×

Homoclinic solutions of discrete \(\phi\)-Laplacian equations with mixed nonlinearities. (English) Zbl 1396.39008

Summary: By using critical point theory, we obtain some new sufficient conditions on the existence of homoclinic solutions of a class of nonlinear discrete \(\phi\)-Laplacian equations with mixed nonlinearities for the potentials being periodic or being unbounded, respectively. And we prove it is also necessary in some special cases. In addition, multiplicity results of homoclinic solutions for nonlinear discrete \(\phi\)-Laplacian equations with unbounded potentials have also been considered. In our paper, the nonlinearities can be mixed super \(p\)-linear with asymptotically \(p\)-linear at \(\infty\) for \(p\geq 1\). To the best of our knowledge, there is no such result for the existence of homoclinic solutions with discrete \(\phi\)-Laplacian before. Finally, an extension has also been considered.

MSC:

39A14 Partial difference equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
39A12 Discrete version of topics in analysis
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] G. Arioli; F. Gazzola, Periodic motions of an infinite lattice of particles with nearest neighbor interaction, Nonlinear Anal., 26, 1103, (1996) · Zbl 0867.70004
[2] S. Aubry, Breathers in nonlinear lattices: existence, linear stability and quantization, Physica D, 103, 201, (1997) · Zbl 1194.34059
[3] S. Aubry, Discrete breathers: localization and transfer of energy in discrete Hamiltonian nonlinear systems, Physica D, 216, 1, (2006) · Zbl 1159.82312
[4] G. Chen; S. Ma, Discrete nonlinear Schrödinger equations with superlinear nonlinearities, Appl. Math. Comput., 218, 5496, (2012) · Zbl 1254.39006
[5] G. Chen; S. Ma, Homoclinic solutions of discrete nonlinear Schrödinger equations with asymptotically or super linear terms, Appl. Math. Comput., 232, 787, (2014) · Zbl 1410.39008
[6] W. Chen; M. Yang, Standing waves for periodic discrete nonlinear Schrödinger equations with asymptotically linear terms, Acta Math. Appl. Sin. Engl. Ser., 28, 351, (2012) · Zbl 1359.35175
[7] J. Cuevas; P. G. Kevrekidis; D. J. Frantzeskakis; B. A. Malomed, Discrete solitons in nonlinear Schrödinger lattices with a power-law nonlinearity, Physica D, 238, 67, (2009) · Zbl 1153.82321
[8] S. Flach; A. V. Gorbach, Discrete breathers-advance in theory and applications, Phys. Rep., 467, 1, (2008)
[9] J. W. Fleischer; M. Segev; N. K. Efremidis; D. N. Christodoulides, Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices, Nature, 422, 147, (2003)
[10] A. V. Gorbach; M. Johansson, Gap and out-gap breathers in a binary modulated discrete nonlinear Schrödinger model, Eur. Phys. J. D, 29, 77, (2004)
[11] G. James, Centre manifold reduction for quasilinear discrete systems, J. Nonlinear Sci., 13, 27, (2003) · Zbl 1185.37158
[12] A. Khare; K. Rasmussen; M. Samuelsen; A. Saxena, Exact solutions of the saturable discrete nonlinear Schrödinger equation, J. Phys. A, 38, 807, (2005) · Zbl 1069.81016
[13] G. Kopidakis, S. Aubry and G. P. Tsironis, Targeted energy transfer through discrete breathers in nonlinear systems, Phys. Rev. Lett., 87 (2001), Art. ID 165501.
[14] W. Krolikowski; B. L. Davies; C. Denz, Photorefractive solitons, IEEE J. Quant. Electron., 39, 3, (2003)
[15] J. Kuang; Z. Guo, Homoclinic solutions of a class of periodic difference equations with asymptotically linear nonlinearities, Nonlinear Anal., 89, 208, (2013) · Zbl 1325.39004
[16] G. Lin and Z. Zhou, Periodic and subharmonic solutions for a \begin{document}\( 2n \)\end{document}th-order difference equation containing both advance and retardation with \begin{document}\( ϕ \)\end{document}-Laplacian, Adv. Difference Equ., 2014(2014), Art. ID 74.
