Remarks on Finding Critical Points

In the course of writing a chapter of a book we observed some simple facts dealing with the Palais SmaIe property and critical points of functions. Some of these facts turned out to be known, though not well-known, and we think it worthwhile to make them more available. In addition, we present some other recent results which we believe will prove to be useful-in particular, a result of Ghoussoub and Preiss; see [ 91, [ 81. There are two useful techniques used in obtaining critical points. One is Ekeland’s Principle (see below), the other is based on deformation arguments. We will use versions of both of them. In particular we present a rather general deformation result. Throughout this paper we consider real C’ functions F defined on a Banach space X. When looking for critical points of F it has become standard to assume the following “compactness condition”: