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The structure of the critical set in the mountain pass theorem. (English) Zbl 0611.58019
Let I be a real-valued $$C^ 1$$ functional defined on a real Banach space X. Suppose I satisfies the Palais-Smale condition, and put $$K_ b=\{x\in X:$$ $$I(x)=b$$ and $$I'(x)=0\}$$ for any real number b. Under some assumptions, it is well-known that $$K_ b$$ is not empty by the mountain pass theorem. In this paper the authors consider the nature of $$K_ b.$$
For the infinite dimensional case they show that either the critical set $$K_ b$$ contains a point which is simultaneously of saddle and mountain- pass type, or the set of local minima in $$K_ b$$ is nonempty and its closure intersects at least two components of the set of saddle points. In the case of finite dimensional spaces, they obtain a weaker version. These results improve those in the papers by H. Hofer [J. Lond. Math. Soc., II. Ser. 31, 566-570 (1985; Zbl 0573.58007)] and P. Pucci and J. Serrin [J. Funct. Anal. 59, 185-210 (1984; Zbl 0564.58012)].
Reviewer: Duong Minh Duc

##### MSC:
 5.8e+06 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
##### Keywords:
structure of the critical set; Mountain pass theorem
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##### References:
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