Multiplicity of periodic solutions for second-order nonlinear partial difference equations with $ \phi $-Laplacian

Ziying Guo; Juping Ji; Genghong Lin; Ziying Guo; Juping Ji; Genghong Lin

doi:10.3934/math.2025965

AIMS Mathematics

2025, Volume 10, Issue 9: 21721-21736. doi: 10.3934/math.2025965

Previous Article Next Article

Research article Special Issues

Multiplicity of periodic solutions for second-order nonlinear partial difference equations with $ \phi $-Laplacian

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

Received: 23 June 2025 Revised: 28 August 2025 Accepted: 09 September 2025 Published: 18 September 2025
MSC : 34C37, 39A14

In this paper, we primarily employ the critical point theory to examine the multiplicity of periodic solutions for a specific class of second-order nonlinear partial difference equations with $ \phi $-Laplacian. We can study the more general $ \phi $-Laplacian, obtain clearer sufficient conditions for the multiplicity of periodic solutions of the equation, and compare our results with existing works. Our result can be applied to exploring the spacetime periodic solutions of the two-dimensional discrete nonlinear Schrödinger equation.
- partial difference equation,
- periodic solution,
- $ \phi $-Laplacian,
- critical point theory
Citation: Ziying Guo, Juping Ji, Genghong Lin. Multiplicity of periodic solutions for second-order nonlinear partial difference equations with $ \phi $-Laplacian[J]. AIMS Mathematics, 2025, 10(9): 21721-21736. doi: 10.3934/math.2025965

Related Papers:

Abstract

In this paper, we primarily employ the critical point theory to examine the multiplicity of periodic solutions for a specific class of second-order nonlinear partial difference equations with $ \phi $-Laplacian. We can study the more general $ \phi $-Laplacian, obtain clearer sufficient conditions for the multiplicity of periodic solutions of the equation, and compare our results with existing works. Our result can be applied to exploring the spacetime periodic solutions of the two-dimensional discrete nonlinear Schrödinger equation.

