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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.