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Morse-type information on Palais-Smale sequences obtained by min-max principles. (English) Zbl 0829.58008

The authors consider a \({\mathcal C}^2\)-functional \(\varphi: X\to \mathbb{R}\) defined on a Hilbert manifold \(X\) with \(d\varphi\) and \(d^2 \varphi\) being Hölder continuous. Let \(c\in \mathbb{R}\) be a potential critical value obtained via a minimax description like in the mountain pass theorem or a linking theorem. In order to prove that \(c\) is a critical value using Ekeland’s variational principle, for instance, one can first construct a Palais-Smale sequence \((x_n)\) in \(X\), that is, \(\varphi (x_n)\to c\) and \(\varphi' (x_n)\to 0\). Then it remains to show that this sequence has an accumulation point which will be a critical value at the level \(c\).
The authors use the minimax description in order to obtain information on the approximate Morse indices of \(x_n\) as well as on the location of \((x_n)\) in the space \(X\). This additional information can potentially be useful to prove the existence of an accumulation point even if arbitrary Palais-Smale sequences need not have one. Several results in this direction are proved in detail. As an application of this abstract critical point theory a proof of a result of P. L. Lions on the existence of infinitely many solutions of the Hartree-Fock equation is included.

MSC:

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
49Q20 Variational problems in a geometric measure-theoretic setting
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