Homoclinic orbits for discrete Hamiltonian systems with local super-quadratic conditions

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January 2019, 18(1): 425-434. doi: 10.3934/cpaa.2019021

Homoclinic orbits for discrete Hamiltonian systems with local super-quadratic conditions

Qinqin Zhang ^,

Center for Applied Mathematics, Guangzhou University, Guangzhou, Guangdong, 510006, China

^* Corresponding author

Received February 2018 Revised April 2018 Published August 2018

Fund Project: This work was partially supported by National Natural Science Foundation of China grant 11471085 and the Program for Changjiang Scholars and Innovative Research Team in University grant IRT1226

Consider the second order self-adjoint discrete Hamiltonian system

$\triangle [p(n)\triangle u(n-1)]-L(n)u(n)+\nabla W(n, u(n)) = 0, $

where

$p(n), L(n)$

and

$W(n, x)$

are

$N$

-periodic on

$n$

, and 0 lies in a gap of the spectrum

$σ(\mathcal{A})$

of the operator

$\mathcal{A}$

, which is bounded self-adjoint in

$l^2(\mathbb{Z}, \mathbb{R}^{\mathcal{N}})$

defined by

$(\mathcal{A}u)(n) = \triangle [p(n)\triangle u(n-1)]-L(n)u(n)$

. We obtain a sufficient condition on the existence of nontrivial homoclinic orbits for the above system under a much weaker condition than

$\lim_{|x|\to ∞}\frac{W(n, x)}{|x|^2} = ∞$

uniformly in

$ n∈ \mathbb{Z}$

, which has been a common condition used in the existing literature. We also give three examples to illustrate our result.

Keywords: Discrete Hamiltonian system, homoclinic orbit, strongly indefinite functional, local super-quadratic condition.

Mathematics Subject Classification: Primary: 39A11, 58E05; Secondary: 70H05.

Citation: Qinqin Zhang. Homoclinic orbits for discrete Hamiltonian systems with local super-quadratic conditions. Communications on Pure and Applied Analysis, 2019, 18 (1) : 425-434. doi: 10.3934/cpaa.2019021

References:

