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Location, multiplicity and Morse indices of min-max critical points. (English) Zbl 0736.58011
Let \(\phi\) be a \(C^ 1\)-functional verifying a compactness condition of Palais-Smale type on a smooth Banach manifold \(X\). A well established procedure for exhibiting critical points for \(\phi\) consists of finding an appropriate class \(\mathcal F\) of compact subsets of \(X\), all containing a fixed boundary \(B\), and then showing that the number \[ \textstyle c=c(\phi,{\mathcal F})=\inf_{A\in{\mathcal F}}\sup_{x\in A}\phi(x) \] is actually a critical value for \(\phi\), provided it satisfies (F0) \(\sup\phi(B)<c\). The goal in this paper is, first to relax the boundary condition (F0) (i.e. to allow \(\sup\phi(B)=c\) sometimes) and second to get some more information about the location of the critical points obtained by such a procedure.
Then it is shown that, once such a refined version of the min-max principle is proved, it can be used to derive — besides existence results in the “limiting case” — various old and new results concerning the multiplicity and the Morse indices of the critical points. One is a “min-max counterpart” of Eckeland’s minimization principle and another is reminiscent of the Ljusternik-Schnirelmann theory.
Reviewer: M.Adachi (Kyoto)

MSC:
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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