT_16R16)

Abstract

Abstract

We obtain sufficient conditions on the existence of infinitely many homoclinic solutions for a class of discrete Schrödinger equations when the nonlinearities are assumed just to be sublinear near the origin. The problem we are going to study in this paper has two main difficulties, one is that the nonlinear terms are locally sublinear and the other is that the associated variational functional is indefinite. Some new techniques including cutoff methods and compact inclusions are applied here to overcome these two difficulties. Our results also improve some existing ones in the literature.
- Discrete nonlinear Schr?dinger equation /
- homoclinic solution /
- multiplicity /
- local sublinear term /
- critical point theory
MSC: 39A12, 39A70, 35Q51, 35Q55

FullText(HTML)

References

[1]	S. Ai, J. Li, J. Yu and B. Zheng, Stage-structured models for interactive wild and periodically and impulsively released sterile mosquitoes, Discrete Contin. Dyn. Syst., 2022. doi: 10.3934/dcdsb.2021172. CrossRef Google Scholar
[2]	Z. Balanov, C. G. Azpeitia and W. Krawcewicz, On variational and topological methods in nonlinear difference equations, Comm. Pure Appl. Anal., 2018, 17, 2813-2844. doi: 10.3934/cpaa.2018133 CrossRef Google Scholar
[3]	J. Barrow and P. Parsons, Inflationary models with logarithmic potentials, Phys. Rev. D, 1995, 52, 5576-5587. doi: 10.1103/PhysRevD.52.5576 CrossRef Google Scholar
[4]	G. Chen, Homoclinic solutions for perturbed discrete Schrödinger systems in infinite lattices: sublinear and asymptotically linear cases, Appl. Math. Lett., 2021, 117, 107062. doi: 10.1016/j.aml.2021.107062 CrossRef Google Scholar
[5]	G. Chen and S. Ma, Perturbed Schrödinger lattice systems: existence of homoclinic solutions, Proc. R. Soc. Edinb. Sect. A, 2019, 149, 1083-1096. doi: 10.1017/prm.2018.106 CrossRef Google Scholar
[6]	G. Chen and M. Schechter, Multiple homoclinic solutions for discrete Schrödinger equations with perturbed and sublinear terms, Z. Angew. Math. Phys., 2021, 72, 63. doi: 10.1007/s00033-021-01503-z CrossRef Google Scholar
[7]	G. Chen and J. Sun, Infinitely many homoclinic solutions for sublinear and nonperiodic Schrödinger lattice systems, Bound. Value Probl., 2021, 2021, 6. doi: 10.1186/s13661-020-01479-1 CrossRef Google Scholar
[8]	P. Chen and X. He, Existence and multiplicity of homoclinic solutions for second order nonlinear difference equations with Jacobi operators, Math. Methods Appl. Sci., 2016, 39, 5705-5719. doi: 10.1002/mma.3955 CrossRef Google Scholar
[9]	S. Chen, X. Tang and J. Yu, Sign-changing ground state solutions for discrete nonlinear Schrödinger equations, J. Differ. Equ. Appl., 2019, 25, 202-218. doi: 10.1080/10236198.2018.1563601 CrossRef Google Scholar
[10]	D. N. Christodoulides, F. Lederer and Y. Silberberg, Discretizing light behaviour in linear and nonlinear waveguide lattices, Nature, 2003, 424, 817-823. doi: 10.1038/nature01936 CrossRef Google Scholar
[11]	A. Comecha, J. Cuevasb and P. G. Kevrekidis, Discrete peakons, Physica D, 2005, 207, 137-160. doi: 10.1016/j.physd.2005.05.019 CrossRef Google Scholar
[12]	L. Ding and J. Wei, Notes on gap solitons for periodic discrete nonlinear Schrödinger equations, Math. Methods Appl. Sci., 2018, 41, 6673-6682. doi: 10.1002/mma.5183 CrossRef Google Scholar
[13]	K. Enqvist and J. McDonald, Q-balls and baryogenesis in the MSSM, Phys. Lett. B, 1998, 425, 309-321. doi: 10.1016/S0370-2693(98)00271-8 CrossRef Google Scholar
[14]	L. Erbe, B. Jia and Q. Zhang, Homoclinic solutions of discrete nonlinear systems via variational method, J. Appl. Anal. Comput., 2019, 9, 271-294. Google Scholar
