The Method of Invariant Sets of Descending Flow for Locally Lipschitz Functionals

TL;DR

This paper extends the invariant sets method of descending flow to locally Lipschitz functionals, establishing existence of various critical points and applying these results to differential inclusion problems with p-Laplacian.

Contribution

It introduces new techniques for extending pseudo-gradient fields and invariance of sets, and develops a new (PS) condition for locally Lipschitz functionals.

Findings

Existence of positive, negative, and sign-changing critical points established.

New methods for extending pseudo-gradient fields preserving local information.

Application to differential inclusion problems with p-Laplacian demonstrated.

Abstract

In this paper, we extend the method of invariant sets of descending flow that proposed by Sun Jingxian for smooth functionals to the locally Lipschitz functionals. By this way, we obtain the existence results for the positive, negative and sign-changing critical points of the locally Lipschitz functionals, and apply these theoretical results to the study of differential inclusion problems with p-Laplacian. In order to obtain the above results,we develop some new techniques: 1) We establish the method of how to extend the pseudo-gradient field to the whole space on the premise of preserving the useful information of the local pseudo-gradient field; 2) In the case of set-valued mapping, a pseudo-gradient field is established to make both the cone and the negative cone being invariant sets of descending flow. To obtain our main results, a new class of (PS) condition is also proposed.