Subharmonic solutions of bounded coupled Hamiltonian systems with sublinear growth

Fanfan Chen; Dingbian Qian; Xiying Sun; Yinyin Wu

doi:10.3934/cpaa.2021180

Article Contents

2022, Volume 21, Issue 1: 337-354. Doi: 10.3934/cpaa.2021180

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Subharmonic solutions of bounded coupled Hamiltonian systems with sublinear growth

1.
School of Mathematical Sciences, Soochow University, Suzhou 215006, China
2.
Department of Fundamental Courses, Wuxi Institute of Technology, Wuxi 214121, China

^* Corresponding author
^* Corresponding author

Received: March 31, 2021

Revised: August 31, 2021

Early access: October 2021

Published: January 2022

This work is supported by the National Natural Science Foundation of China (No. 12071327, No. 11671287)

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Abstract

We prove the existence and multiplicity of subharmonic solutions for bounded coupled Hamiltonian systems. The nonlinearities are assumed to satisfy Landesman-Lazer conditions at the zero eigenvalue, and to have some kind of sublinear behavior at infinity. The proof is based on phase plane analysis and a higher dimensional version of the Poincaré-Birkhoff twist theorem by Fonda and Ureña. The results obtained generalize the previous works for scalar second-order differential equations or relativistic equations to higher dimensional systems.

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Mathematics Subject Classification: Primary: 34C25; Secondary: 34C15, 46N20.

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Figure 1. $ z_k(t)\in \mathcal{D}_{1} $ and $ z_k(t)\in\mathcal{D}_{1}\cup \mathcal{E}_{k, y} $

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Figure 2. $ z_k(t)\in \mathcal{D}_{2} $

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