HOMOCLINIC SOLUTIONS NEAR THE ORIGIN FOR A CLASS OF FIRST ORDER HAMILTONIAN SYSTEMS*

doi:10.1007/s10473-023-0312-3

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (3): 1195-1210.doi: 10.1007/s10473-023-0312-3

HOMOCLINIC SOLUTIONS NEAR THE ORIGIN FOR A CLASS OF FIRST ORDER HAMILTONIAN SYSTEMS^*

Qingye Zhang¹, Chungen Liu^2,†

1. School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022, China;
2. School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

收稿日期:2021-09-29 出版日期:2023-06-25 发布日期:2023-06-06
通讯作者: ^† Chungen Liu, E-mail: liucg@nankai.edu.cn
作者简介:Qingye Zhang, E-mail: zhangqy@jxnu.edu.cn
基金资助:
National Natural Science Foundation of China (11761036, 11201196) and the Natural Science Foundation of Jiangxi Province (20171BAB211002); The second author was supported by the National Natural Science Foundation of China (11790271, 12171108), the Guangdong Basic and Applied basic Research Foundation (2020A1515011019), the Innovation and Development Project of Guangzhou University, and the Nankai Zhide Foundation.

HOMOCLINIC SOLUTIONS NEAR THE ORIGIN FOR A CLASS OF FIRST ORDER HAMILTONIAN SYSTEMS^*

Qingye Zhang¹, Chungen Liu^2,†

1. School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022, China;
2. School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

Received:2021-09-29 Online:2023-06-25 Published:2023-06-06
Contact: ^† Chungen Liu, E-mail: liucg@nankai.edu.cn
About author:Qingye Zhang, E-mail: zhangqy@jxnu.edu.cn
Supported by:
National Natural Science Foundation of China (11761036, 11201196) and the Natural Science Foundation of Jiangxi Province (20171BAB211002); The second author was supported by the National Natural Science Foundation of China (11790271, 12171108), the Guangdong Basic and Applied basic Research Foundation (2020A1515011019), the Innovation and Development Project of Guangzhou University, and the Nankai Zhide Foundation.

摘要/Abstract

摘要： In this paper, we study the existence of infinitely many homoclinic solutions for a class of first order Hamiltonian systems $\dot{z}=JH_z(t,z)$, where the Hamiltonian function $H$ possesses the form $H(t,z)=\frac{1}{2}L(t)z\cdot z +G(t,z)$, and $G(t,z)$ is only locally defined near the origin with respect to $z$. Under some mild conditions on $L$ and $G$, we show that the existence of a sequence of homoclinic solutions is actually a local phenomenon in some sense, which is essentially forced by the subquadraticity of $G$ near the origin with respect to $z$.

关键词: Hamiltonian systems, homoclinic solutions, variational method

Abstract: In this paper, we study the existence of infinitely many homoclinic solutions for a class of first order Hamiltonian systems $\dot{z}=JH_z(t,z)$, where the Hamiltonian function $H$ possesses the form $H(t,z)=\frac{1}{2}L(t)z\cdot z +G(t,z)$, and $G(t,z)$ is only locally defined near the origin with respect to $z$. Under some mild conditions on $L$ and $G$, we show that the existence of a sequence of homoclinic solutions is actually a local phenomenon in some sense, which is essentially forced by the subquadraticity of $G$ near the origin with respect to $z$.

Key words: Hamiltonian systems, homoclinic solutions, variational method

Qingye Zhang, Chungen Liu. HOMOCLINIC SOLUTIONS NEAR THE ORIGIN FOR A CLASS OF FIRST ORDER HAMILTONIAN SYSTEMS^*[J]. 数学物理学报(英文版), 2023, 43(3): 1195-1210.

Qingye Zhang, Chungen Liu. HOMOCLINIC SOLUTIONS NEAR THE ORIGIN FOR A CLASS OF FIRST ORDER HAMILTONIAN SYSTEMS^*[J]. Acta mathematica scientia,Series B, 2023, 43(3): 1195-1210.

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HOMOCLINIC SOLUTIONS NEAR THE ORIGIN FOR A CLASS OF FIRST ORDER HAMILTONIAN SYSTEMS^*

HOMOCLINIC SOLUTIONS NEAR THE ORIGIN FOR A CLASS OF FIRST ORDER HAMILTONIAN SYSTEMS^*

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