Periodic solutions with prescribed minimal period for second order even Hamiltonian systems

Juhong Kuang; Weiyi Chen; Zhiming Guo

doi:10.3934/cpaa.2021166

Article Contents

2022, Volume 21, Issue 1: 47-59. Doi: 10.3934/cpaa.2021166

This issue Previous Article Radial quasilinear elliptic problems with singular or vanishing potentials Next Article Singular limit for reactive transport through a thin heterogeneous layer including a nonlinear diffusion coefficient

Periodic solutions with prescribed minimal period for second order even Hamiltonian systems

1.
School of Mathematics and Computational Sciences, Wuyi University, Jiangmen, 529020, China
2.
School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China

^* Corresponding author
^* Corresponding author

Received: March 31, 2021

Revised: May 31, 2021

Early access: September 2021

Published: January 2022

The first author is supported by National Natural Science Foundation of China grant 11901438 and Natural Science Foundation of Guangdong Province, China grant 2018A0303130058, 2021A1515010062. The third author is supported by National Natural Science Foundation of China grant 11771104, 12171110 and Science and Technology Planning Project of Guangdong Province of China grant 2020A1414010106

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Abstract

In this paper, we develop a new method to study Rabinowitz's conjecture on the existence of periodic solutions with prescribed minimal period for second order even Hamiltonian system without any convexity assumptions. Specifically, we first study the associated homogenous Dirichlet boundary value problems for the discretization of the Hamiltonian system with given step length and obtain a sequence of nonnegative solutions corresponding to different step lengths by using discrete variational methods. Then, using the sequence of nonnegative solutions, we construct a sequence of continuous functions which can be shown to be precompact. Finally, by utilizing the limit function of convergent subsequence and the symmetry of the potential, we will obtain the desired periodic solution. In particular, we prove Rabinowitz's conjecture in the case when the potential satisfies a certain symmetric assumption. Moreover, our main result greatly improves the related results in the literature in the case where $ N = 1 $.

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Mathematics Subject Classification: Primary: 34A12, 39A23, 39A70.

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