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.

摘要

Ambrosetti-Rabinowitz 的经典山口引理已在多个方向上进行了研究、扩展和修改。值得注意的例子当然包括 KC Chang 对局部 Lipschitz 泛函的推广，在 Hofer、Pucci-Serrin 和 Tian 的著作中分析了山口定理中临界集的结构，以及 Ghoussoub-Preiss 对封闭子集的扩展具有最近变化的 Banach 空间。在本文中，我们利用 Clarke 的广义梯度和 Ekeland 变分原理在局部 Lipschitz 泛函的设置中推广 Ghoussoub-Preiss 定理。我们给出了哈密顿系统周期解的应用。