Periodic solutions with prescribed minimal period for second order even Hamiltonian systems
Periodic solutions with prescribed minimal period for second order even Hamiltonian systems
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DOI：
10.3934/cpaa.2021166
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发表时间：
2021
影响因子：
1
通讯作者：
Juhong Kuang;Weiyi Chen;Zhiming Guo
中科院分区：
文献类型：
--
作者：
Juhong Kuang;Weiyi Chen;Zhiming Guo
本文发展了一种新的方法来研究二阶甚至哈密顿系统在没有任何凸性假设的情况下，Rabinowitz关于具有给定最小周期的周期解的存在性的猜想。具体地说，我们首先研究了具有给定步长的哈密顿系统离散化的相关齐次Dirichlet边值问题，并利用离散变分方法得到了对应于不同步长的非负解序列。然后，利用非负解序列，我们构造了一个可证明为预紧的连续函数序列。最后，利用收敛子序列的极限函数和势的对称性，得到期望的周期解。特别地，我们在势满足一定对称假设的情况下证明了Rabinowitz猜想。此外，在Begin{DOCUMENT}$N=1$END{DOCUMENT}的情况下，我们的主要结果大大改进了文献中的相关结果。
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 \begin{document}$ N = 1 $\end{document}.