[1]	R. P. Agarwal, Difference Equations and Inequalities: Theory, Metho and Applications, second edition, Marcel Dekker, Inc. 2000.
[2]	C. D. Ahlbran and A. C. Peterson, Discrete Hamiltonian Systems: Difference Equations, Continued Fraction and Riccati Equations, Kluwer Academic, Dordrecht, 1996. doi: 10.1007/978-1-4757-2467-7.
[3]	Z. Balanov, C. Carcía - Azpeitia and W. Krawcewicz, On Variational and Topological Methods in Nonlinear Difference Equations, Commun. Pure Appl. Anal., 17 (2018), 2813-2844. doi: 10.3934/cpaa.2018133.
[4]	V. Coti Zelati, I. Ekeland and E. Sere, A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann., 288 (1990), 133-160. doi: 10.1007/BF01444526.
[5]	V. Coti Zelati and P. H. Rabinowitz, Homoclinic orbits for second second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), 693-727. doi: 10.2307/2939286.
[6]	D. E. Edmunds, W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987.
[7]	Z. M. Guo and J. S. Yu, The existence of periodic and subharmonic solutions of subquadratic second order difference equation, J. London Math. Soc., 68 (2003), 419-430. doi: 10.1112/S0024610703004563.
[8]	M. Izydorek and J. Janczewska, Homoclinic solutions for a class of second order Hamiltonian systems, J. Differential Equations, 219 (2005), 375-389. doi: 10.1016/j.jde.2005.06.029.
[9]	G. B. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776. doi: 10.1142/S0219199702000853.
[10]	G. H. Lin and Z. Zhou, Homoclinic solutions of discrete φ- Laplacian equations with mixed nonlinearities, Commun. Pure Appl. Anal., 17 (2018), 1723-1747. doi: 10.3934/cpaa.2018082.
[11]	X. Y. Lin and X. H. Tang, Existence of infinitely many homoclinic orbits in discrete Hamiltonian systems, J. Math. Anal. Appl., 373 (2011), 59-72. doi: 10.1016/j.jmaa.2010.06.008.
[12]	Z. L. Liu and Z.-Q. Wang, On the Ambrosetti-Rabinowitz superlinear condition, Adv. Nonlinear Stud., 4 (2004), 561-572. doi: 10.1515/ans-2004-0411.
[13]	W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5 (1992), 1115-1120.
[14]	P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc, Roy. Soc. Edinburgh Sect. A, 114 (1990), 33-38. doi: 10.1017/S0308210500024240.
[15]	P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206 (1991), 473-499. doi: 10.1007/BF02571356.
[16]	X. H. Tang, Non-Nehari manifold method for asymptotically linear Schr¨odinger equation, J. Aust. Math. Soc., 98 (2015), 104-116. doi: 10.1017/S144678871400041X.
[17]	X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohozaev type for SchrödingerPoisson problems with general potentials, Disc. Contin. Dyn. Syst.-Series A., 37 (2017), 4973-5002. doi: 10.3934/dcds.2017214.
[18]	X. H. Tang and X. Y. Lin, Existence and multiplicity of homoclinic solutions for second-order discrete Hamiltonian systems with subquadratic potential, J. Differ. Equ. Appl., 17 (2011), 1617-1634. doi: 10.1080/10236191003730514.
[19]	X. H. Tang, X. Y. Lin and J. S. Yu, Nontrivial solutions for Schrödinger equation with local super-quadratic conditions, J. Dyn. Diff. Equat.. doi: 10.1007/s10884-018-9662-2.
[20]	H. F. Xiao and J. S. Yu, Heteroclinic orbits for a discrete pendulum equation, J. Difference Equ. Appl., 17 (2011), 1267-1280. doi: 10.1080/10236190903167991.
[21]	J. S. Yu and Z. M. Guo, Homoclinic orbits for nonlinear difference equations containing both advance and retardation, J. Math. Anal. Appl., 352 (2009), 799-806. doi: 10.1016/j.jmaa.2008.11.043.
[22]	Q. Zhang, Homoclinic orbits for a class of discrete periodic Hamiltonian systems, Proc. Amer. Math. Soc., 143 (2015), 3155-3163. doi: 10.1090/S0002-9939-2015-12107-7.
[23]	Q. Zhang, Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part and super linear terms, Commun. Pure Appl. Anal., 14 (2015), 1929-1940. doi: 10.3934/cpaa.2015.14.1929.
[24]	Z. Zhou and J. S. Yu, On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems, J. Differential Equations, 249 (2010), 1199-1212. doi: 10.1016/j.jde.2010.03.010.
[25]	Z. Zhou, J. S. Yu and Y. Chen, Homoclinic solutions in periodic difference equations with saturable nonlinearity, Science China, Mathematics, 54 (2011), 83-93. doi: 10.1007/s11425-010-4101-9.

show all references

References:

