A generalized mountain pass lemma with a closed subset for locally Lipschitz functionals. Issue 16 (2nd November 2022)

• Record Type: Journal Article
• Authors: Li, Fengying; Li, Bingyu; Zhang, Shiqing
• Is Part Of: Applicable analysis. Volume 101:Issue 16(2022)
• Page Start: 5643
• Page End: 5659
• Publication Date: 2022-11-02
• Subjects: Mountain pass lemma of Ambrosetti-Rabinowitz -- Ekeland's variational principle -- locally Lipschitz functionals -- Clarke's generalized gradient -- generalized mountain pass lemma; 34A34 -- 34C25 -- 35A15; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2021.1903441
• Languages: English
• ISSNs: 0003-6811

Abstract

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.