Generalized Mountain Pass Lemma Related with a Closed Subset and 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, broadening its applicability in critical point theory.

Contribution

It generalizes Ghoussoub-Preiss's theorem for closed subsets in Banach spaces to the setting of locally Lipschitz functionals.

Findings

Generalization of the mountain pass theorem to locally Lipschitz functionals.

Application of Clarke's generalized gradient and Ekeland's variational principle.

Broader framework for critical point analysis in nonsmooth analysis.

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 in K.C. Chang, analysis of the structure of the critical set in the mountain pass theorem by Hofer and Pucci-Serrin and Tian, the extension by Ghoussoub-Preiss to closed subsets in a Banach space, and variations found in the recent Peral . In this paper, we utilize the generalized gradient of Clarke and Ekeland's variational principle to generalize the Ghoussoub-Preiss's Theorem in the setting of locally Lipschitz functionals.