Generalized Mountain Pass Lemma Related with a Closed Subset and Locally Lipschitz Functionals

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.