# American Institute of Mathematical Sciences

September  2018, 17(5): 1723-1747. doi: 10.3934/cpaa.2018082

## Homoclinic solutions of discrete $\phi$-Laplacian equations with mixed nonlinearities

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

* Corresponding author

Received  March 2017 Revised  November 2017 Published  April 2018

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 $∞$ for $p≥ 1$. 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.

Citation: Genghong Lin, Zhan Zhou. Homoclinic solutions of discrete $\phi$-Laplacian equations with mixed nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1723-1747. doi: 10.3934/cpaa.2018082
##### References:
 [1] G. Arioli and F. Gazzola, Periodic motions of an infinite lattice of particles with nearest neighbor interaction, Nonlinear Anal., 26 (1996), 1103-1114.   Google Scholar [2] S. Aubry, Breathers in nonlinear lattices: existence, linear stability and quantization, Physica D, 103 (1997), 201-250.   Google Scholar [3] S. Aubry, Discrete breathers: localization and transfer of energy in discrete Hamiltonian nonlinear systems, Physica D, 216 (2006), 1-30.   Google Scholar [4] G. Chen and S. Ma, Discrete nonlinear Schrödinger equations with superlinear nonlinearities, Appl. Math. Comput., 218 (2012), 5496-5507.   Google Scholar [5] G. Chen and S. Ma, Homoclinic solutions of discrete nonlinear Schrödinger equations with asymptotically or super linear terms, Appl. Math. Comput., 232 (2014), 787-798.   Google Scholar [6] W. Chen and M. Yang, Standing waves for periodic discrete nonlinear Schrödinger equations with asymptotically linear terms, Acta Math. Appl. Sin. Engl. Ser., 28 (2012), 351-360.   Google Scholar [7] J. Cuevas, P. G. Kevrekidis, D. J. Frantzeskakis and B. A. Malomed, Discrete solitons in nonlinear Schrödinger lattices with a power-law nonlinearity, Physica D, 238 (2009), 67-76.   Google Scholar [8] S. Flach and A. V. Gorbach, Discrete breathers-advance in theory and applications, Phys. Rep., 467 (2008), 1-116.   Google Scholar [9] 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, 422 (2003), 147-150.   Google Scholar [10] A. V. Gorbach and M. Johansson, Gap and out-gap breathers in a binary modulated discrete nonlinear Schrödinger model, Eur. Phys. J. D, 29 (2004), 77-93.   Google Scholar [11] G. James, Centre manifold reduction for quasilinear discrete systems, J. Nonlinear Sci., 13 (2003), 27-63.   Google Scholar [12] A. Khare, K. Rasmussen, M. Samuelsen and A. Saxena, Exact solutions of the saturable discrete nonlinear Schrödinger equation, J. Phys. A, 38 (2005), 807-814.   Google Scholar [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. Google Scholar [14] W. Krolikowski, B. L. Davies and C. Denz, Photorefractive solitons, IEEE J. Quant. Electron., 39 (2003), 3-12.   Google Scholar [15] J. Kuang and Z. Guo, Homoclinic solutions of a class of periodic difference equations with asymptotically linear nonlinearities, Nonlinear Anal., 89 (2013), 208-218.   Google Scholar [16] G. Lin and Z. Zhou, Periodic and subharmonic solutions for a $2n$th-order difference equation containing both advance and retardation with $\phi$-Laplacian, Adv. Difference Equ., 2014(2014), Art. ID 74. Google Scholar [17] G. Lin and Z. Zhou, Homoclinic solutions in periodic difference equations with mixed nonlinearities, Math. Method Appl. Sci., 39 (2016), 245-260.   