[1]	R. P. Agarwal, Difference Equations and Inequalities: Theory, Metho and Applications, second edition, Marcel Dekker, Inc. 2000.
[2]	C. D. Ahlbran and A. C. Peterson, Discrete Hamiltonian Systems: Difference Equations, Continued Fraction and Riccati Equations, Kluwer Academic, Dordrecht, 1996. doi: 10.1007/978-1-4757-2467-7.
[3]	Z. Balanov, C. Carcía - Azpeitia and W. Krawcewicz, On Variational and Topological Methods in Nonlinear Difference Equations, Commun. Pure Appl. Anal., 17 (2018), 2813-2844. doi: 10.3934/cpaa.2018133.
[4]	V. Coti Zelati, I. Ekeland and E. Sere, A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann., 288 (1990), 133-160. doi: 10.1007/BF01444526.
[5]	V. Coti Zelati and P. H. Rabinowitz, Homoclinic orbits for second second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), 693-727. doi: 10.2307/2939286.
[6]	D. E. Edmunds, W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987.
[7]	Z. M. Guo and J. S. Yu, The existence of periodic and subharmonic solutions of subquadratic second order difference equation, J. London Math. Soc., 68 (2003), 419-430. doi: 10.1112/S0024610703004563.
[8]	M. Izydorek and J. Janczewska, Homoclinic solutions for a class of second order Hamiltonian systems, J. Differential Equations, 219 (2005), 375-389. doi: 10.1016/j.jde.2005.06.029.
[9]	G. B. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776. doi: 10.1142/S0219199702000853.
[10]	G. H. Lin and Z. Zhou, Homoclinic solutions of discrete φ- Laplacian equations with mixed nonlinearities, Commun. Pure Appl. Anal., 17 (2018), 1723-1747. doi: 10.3934/cpaa.2018082.
[11]	X. Y. Lin and X. H. Tang, Existence of infinitely many homoclinic orbits in discrete Hamiltonian systems, J. Math. Anal. Appl., 373 (2011), 59-72. doi: 10.1016/j.jmaa.2010.06.008.
[12]	Z. L. Liu and Z.-Q. Wang, On the Ambrosetti-Rabinowitz superlinear condition, Adv. Nonlinear Stud., 4 (2004), 561-572. doi: 10.1515/ans-2004-0411.
[13]	W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5 (1992), 1115-1120.
[14]	P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc, Roy. Soc. Edinburgh Sect. A, 114 (1990), 33-38. doi: 10.1017/S0308210500024240.
[15]	P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206 (1991), 473-499. doi: 10.1007/BF02571356.
[16]	X. H. Tang, Non-Nehari manifold method for asymptotically linear Schr¨odinger equation, J. Aust. Math. Soc., 98 (2015), 104-116. doi: 10.1017/S144678871400041X.
[17]	X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohozaev type for SchrödingerPoisson problems with general potentials, Disc. Contin. Dyn. Syst.-Series A., 37 (2017), 4973-5002. doi: 10.3934/dcds.2017214.
[18]	X. H. Tang and X. Y. Lin, Existence and multiplicity of homoclinic solutions for second-order discrete Hamiltonian systems with subquadratic potential, J. Differ. Equ. Appl., 17 (2011), 1617-1634. doi: 10.1080/10236191003730514.
[19]	X. H. Tang, X. Y. Lin and J. S. Yu, Nontrivial solutions for Schrödinger equation with local super-quadratic conditions, J. Dyn. Diff. Equat.. doi: 10.1007/s10884-018-9662-2.
[20]	H. F. Xiao and J. S. Yu, Heteroclinic orbits for a discrete pendulum equation, J. Difference Equ. Appl., 17 (2011), 1267-1280. doi: 10.1080/10236190903167991.
[21]	J. S. Yu and Z. M. Guo, Homoclinic orbits for nonlinear difference equations containing both advance and retardation, J. Math. Anal. Appl., 352 (2009), 799-806. doi: 10.1016/j.jmaa.2008.11.043.
[22]	Q. Zhang, Homoclinic orbits for a class of discrete periodic Hamiltonian systems, Proc. Amer. Math. Soc., 143 (2015), 3155-3163. doi: 10.1090/S0002-9939-2015-12107-7.
[23]	Q. Zhang, Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part and super linear terms, Commun. Pure Appl. Anal., 14 (2015), 1929-1940. doi: 10.3934/cpaa.2015.14.1929.
[24]	Z. Zhou and J. S. Yu, On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems, J. Differential Equations, 249 (2010), 1199-1212. doi: 10.1016/j.jde.2010.03.010.
[25]	Z. Zhou, J. S. Yu and Y. Chen, Homoclinic solutions in periodic difference equations with saturable nonlinearity, Science China, Mathematics, 54 (2011), 83-93. doi: 10.1007/s11425-010-4101-9.

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