×

zbMATH — the first resource for mathematics

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:
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349 – 381. · Zbl 0273.49063
[2] David C. Clark, A variant of the Lusternik-Schnirelman theory, Indiana Univ. Math. J. 22 (1972/73), 65 – 74. · Zbl 0228.58006 · doi:10.1512/iumj.1972.22.22008 · doi.org
[3] I. Ekeland and H. Hofer, Periodic solutions with prescribed minimal period for convex autonomous Hamiltonian systems, Invent. Math. 81 (1985), no. 1, 155 – 188. · Zbl 0594.58035 · doi:10.1007/BF01388776 · doi.org
[4] Helmut Hofer, A geometric description of the neighbourhood of a critical point given by the mountain-pass theorem, J. London Math. Soc. (2) 31 (1985), no. 3, 566 – 570. · Zbl 0573.58007 · doi:10.1112/jlms/s2-31.3.566 · doi.org
[5] L. Nirenberg, Variational and topological methods in nonlinear problems, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 3, 267 – 302. · Zbl 0468.47040
[6] Patrizia Pucci and James Serrin, A mountain pass theorem, J. Differential Equations 60 (1985), no. 1, 142 – 149. · Zbl 0585.58006 · doi:10.1016/0022-0396(85)90125-1 · doi.org
[7] Patrizia Pucci and James Serrin, Extensions of the mountain pass theorem, J. Funct. Anal. 59 (1984), no. 2, 185 – 210. · Zbl 0564.58012 · doi:10.1016/0022-1236(84)90072-7 · doi.org
[8] P. H. Rabinowitz, Variational methods for nonlinear eigenvalue problems, Eigenvalues of non-linear problems (Centro Internaz. Mat. Estivo (C.I.M.E.), III Ciclo, Varenna, 1974) Edizioni Cremonese, Rome, 1974, pp. 139 – 195.
[9] -, Some aspects of critical point theory, Univ. of Wisconsin, MRC Tech. Rep. No. 2465, 1983.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.