Google Scholar [18] G. Lin and Z. Zhou, Homoclinic solutions in non-periodic discrete $\phi$-Laplacian equations with mixed nonlinearities, Appl. Math. lett., 64 (2017), 15-20.   Google Scholar [19] S. Liu and S. Li, Infinitely many solutions for a superlinear elliptic equation, Acta Math. Sinica Chin. Ser., 46 (2003), 625-630.   Google Scholar [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. Google Scholar [21] S. Ma and Z. Wang, Multibump solutions for discrete periodic nonlinear Schrödinger equations, Z. Angew. Math. Phys., 64 (2013), 1413-1442.   Google Scholar [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. Google Scholar [23] J. Mawhin, Periodic solutions of second order nonlinear difference systems with $\phi$-Laplacian: a variational approach, Nonlinear Anal., 75 (2012), 4672-4687.   Google Scholar [24] J. Mawhin, Periodic solutions of second order Lagrangian difference systems with bounded or singular $\phi$-Laplacian and periodic potential, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 1065-1076.   Google Scholar [25] A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations with saturable nonlinearities, J. Math. Anal. Appl., 371 (2010), 254-265.   Google Scholar [26] A. Pankov and V. Rothos, Periodic and decaying solutions in discrete nonlinear Schrödinger with saturable nonlinearity, Proc. R. Soc. A, 464 (2008), 3219-3236.   Google Scholar [27] H. Shi, Gap solitons in periodic discrete Schrödinger equations with nonlinearity, Acta Appl. Math., 109 (2010), 1065-1075.   Google Scholar [28] H. Shi and H. Zhang, Existence of gap solitons in a periodic discrete nonlinear Schrödinger equations, J. Math. Anal. Appl., 361 (2010), 411-419.   Google Scholar [29] C. A. Stuart, Locating Cerami sequences in a mountain pass geometry, Commun. Appl. Anal., 2-4 (2011), 569-588.   Google Scholar [30] A. A. Sukhorukov and Y. S. Kivshar, Generation and stability of discrete gap solitons, Opt. Lett., 28 (2003), 2345-2347.   Google Scholar [31] X. Tang, Non-Nehari manifold method for periodic discrete superlinear Schrödinger equation, Acta Math. Sin. Engl. Ser., 32 (2016), 463-473.   Google Scholar [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. Google Scholar [33] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. Google Scholar [34] M. Yang, W. Chen and Y. Ding, Solutions for discrete periodic Schrödinger equations with spectrum $0$, Acta. Appl. Math., 110 (2010), 1475-1488.   Google Scholar [35] 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 [36] 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.   Google Scholar [37] 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.   Google Scholar [38] Z. Zhou and J. Yu, Homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity, Acta Math. Appl. Sin. Engl. Ser., 29 (2013), 1809-1822.   Google Scholar [39] Z. Zhou, J. Yu and Y. Chen, Homoclinic solutions in periodic diffrence equations with saturable nonlinearity, Sci. China Math., 54 (2011), 83-93.   Google Scholar [40] W. Zou, Variant fountain theorems and their applications, Manuscripta Math., 104 (2001), 343-358.   Google Scholar

