A generalized mountain pass lemma with a closed subset for locally Lipschitz functionals

ABSTRACT

The classical Mountain Pass Lemma of Ambrosetti-Rabinowitz has been studied, extended and modified in several directions. Notable examples would certainly include the generalization to locally Lipschitz functionals by K. C. Chang, analyzing the structure of the critical set in the mountain pass theorem in the works of Hofer, Pucci-Serrin and Tian, and the extension by Ghoussoub-Preiss to closed subsets in a Banach space with recent variations. In this paper, we utilize the generalized gradient of Clarke and Ekeland's variatonal principle to generalize the Ghoussoub-Preiss's Theorem in the setting of locally Lipschitz functionals. We give an application to periodic solutions of Hamiltonian systems.

Keywords:

• Mountain pass lemma of Ambrosetti-Rabinowitz
• Ekeland's variational principle
• locally Lipschitz functionals
• Clarke's generalized gradient
• generalized mountain pass lemma

2000 Mathematical Subject Classifications:

• 34A34
• 34C25
• 35A15

Acknowledgments

This research was partially supported by NSF of China (No. 11701463, No. 11671278 and No. 12071316). The authors sincerely thank the referees and the editors for their many valuable comments and suggestions which help us improving the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by National Natural Science Foundation of China [11671278,11701463,12071316].