# American Institute of Mathematical Sciences

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Radial quasilinear elliptic problems with singular or vanishing potentials
January  2022, 21(1): 47-59. doi: 10.3934/cpaa.2021166

## 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

Received  April 2021 Revised  June 2021 Published  January 2022 Early access  September 2021

Fund Project: 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

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 $N = 1$.

Citation: Juhong Kuang, Weiyi Chen, Zhiming Guo. Periodic solutions with prescribed minimal period for second order even Hamiltonian systems. Communications on Pure & Applied Analysis, 2022, 21 (1) : 47-59. doi: 10.3934/cpaa.2021166
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