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Remarks on finding critical points. (English) Zbl 0751.58006

Let \(F\) be a real \(C^ 1\) function defined on a Banach space \(X\). In the first part of the paper there are presented some applications of Ekeland’s Principle in obtaining critical points of functions \(F\) which satisfy the Palais-Smale property. The main result is given by Theorem 1, proved in this part by using Ekeland’s Principle. In the second part is presented a general deformation theorem (Theorem 3). Next a new proof of a recent theorem of Ghoussoub (Theorem 2) is given by using deformation Theorem 3. In the third part, the authors apply Theorem 2 to functions \(F\) which are bounded below and satisfy the Palais-Smale property. Finally, in the Appendix, the authors give a new proof of Theorem 1 based on deformation Theorem 3.

MSC:

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
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