References

[1]	D. N. Christodoulides, F. Lederer, Y. Silberberg, Discretizing light behavior in linear and nonlinear waveguide lattices, Nature, 424 (2003), 817–823. http://dx.doi.org/10.1038/nature01936 doi: 10.1038/nature01936
[2]	S. Flach, A. V. Gorbach, Discrete breathers–Advance in theory and applications, Phys. Rep., 467 (2008), 1–116. http://dx.doi.org/10.1016/j.physrep.2008.05.002 doi: 10.1016/j.physrep.2008.05.002
[3]	G. Kopidakis, S. Aubry, G. P. Tsironis, Targeted energy transfer through discrete breathers in nonlinear systems, Phys. Rev. Lett., 87 (2001), 165501. https://doi.org/10.1103/PhysRevLett.87.165501 doi: 10.1103/PhysRevLett.87.165501
[4]	R. Livi, R. Franzosi, G. L. Oppo, Self-localization of Bose-Einstein condensates in optical lattices via boundary dissipation, Phys. Rev. Lett., 97 (2006), 060401. https://doi.org/10.1103/PhysRevLett.97.060401 doi: 10.1103/PhysRevLett.97.060401
[5]	G. Lin, J. Yu, Homoclinic solutions of periodic discrete Schrödinger equations with local superquadratic conditions, SIAM J. Math. Anal., 54 (2022), 1966–2005. https://doi.org/10.1137/21M1413201 doi: 10.1137/21M1413201
[6]	G. Lin, 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., 32 (2022), 127. https://doi.org/10.1007/s12220-022-00866-7 doi: 10.1007/s12220-022-00866-7
[7]	G. Lin, Z. Zhou, Z. Shen, J. Yu, Existence of uncountably many periodic solutions for second-order superlinear difference equations with continuous time, Bull. Math. Sci., 15 (2025), 2450010. https://doi.org/10.1142/S1664360724500103 doi: 10.1142/S1664360724500103
[8]	G. Lin, Z. Zhou, J. Yu, Ground state solutions of discrete asymptotically linear Schrödinger equations with bounded and non-periodic potentials, J. Dyn. Diff. Equat., 32 (2020), 527–555. https://doi.org/10.1007/s10884-019-09743-4 doi: 10.1007/s10884-019-09743-4
[9]	X. Xu, H. Chen, Z. Ouyang, New results for periodic discrete nonlinear Schrödinger equations, Math. Methods Appl. Sci., 48 (2025), 5768–5780. https://doi.org/10.1002/mma.10635 doi: 10.1002/mma.10635
[10]	F. Kh. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, M. Salerno, Nonlinear excitations in arrays of Bose-Einstein condensates, Phys. Rev. A, 64 (2001), 043606. https://doi.org/10.1103/physreva.64.043606 doi: 10.1103/physreva.64.043606
[11]	S. Liu, Multiple periodic solutions for non-linear difference systems involving the $p$-Laplacian, J. Difference Equ. Appl., 17 (2011), 1591–1598. http://dx.doi.org/10.1080/10236191003730480 doi: 10.1080/10236191003730480
[12]	X. Liu, Y. Zhang, H. Shi, X. Deng, Existence of periodic solutions for a $2n$ th-order difference equation involving $p$-Laplacian, Bull. Malays. Math. Sci. Soc., 38 (2015), 1107–1125. https://doi.org/10.1007/s40840-014-0066-0 doi: 10.1007/s40840-014-0066-0
[13]	C. Gao, Solutions to discrete multiparameter periodic boundary value problems involving the $p$-Laplacian via critical point theory, Acta Math. Sci., 34 (2014), 1225–1236. https://doi.org/10.1016/S0252-9602(14)60081-3 doi: 10.1016/S0252-9602(14)60081-3
[14]	P. Mei, Z. Zhou, Periodic and subharmonic solutions for a $2n$th-order $p$-Laplacian difference equation containing both advances and retardations, Open Math., 16 (2018), 1435–1444. https://doi.org/10.1515/math-2018-0123 doi: 10.1515/math-2018-0123
[15]	Y. Long, D. Li, Multiple periodic solutions of a second-order partial difference equation involving $p$-Laplacian, J. Appl. Math. Comput., 69 (2023), 3489–3508. https://doi.org/10.1007/s12190-023-01891-7 doi: 10.1007/s12190-023-01891-7
[16]	P. Mei, Z. Zhou, G. Lin, Periodic and subharmonic solutions for a $2n$th-order $\phi_c$-Laplacian difference equation containing both advances and retardations, Discret. Contin. Dyn. Syst. -S, 12 (2019), 2085–2095. https://doi.org/10.3934/dcdss.2019134 doi: 10.3934/dcdss.2019134
[17]	Z. Zhou, J. Ling, Infinitely many positive solutions for a discrete two point nonlinear boundary value problem with $\phi_c$-Laplacian, Appl. Math. Lett., 91 (2019), 28–34. https://doi.org/10.1016/j.aml.2018.11.016 doi: 10.1016/j.aml.2018.11.016
[18]	Y. Chen, Z. Zhou, Existence of three solutions for a nonlinear discrete boundary value problem with $\phi_c$-Laplacian, Symmetry, 12 (2020), 1839. https://doi.org/10.3390/sym12111839 doi: 10.3390/sym12111839
[19]	F. Xiong, Infinitely many solutions for a perturbed partial discrete dirichlet problem involving $\phi_{c}$-Laplacian, Axioms, 12 (2023), 909. https://doi.org/10.3390/axioms12100909 doi: 10.3390/axioms12100909
[20]	G. Lin, Z. Zhou, Periodic and subharmonic solutions for a 2nth-order difference equation containing both advance and retardation with $\phi$-Laplacian, Adv. Differ. Equ., 2014 (2014), 74. https://doi.org/10.1186/1687-1847-2014-74 doi: 10.1186/1687-1847-2014-74
[21]	P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, American Mathematical Society, 1986.

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)