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Remarks on finding critical points

โœ Scribed by Haim Brezis; Louis Nirenberg

Publisher
John Wiley and Sons
Year
1991
Tongue
English
Weight
882 KB
Volume
44
Category
Article
ISSN
0010-3640

No coin nor oath required. For personal study only.

โœฆ Synopsis

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": any sequence (u,) in X such that F( u,) --* a and (1 F'( u,) I( + 0 has a convergent subsequence .

If this holds for every a E Iw one says that F satisfies (PS)-a condition originally introduced by Palais and Smale; see [ 131.

1. Some Applications of Ekeland's Principle

We start with an elementary statement in which (PS) is not assumed. PROPOSITION 1. If a = lim inf F( u ) I I U I I + ~ isjinite then there exists a sequence (u,) in X such that llu,,~l + 03, F( u,,) + a, and II F'( u,) II + 0.

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ON CRITICAL POINTS t h e s e c o n d order, t h e s e p o i n t s are to be c o n s i d e r e d as critical p o i n t s in t h e sense d e f i n e d here. T h e r e s e e m s t o be no o b j e c t i o n a g a i n s t t h e i n d i c a t i o n " c r i t i c a l triple p o i n t " , as t h i s t e r m