[17] G. Lin; Z. Zhou, Homoclinic solutions in periodic difference equations with mixed nonlinearities, Math. Method Appl. Sci., 39, 245, (2016) · Zbl 1348.39003
[18] G. Lin; Z. Zhou, Homoclinic solutions in non-periodic discrete \begin{document}\( ϕ \)\end{document}-Laplacian equations with mixed nonlinearities, Appl. Math. lett., 64, 15, (2017)
[19] S. Liu; S. Li, Infinitely many solutions for a superlinear elliptic equation, Acta Math. Sinica Chin. Ser., 46, 625, (2003) · Zbl 1081.35043
[20] R. Livi, R. Franzosi and G. L. Oppo, Self-localization of Bose-Einstein condensates in optical lattices via boundary dissipation, Phys. Rev. Lett., 97 (2006), Art. ID 060401.
[21] S. Ma; Z. Wang, Multibump solutions for discrete periodic nonlinear Schrödinger equations, Z. Angew. Math. Phys., 64, 1413, (2013) · Zbl 1280.35139
[22] A. Mai and Z. Zhou, Ground state solutions for the periodic discrete nonlinear Schrödinger equations with superlinear nonlinearities, Abstr. Appl. Anal., 2013 (2013), Art. ID 317139.
[23] J. Mawhin, Periodic solutions of second order nonlinear difference systems with \begin{document}\( ϕ \)\end{document}-Laplacian: a variational approach, Nonlinear Anal., 75, 4672, (2012) · Zbl 1252.39016
[24] J. Mawhin, Periodic solutions of second order Lagrangian difference systems with bounded or singular \begin{document}\( ϕ \)\end{document}-Laplacian and periodic potential, Discrete Contin. Dyn. Syst. Ser. S, 6, 1065, (2013) · Zbl 1261.39017
[25] A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations with saturable nonlinearities, J. Math. Anal. Appl., 371, 254, (2010) · Zbl 1197.35273
[26] A. Pankov; V. Rothos, Periodic and decaying solutions in discrete nonlinear Schrödinger with saturable nonlinearity, Proc. R. Soc. A, 464, 3219, (2008) · Zbl 1186.35206
[27] H. Shi, Gap solitons in periodic discrete Schrödinger equations with nonlinearity, Acta Appl. Math., 109, 1065, (2010) · Zbl 1190.35192
[28] H. Shi; H. Zhang, Existence of gap solitons in a periodic discrete nonlinear Schrödinger equations, J. Math. Anal. Appl., 361, 411, (2010) · Zbl 1178.35351
[29] C. A. Stuart, Locating Cerami sequences in a mountain pass geometry, Commun. Appl. Anal., 2-4, 569, (2011) · Zbl 1232.58007
[30] A. A. Sukhorukov; Y. S. Kivshar, Generation and stability of discrete gap solitons, Opt. Lett., 28, 2345, (2003)
[31] X. Tang, Non-Nehari manifold method for periodic discrete superlinear Schrödinger equation, Acta Math. Sin. Engl. Ser., 32, 463, (2016) · Zbl 1386.39015
[32] X. Tang and J. Chen, Infinitely many homoclinic orbits for a class of discrete Hamiltonian systems, Adv. Difference Equ., 2013 (2013), Art. ID 242. · Zbl 1375.39020
[33] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
[34] M. Yang; W. Chen; Y. Ding, Solutions for discrete periodic Schrödinger equations with spectrum \begin{document}\( 0 \)\end{document}, Acta. Appl. Math., 110, 1475, (2010) · Zbl 1191.35260
[35] G. Zhang; A. Pankov, Standing waves of the discrete nonlinear Schrödinger equations with growing potentials, Commun. Math. Anal., 5, 38, (2008) · Zbl 1168.35437
[36] Z. Zhou; D. Ma, Multiplicity results of breathers for the discrete nonlinear Schrödinger equations with unbounded potentials, Sci. China Math., 58, 781, (2015) · Zbl 1328.39011
[37] Z. Zhou; J. Yu, On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems, J. Differential Equations, 249, 1199, (2010) · Zbl 1200.39001
[38] Z. Zhou; J. Yu, Homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity, Acta Math. Appl. Sin. Engl. Ser., 29, 1809, (2013) · Zbl 1284.39006
[39] Z. Zhou; J. Yu; Y. Chen, Homoclinic solutions in periodic diffrence equations with saturable nonlinearity, Sci. China Math., 54, 83, (2011) · Zbl 1239.39010
[40] W. Zou, Variant Fountain theorems and their applications, Manuscripta Math., 104, 343, (2001) · Zbl 0976.35026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.