Deformation properties for continuous functionals and critical point theory (Q1310500)

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.