×

Remarks on finding critical points. (English) Zbl 0751.58006

Let \(F\) be a real \(C^ 1\) function defined on a Banach space \(X\). In the first part of the paper there are presented some applications of Ekeland’s Principle in obtaining critical points of functions \(F\) which satisfy the Palais-Smale property. The main result is given by Theorem 1, proved in this part by using Ekeland’s Principle. In the second part is presented a general deformation theorem (Theorem 3). Next a new proof of a recent theorem of Ghoussoub (Theorem 2) is given by using deformation Theorem 3. In the third part, the authors apply Theorem 2 to functions \(F\) which are bounded below and satisfy the Palais-Smale property. Finally, in the Appendix, the authors give a new proof of Theorem 1 based on deformation Theorem 3.

MSC:

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ambrosetti, J. Nonlinear Anal. 8 pp 1145– (1984)
[2] Ambrosetti, J. Funct. Anal. 14 pp 349– (1973)
[3] and , Applied Nonlinear Analysis, Wiley-Interscience, New York, 1984.
[4] Chang, Comm. Pure Appl. Math. 34 pp 693– (1981)
[5] Infinite Dimensional Morse Theory and its Applications, Presses Univ. Montréal, 1985.
[6] Costa, Nonlinear Anal., Theory, Methods & Appl. 16 pp 371– (1991)
[7] Ekeland, J. Math. Anal. Applic. 47 pp 324– (1974)
[8] Location, multiplicity and Morse indices of min-max critical points, Z. Reine und Angew. Math, in press. · Zbl 0736.58011
[9] Ghoussoub, Ann. I. H. P. Analyse Nonlinéaire 6 pp 321– (1989)
[10] Some aspects of critical point theory, to appear.
[11] The Morse index of a saddle point, to appear. · Zbl 0732.58011
[12] Liu, Kexue Tongbao 17 pp 1025– (1984)
[13] Palais, Bull. Amer. Math. Soc. 70 pp 165– (1964)
[14] Pucci, J. Diff. Eqns. 60 pp 142– (1985)
[15] Minimax methods in critical point theory with applications to differential equations, CBMS Reg. Conf., Vol. 65, Amer. Math. Soc., 1986.
[16] Elves Alves de B. e, Nonlinear Anal., Theory, Methods & Appl. 16 pp 455– (1991)
[17] Taubes, J. Diff. Geom. 19 pp 337– (1984)
[18] Caklovic, Diff. Int. Eqn. 3 pp 799– (1990)
[19] Lectures on critical point theory, Fund. Universidade de Brasília, No. 199, 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.