show all references

##### References:
 [1] G. Arioli and F. Gazzola, Periodic motions of an infinite lattice of particles with nearest neighbor interaction, Nonlinear Anal., 26 (1996), 1103-1114.   Google Scholar [2] S. Aubry, Breathers in nonlinear lattices: existence, linear stability and quantization, Physica D, 103 (1997), 201-250.   Google Scholar [3] S. Aubry, Discrete breathers: localization and transfer of energy in discrete Hamiltonian nonlinear systems, Physica D, 216 (2006), 1-30.   Google Scholar [4] G. Chen and S. Ma, Discrete nonlinear Schrödinger equations with superlinear nonlinearities, Appl. Math. Comput., 218 (2012), 5496-5507.   Google Scholar [5] G. Chen and S. Ma, Homoclinic solutions of discrete nonlinear Schrödinger equations with asymptotically or super linear terms, Appl. Math. Comput., 232 (2014), 787-798.   Google Scholar [6] W. Chen and M. Yang, Standing waves for periodic discrete nonlinear Schrödinger equations with asymptotically linear terms, Acta Math. Appl. Sin. Engl. Ser., 28 (2012), 351-360.   Google Scholar [7] J. Cuevas, P. G. Kevrekidis, D. J. Frantzeskakis and B. A. Malomed, Discrete solitons in nonlinear Schrödinger lattices with a power-law nonlinearity, Physica D, 238 (2009), 67-76.   Google Scholar [8] S. Flach and A. V. Gorbach, Discrete breathers-advance in theory and applications, Phys. Rep., 467 (2008), 1-116.   Google Scholar [9] 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, 422 (2003), 147-150.   Google Scholar [10] A. V. Gorbach and M. Johansson, Gap and out-gap breathers in a binary modulated discrete nonlinear Schrödinger model, Eur. Phys. J. D, 29 (2004), 77-93.   Google Scholar [11] G. James, Centre manifold reduction for quasilinear discrete systems, J. Nonlinear Sci., 13 (2003), 27-63.   Google Scholar [12] A. Khare, K. Rasmussen, M. Samuelsen and A. Saxena, Exact solutions of the saturable discrete nonlinear Schrödinger equation, J. Phys. A, 38 (2005), 807-814.   Google Scholar [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. Google Scholar [14] W. Krolikowski, B. L. Davies and C. Denz, Photorefractive solitons, IEEE J. Quant. Electron., 39 (2003), 3-12.   Google Scholar [15] J. Kuang and Z. Guo, Homoclinic solutions of a class of periodic difference equations with asymptotically linear nonlinearities, Nonlinear Anal., 89 (2013), 208-218.   Google Scholar [16] G. Lin and Z. Zhou, Periodic and subharmonic solutions for a $2n$th-order difference equation containing both advance and retardation with $\phi$-Laplacian, Adv. Difference Equ., 2014(2014), Art. ID 74. Google Scholar [17] G. Lin and Z. Zhou, Homoclinic solutions in periodic difference equations with mixed nonlinearities, Math. Method Appl. Sci., 39 (2016), 245-260.   Google Scholar [18] G. Lin and Z. Zhou, Homoclinic solutions in non-periodic discrete $\phi$-Laplacian equations with mixed nonlinearities, Appl. Math. lett., 64 (2017), 15-20.   Google Scholar [19] S. Liu and S. Li, Infinitely many solutions for a superlinear elliptic equation, Acta Math. Sinica Chin. Ser., 46 (2003), 625-630.   Google Scholar [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. Google Scholar [21] S. Ma and Z. Wang, Multibump solutions for discrete periodic nonlinear Schrödinger equations, Z. Angew. Math. Phys., 64 (2013), 1413-1442.   Google Scholar [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. Google Scholar [23] J. Mawhin, Periodic solutions of second order nonlinear difference systems with $\phi$-Laplacian: a variational approach, Nonlinear Anal., 75 (2012), 4672-4687.   Google Scholar [24] J. Mawhin, Periodic solutions of second order Lagrangian difference systems with bounded or singular $\phi$-Laplacian and periodic potential, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 1065-1076.   Google Scholar [25] A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations with saturable nonlinearities, J. Math. Anal. Appl., 371 (2010), 254-265.   Google Scholar [26] A. Pankov and V. Rothos, Periodic and decaying solutions in discrete nonlinear Schrödinger with saturable nonlinearity, Proc. R. Soc. A, 464 (2008), 3219-3236.   Google Scholar [27] H. Shi, Gap solitons in periodic discrete Schrödinger equations with nonlinearity, Acta Appl. Math., 109 (2010), 1065-1075.   Google Scholar [28] H. Shi and H. Zhang, Existence of gap solitons in a periodic discrete nonlinear Schrödinger equations, J. Math. Anal. Appl., 361 (2010), 411-419.   Google Scholar [29] C. A. Stuart, Locating Cerami sequences in a mountain pass geometry, Commun. Appl. Anal., 2-4 (2011), 569-588.   Google Scholar [30] A. A. Sukhorukov and Y. S. Kivshar, Generation and stability of discrete gap solitons, Opt. Lett., 28 (2003), 2345-2347.   Google Scholar [31] X. Tang, Non-Nehari manifold method for periodic discrete superlinear Schrödinger equation, Acta Math. Sin. Engl. Ser., 32 (2016), 463-473.   Google Scholar [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. Google Scholar [33] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. Google Scholar [34] M. Yang, W. Chen and Y. Ding, Solutions for discrete periodic Schrödinger equations with spectrum $0$, Acta. Appl. Math., 110 (2010), 1475-1488.   Google Scholar [35] 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 [36] 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.   Google Scholar [37] 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.   Google Scholar [38] Z. Zhou and J. Yu, Homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity, Acta Math. Appl. Sin. Engl. Ser., 29 (2013), 1809-1822.   Google Scholar [39] Z. Zhou, J. Yu and Y. Chen, Homoclinic solutions in periodic diffrence equations with saturable nonlinearity, Sci. China Math., 54 (2011), 83-93.   Google Scholar [40] W. Zou, Variant fountain theorems and their applications, Manuscripta Math., 104 (2001), 343-358.   Google Scholar
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