[15]	S. Flach and A. V. Gorbach, Discrete breathers—Advance in theory and applications, Phys. Rep., 2008, 467, 1-116. Google Scholar
[16]	J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis and D. N. Christodoulides, Observation of discrete solitons in optically induced real time waveguide arrays, Phys. Rev. Lett., 2003, 90, 023902. doi: 10.1103/PhysRevLett.90.023902 CrossRef Google Scholar
[17]	J. W. Fleischer, M. Segev, N. K. Efremidis and D. N. Christodoulides, Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices, Nature, 2003, 422, 147-150. doi: 10.1038/nature01452 CrossRef Google Scholar
[18]	G. Kopidakis, S. Aubry and G. P. Tsironis, Targeted energy transfer through discrete breathers in nonlinear systems, Phys. Rev. Lett., 2001, 87, 165501. doi: 10.1103/PhysRevLett.87.165501 CrossRef Google Scholar
[19]	J. Kuang and Z. Guo, Homoclinic solutions of a class of periodic difference equations with asymptotically linear nonlinearities, Nonlinear Anal., 2013, 89, 208-218. doi: 10.1016/j.na.2013.05.012 CrossRef Google Scholar
[20]	G. Lin and Y. Hui, Stability analysis in a mosquito population suppression model, J. Biol. Dynam., 2020, 14, 578-589. doi: 10.1080/17513758.2020.1792565 CrossRef Google Scholar
[21]	G. Lin, J. Yu and Z. Zhou, Homoclinic solutions of discrete nonlinear Schrödinger equations with partially sublinear nonlinearities, Electron. J. Differ. Equ., 2019, 96, 1-14. Google Scholar
[22]	G. Lin and Z. Zhou, Homoclinic solutions in periodic difference equations with mixed nonlinearities, Math. Methods Appl. Sci., 2016, 39, 245-260. doi: 10.1002/mma.3474 CrossRef Google Scholar
[23]	G. Lin and Z. Zhou, Homoclinic solutions in non-periodic discrete $\phi$-Laplacian equations with mixed nonlinearities, Appl. Math. Lett., 2017, 64, 15-20. doi: 10.1016/j.aml.2016.08.001 CrossRef Google Scholar
[24]	G. Lin and Z. Zhou, Homoclinic solutions of discrete $\phi$-Laplacian equations with mixed nonlinearities, Comm. Pure Appl. Anal., 2018, 17, 1723-1747. doi: 10.3934/cpaa.2018082 CrossRef Google Scholar
[25]	G. Lin, Z. Zhou and J. Yu, Ground state solutions of discrete asymptotically linear Schrödinger equations with bounded and non-periodic potentials, J. Dyn. Differ. Equ., 2020, 32, 527-555. doi: 10.1007/s10884-019-09743-4 CrossRef Google Scholar
[26]	G. Lin, J. Ji, L. Wang and J. Yu, Multitype bistability and long transients in a delayed spruce budworm population model, J. Differential Equations, 2021, 283, 263-289. doi: 10.1016/j.jde.2021.02.034 CrossRef Google Scholar
[27]	G. Lin and J. Yu, Existence of a ground state and infinitely many homoclinic solutions for a periodic discrete system with sign-changing mixed nonlinearities, J. Geom. Anal., 2022, 32, 127. doi: 10.1007/s12220-022-00866-7. CrossRef Google Scholar
[28]	G. Lin and J. Yu, Homoclinic solutions of periodic discrete Schrödinger equations with local superquadratic conditions, SIAM J. Math. Anal., 2022, 54, 1966-2005. doi: 10.1137/21M1413201 CrossRef Google Scholar
[29]	S. Ma and Z. Wang, Multibump solutions for discrete periodic nonlinear Schrödinger equations, Z. Angew. Math. Phys., 2013, 64, 1413-1442. doi: 10.1007/s00033-012-0295-8 CrossRef Google Scholar
[30]	A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations, Nonlinearity, 2006, 19, 27-40. doi: 10.1088/0951-7715/19/1/002 CrossRef Google Scholar
[31]	A. Pankov and V. Rothos, Periodic and decaying solutions in discrete nonlinear Schrödinger with saturable nonlinearity, Proc. R. Soc. A, 2008, 464, 3219-3236. doi: 10.1098/rspa.2008.0255 CrossRef Google Scholar
[32]	H. Shi and Y. Zhang, Standing wave solutions for the discrete nonlinear Schrödinger equations with indefinite sign subquadratic potentials, Appl. Math. Lett., 2016, 58, 95-102. doi: 10.1016/j.aml.2016.02.010 CrossRef Google Scholar
