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. CuevasP. G. KevrekidisD. 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. FleischerM. SegevN. 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. KhareK. RasmussenM. 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. KrolikowskiB. 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. YangW. 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. ZhouJ. 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. CuevasP. G. KevrekidisD. 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. FleischerM. SegevN. 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. KhareK. RasmussenM. 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. KrolikowskiB. 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. YangW. 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. ZhouJ. 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

[1]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[2]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445

[3]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[4]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[5]

Federico Rodriguez Hertz, Zhiren Wang. On $ \epsilon $-escaping trajectories in homogeneous spaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 329-357. doi: 10.3934/dcds.2020365

[6]

Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246

[7]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378

[8]

Aihua Fan, Jörg Schmeling, Weixiao Shen. $ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363

[9]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447

[10]

Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $. Electronic Research Archive, , () : -. doi: 10.3934/era.2020123

[11]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

[12]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076

[13]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051

[14]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020264

[15]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[16]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

[17]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[18]

Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020381

[19]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[20]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (263)
  • HTML views (307)
  • Cited by (19)

Other articles
by authors

[Back to Top]