Deformation properties for continuous functionals and critical point theory. (English) Zbl 0789.58021

Corvellec, Jean-Noël; Degiovanni, Marco; Marzocchi, Marco

Topol. Methods Nonlinear Anal. 1, No. 1, 151-171 (1993).

Let \(X\) be a complete metric space and \(f: X \to \mathbb{R}\) be continuous. The authors define the notion of weak slope \(| df| (u) \in [0,\infty]\), \(u\in X\), which corresponds to \(\| df(u)\|\) if \(X\) and \(f\) are of class \(C^ 1\). The definition is based on the existence of certain deformations of neighborhoods of \(u\) along which \(f\) decreases. Using these local deformations the authors prove a version of the deformation lemma. If \(f\) satisfies the Palais-Smale condition abstract critical point theorems follow in a standard way. No use is made of Ekeland’s variational principle. Under additional assumptions the theory can be generalized to lower semi-continuous functions.

Reviewer: Th.J.Bartsch (Heidelberg)

MSC:

58E05

Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

Keywords:

critical point theory; continuous functionals; deformation lemma