[33]	X. Tang, Non-Nehari manifold method for periodic discrete superlinear Schrödinger equation, Acta Math. Sin. Engl. Ser., 2016, 32, 463-473. doi: 10.1007/s10114-016-4262-8 CrossRef Google Scholar
[34]	M. Willem, Minimax theorems, Birkhäuser, Boston, MA, 1996. Google Scholar
[35]	J. Yu and J. Li, Global asymptotic stability in an interactive wild and sterile mosquito model, J. Differential Equations, 2020, 269, 6193-6215. doi: 10.1016/j.jde.2020.04.036 CrossRef Google Scholar
[36]	J. Yu and J. Li, A delay suppression model with sterile mosquitoes release period equal to wild larvae maturation period, J. Math. Biol., 2022, 84, 14. doi: 10.1007/s00285-022-01718-2. CrossRef Google Scholar
[37]	J. Yu and J. Li, Discrete-time models for interactive wild and sterile mosquitoes with general time steps, Math. Biosci., 2022, 346, 108797. doi: 10.1016/j.mbs.2022.108797. CrossRef Google Scholar
[38]	G. Zhang and A. Pankov, Standing waves of discrete nonlinear Schrödinger equations with growing potential, Commun. Math. Anal., 2008, 5, 38-49. Google Scholar
[39]	G. Zhang, Breather solutions of the discrete nonlinear Schrödinger equations with unbounded potentials, J. Math. Phys., 2009, 50, 013505. doi: 10.1063/1.3036182 CrossRef Google Scholar
[40]	Q. Zhang, Homoclinic orbits for discrete Hamiltonian systems with local super-quadratic conditions, Comm. Pure Appl. Anal., 2019, 18, 425-434. doi: 10.3934/cpaa.2019021 CrossRef Google Scholar
[41]	B. Zheng, J. Li and J. Yu, One discrete dynamical model on Wolbachia infection frequency in mosquito populations, Sci. China Math., 2021. doi: 10.1007/s11425-021-1891-7. CrossRef Google Scholar
[42]	B. Zheng, J. Li and J. Yu, Existence and stability of periodic solutions in a mosquito population suppression model with time delay, J. Differential Equations, 2022, 315, 159-178. doi: 10.1016/j.jde.2022.01.036 CrossRef Google Scholar
[43]	B. Zheng and J. Yu, Existence and uniqueness of periodic orbits in a discrete model on Wolbachia infection frequency, Adv. Nonlinear Anal., 2022, 11, 212-224. Google Scholar
[44]	B. Zheng and J. Yu, At most two periodic solutions for a switching mosquito population suppression model, J. Dyn. Differ. Equ., 2022. doi: 10.1007/s10884-021-10125-y. CrossRef Google Scholar
[45]	B. Zheng, J. Yu and J. Li, Modeling and analysis of the implementation of the Wolbachia incompatible and sterile insect technique for mosquito population suppression, SIAM J. Appl. Math., 2021, 81, 718-740. doi: 10.1137/20M1368367 CrossRef Google Scholar
[46]	Z. Zhou and D. Ma, Multiplicity results of breathers for the discrete nonlinear Schrödinger equations with unbounded potentials, Sci. China Math., 2015, 58, 781-790. doi: 10.1007/s11425-014-4883-2 CrossRef Google Scholar
[47]	Z. Zhou and J. Yu, On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems, J. Differ. Equations, 2010, 249, 1199-1212. doi: 10.1016/j.jde.2010.03.010 CrossRef Google Scholar
[48]	Z. Zhou and J. Yu, Homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity, Acta. Math. Sin. Engl. Ser., 2013, 29, 1809-1822. doi: 10.1007/s10114-013-0736-0 CrossRef Google Scholar
[49]	Z. Zhou, J. Yu and Y. Chen, Homoclinic solutions in periodic difference equations with saturable nonlinearity, Sci. China Math., 2011, 54, 83-93. doi: 10.1007/s11425-010-4101-9 CrossRef Google Scholar
[50]	Q. Zhu, Z. Zhou and L. Wang, Existence and stability of discrete solitons in nonlinear Schrödinger lattices with hard potentials, Physica D, 2020, 403, 132326. doi: 10.1016/j.physd.2019.132326 CrossRef Google Scholar