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On the Existence and the Classification of Critical Points for Non-Smooth Functionals

Published online by Cambridge University Press:  20 November 2018

G. Fang
G. Fang*
Affiliation:
Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012, U.S.A. e-mail: fang@cims. nyu. edu
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Abstract

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We extend the min-max methods used in the critical point theory of differentiable functionals on smooth manifolds to the case of continuous functionals on a complete metric space. We study the topological properties of the min-max generated critical points in this new setting by adopting the methodology developed by Ghoussoub in the smooth case. Many old and new results are extended and unified and some applications are given.

Keywords

58E0549F15
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 4 , 01 August 1995 , pp. 684 - 717
Copyright
Copyright © Canadian Mathematical Society 1995

References

1. Ambrosetti, A. and Rabinowitz, P.H., Dual Variational methods in critical point theory and applications, J. Funct Anal. 14(1973), 349–381.Google Scholar
2. Dancer, E.N., Degenerate critical points, homotopy indices and Morse inequalities HI, Bull. Austral. Math. Soc. 40(1989), 97–108.Google Scholar
3. Degiovanni, M. and Marzocchi, M., A critical point theory for nonsmooth functionals, (1991), preprint.Google Scholar
4. Ekeland, I., Nonconvex minimization problems, Bull. Amer. Math. Soc. 1(1979), 443–474.Google Scholar
5. Ekeland, I., Convexity Methods in Hamiltonian Mechanics, Springer-Verlag, Berlin, Heidelberg, New York, 1990.Google Scholar
6. Fang, G., The structure of the critical set in general mountain pass principle, Ann. Fac. Sci. Toulouse Math. (3) 3(1994).Google Scholar
7. Fang, G., Topics on critical point theory, Ph.D. thesis, the University of British Columbia, 1993.Google Scholar
8. Fang, G. and Ghoussoub, N., Second order information on Palais-Smale sequences in the mountain pass theorem, Manuscripta Math. 75(1992), 81–95.Google Scholar
9. Fang, G., Morse-type information on Palais-Smale sequences obtained by min-max principles, Comm. Pure Appl. Math. 47(1994), 1595–1653.Google Scholar
10. Ghoussoub, N., A Min-Max Principle with a relaxed boundary condition, Proc. Amer. Math. Soc. (2) 117(1993).Google Scholar
11. Ghoussoub, N., Location, multiplicity and Morse Indices of min-max critical points, J. Reine Angew. Math. 417(1991), 27–76.Google Scholar
12. Ghoussoub, N., Duality and perturbation methods in critical point theory, Cambridge Tracts in Math., Cambridge University Press, 1993.Google Scholar
13. Ghoussoub, N. and Preiss, D., A general mountain pass principle for locating and classifying critical points, Ann. Inst. H.Poincaré Anal. Non Linéaire (5) 6(1989), 321–330.Google Scholar
14. Ho fer, H., A geometric description of the neighborhood of a critical point given by the mountain pass theorem, J. London Math. Soc. 31(1985), 566–570.Google Scholar
15. Hu, S.T., Homotopy theory, Academic press, New York, 1959.Google Scholar
16. Kunen, K. and Vaughan, J.E., Handbook of set theoretic topology, North-Holland, 1984.Google Scholar
17. Kuratowski, K., Topology, Vol II, Academic Press, New York and London, 1968.Google Scholar
18. Lazer, A.C. and Solimini, S., Nontrivial solutions of operator equation and Morse indices of critical points of min-max type, Nonlinear Anal. (8) 12(1988), .761-775.Google Scholar
19. Mawhin, J. and Willem, M., Critical point theory and Hamiltonian systems, Springer-Verlag, 1989.Google Scholar
20. Nagata, J., Modern dimension theory, North-Holland Publishing Company-Amsterdam, 1965.Google Scholar
21. Nagata, J., Modern general topology, North-Holland, Second revised edition, 1985.Google Scholar
22. Pucci, P. and Serrin, J., Extensions of the mountain pass theorem, J. Funct. Anal. 59(1984), 185–210.Google Scholar
23. Pucci, P., A mountain pass theorem, J. Differential Equations 60(1985), 142–149.Google Scholar
24. Pucci, P., The structure of the critical set in the mountain pass theorem, Trans. Amer. Math. Soc. (1) 299(1987), 115–132.Google Scholar
25. Solimini, S., Morse index estimates in Min-Max theorems, Manuscripta Math. (4) 63(1989), 421–453.Google Scholar
26. Spanier, E.H., Algebraic topology, McGraw-Hill, New York, 1966.Google Scholar
27. Struwe, M., Plateau s Problem and the Calculus of Variations, Math. Notes, Princeton University Press, 1989.Google Scholar