Some remarks on the Lyusternik-Schnirelman method for non-differentiable functionals invariant with respect to a finite group action (Q805022)

In a previous paper [J. Differ. Equations, 85, No.1, 105-124 (1990; Zbl 0703.58012)] the authors presented a general and explicit formula for the number of critical points of a functional of class \(C^ 1\), invariant with respect to a finite group action. The purpose of this paper is to extend this variational minimax method to the class of locally Lipschitzian functionals. Let G be a finite group, M be a Finsler G-manifold modelled on a reflexive Banach space E and f: \(M\to {\mathbb{R}}\) be a G-invariant locally Lipschitzian functional, bounded below. The generalized gradient of \textit{F. H. Clarke} [Optimization and nonsmooth analysis (1983; Zbl 0582.49001)] is used to extend the concepts of critical point and the Palais-Smale condition. The main result states some lower estimate for the number of critical points of locally Lipschitzian functional, invariant with respect to G- action and satisfying Palais-Smale condition, in terms of the isotropy subgroups of G.