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

Qinqin Zhang

doi:10.3934/cpaa.2019021

Article Contents

2019, Volume 18, Issue 1: 425-434. Doi: 10.3934/cpaa.2019021

This issue Previous Article Spectral expansion series with parenthesis for the nonself-adjoint periodic differential operators Next Article A note on commutators of the fractional sub-Laplacian on Carnot groups

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
^* Corresponding author

Received: January 31, 2018

Revised: March 31, 2018

Published: August 2018

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

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Abstract

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.

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

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