Sign Changing Critical Points for Locally Lipschitz Functionals

TL;DR

This paper develops new theoretical results for finding sign-changing critical points of locally Lipschitz functionals in Banach spaces, relaxing the need for C1 continuity and applying advanced deformation methods.

Contribution

It introduces a novel approach combining invariant sets and quantitative deformation to establish critical point theorems for locally Lipschitz functionals, extending previous results.

Findings

Established existence of sign-changing critical points under weaker conditions

Developed a new quantitative deformation lemma for locally Lipschitz functionals

Applied results to differential inclusion problems

Abstract

In this paper, some existence results for sign-changing critical points of locally Lipschitz functionals in real Banach space are obtained by the method combining the invariant sets of descending ow method with a quantitative deformation. First we assume the locally Lipschitz functionals to be outwardly directed on the the boundary of some closed convex sets of the real Banach space. By using the relation between the critical points on the Banach space and those of the closed convex sets, we construct a quantitative deformation lemma, and then we obtain some linking type of critical points theorems. These theoretical results can be applied to the study of the existence of sign-changing solutions for differential inclusion problems. In contrast with the related results in the literatures, the main results of this paper relax the requirement that the functional being of C1 continuous to locally Lipschitz.