JOURNAL OF DIFFERENTIAL EQUATIONS 85, 105-124 (1990) Lusternik-Schnirelman Method for Functionals Invariant with Respect to a Finite Group Action W. KRAWCEWICZ' Department of Mathematics, University of Alberta, Edmonton, Canada T6G 2GI AND W. MARZANTOWICZ Institute of Mathematics, University of Gdarisk, Gdatisk, Poland Received November 14, 1988; revised May 4, 1989 INTRODUCTION A function f: s” + [w of class C’ has at least two critical points. However, the existence of symmetries for f can give a much larger number of critical points. For example, iffis invariant with respect to a free action of a finite group G, then f has at least ( G I (n + 1) distinct critical points (see [Kl, MA]; we refer also to Corollary (1.5) which is a generalization of this result). Consequently, the existence of a symmetry group G for the func- tionalf, in general defined on a Banach manifold, has a significant impact on the number of critical points. The problem of finding the best possible estimates for the number of orbits of critical points of an invariant func- tional was studied by many authors; for a I!,-action with p prime number, see [EL]; for S’-action, see [Bl, B2, CW2, CW3, FR2, R2]; for a non-free action, where Morse theory is applied, see [Pl, CWl, W]; see also [DA, FH, FRl]. This work is an attempt to give a general and explicit formula (cf. Theorem (1.3)) for the number of critical points of an invariant func- tional f with respect to a finite group action which is not necessarily free. More precisely, let M be a paracompact and complete Banach manifold endowed with an action by diffeomorphisms of a finite group G. We prove (see Theorem (1.3)) that if M\MG contains a G-invariant subset S which ’ Research supported by grant from NSERC Canada. 105 0022-0396190 $3.00 Copyright 6 1990 by Academic Press, Inc. All rights of reproductmn I” any form reserved. brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by Elsevier - Publisher Connector