A generalized mountain pass lemma with a closed subset for 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 by K. C. Chang, analyzing structure critical set mountain pass theorem works Hofer, Pucci-Serrin Tian, extension Ghoussoub-Preiss closed subsets a Banach space with recent variations. In this paper, we utilize generalized gradient Clarke Ekeland's variatonal principle generalize Ghoussoub-Preiss's Theorem setting functionals. We give an application periodic solutions Hamiltonian systems.