A Generalized Mountain Pass Lemma with a Closed Subset for Locally Lipschitz Functionals

TL;DR

This paper extends the Mountain Pass Lemma to locally Lipschitz functionals using Clarke's generalized gradient and Ekeland's variational principle, with applications to Hamiltonian systems.

Contribution

It generalizes Ghoussoub-Preiss's theorem for locally Lipschitz functionals, broadening the scope of the mountain pass approach in nonsmooth analysis.

Findings

Established a generalized mountain pass theorem for locally Lipschitz functionals.

Applied the theorem to find periodic solutions in Hamiltonian systems.

Extended classical results to a broader nonsmooth setting.

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.