Infinitely many non-constant periodic solutions with negative fixed energy for Hamiltonian systems

ABSTRACT

In this paper, we obtain the existence of infinitely many non-constant periodic solutions with negative fixed energy for a class of second-order Hamiltonian systems, which greatly improves the existing results such as Zhang [Periodic solutions for some second order Hamiltonian systems. Nonlinearity. 2009;22(9):2141–2150, Theorem 1.5]. Moreover, we exhibit two simple and instructive examples to make our result more clear, which have not been solved by known results.

COMMUNICATED BY:

• Boris Vainberg

KEYWORDS:

• Periodic solutions
• Hamiltonian systems
• critical point theorem
• Palais symmetry principle
• Egorov theorem

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 34C25
• 58E05

Acknowledgements

The authors express their sincere gratitude to professor Zhang Shiqing for his helpful suggestions, and also sincerely thank the referees and the editors for their helpful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Jinlong Wei http://orcid.org/0000-0002-9182-4587

Additional information

Funding

The first author is partially supported by the Graduate Student's Research and Innovation Fund of Sichuan University (2018YJSY045) and the National Natural Science Foundation of China [grant number 116712787]. The second author is partially supported by the National Natural Science Foundation of China [grant numbers 11501577 and 61773401].