Skip to main content
SpringerLink
Log in

Morse Theory in Differential Equations

  • Conference paper
Proceedings of the International Congress of Mathematicians

  • 3405 Accesses

Abstract

In the study of closed geodesics, Marston Morse developed his theory on the calculus of variations in the large. The Morse inequalities, which link on one hand, the numbers of critical points in various types of a function, and on the other hand, the topological invariants of the underlying manifold, play an important role in Morse theory. Naturally, they provide an estimate for the number of critical points of a function by using the topology of the manifold. Hopefully, this topological method would deal with the existence and the multiplicity of solutions of certain nonlinear differential equations. However, in this theory, the manifold is compact, and the functions are assumed to be C2 and to have only nondegenerate critical points; all of these restrict the applications seriously. In contrast, Leray-Schauder degree theory has become a very useful topological method. In 1946, at the bicentennial conferences of Princeton University, there was much discussion of their contrast. M. Shiffman hoped that the two methods could be brought closer together “so that they may alter and improve each other, and also so that each may fill out the gaps in the scope of the other” [Pr]. Since then, great efforts have been made to extend the Morse theory. We only mention a few names of the pioneers as follows: R. Bott, E. Rothe, R. S. Palais, S. Smale, D. Gromoll, W. Meyer, A. Marino, and G. Prodi.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Amann, H., and Zehnder, E., Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations, Ann. Scuola Norm Sup. Pisa CI. Sci. (4) 7 (1980), 539–630.

    MathSciNet  MATH  Google Scholar 

  2. Bahri, A., Critical point at infinity in some variational problems, Pitman Res. Notes Math. Longman, London, 182 (1989).

    MATH  Google Scholar 

  3. Another proof of the Yamabe conjecture for locally conformally flat manifolds, Nonlinear Anal. TMA 20 (1993), 1261–1278.

    Google Scholar 

  4. The scalar cuvature problem on spheres of dimension larger or equal than 7, preprint (1994).

    Google Scholar 

  5. Bahri, A., and Brezis, H., Elliptic differential equations involving the Sobolev critical exponent on manifolds, preprint (1994).

    MATH  Google Scholar 

  6. Bahri, A., and Coron, J. M., On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math. 41 (1988), 253–294.

    Article  MathSciNet  Google Scholar 

  7. Bahri, A., and Coron, J. M., The scalar–curvature problem on the standard three–dimensional sphere, J. Funct. Anal. 95 (1991), 106–172.

    Article  MathSciNet  Google Scholar 

  8. Bahri, A., and Rabinowitz, P. H., Periodic solutions of Hamiltonian systems of three-body type, Analyse Non Lineaire 6 (1991), 561–649.

    MATH  Google Scholar 

  9. Bartch, T.; Clapp, M.; and Puppe, D., A mountain pass theorem for actions of compact Lie groups, J. Reine Angew. Math. 419 (1991).

    Google Scholar 

  10. Benci, B., A new approach to the Morse-Conley theory and some applications, Ann. Mat. Pura Appl. (IV) 158 (1991), 231–305.

    Article  MathSciNet  Google Scholar 

  11. Benci, V.; Fortunato, D.; and Giannoni, F., On the existence of multiple geodesies in static space-time, Analyse Non Linéaire 8 (1991), 79–102.

    MATH  Google Scholar 

  12. Benci, V., and Giannoni, F., On the existence of closed geodesies on noncompact Riemannian manifolds, Duke Math. J. 68 (1992), 195–215.

    Article  MathSciNet  Google Scholar 

  13. Brézis, H., and Nirenberg, L., H1 versus C1 local minimizers, preprint (1993).

    Google Scholar 

  14. Chang, A. S. Y., and Yang, P., Prescribing Gaussian curvature on S2, Acta Math. 159 (1987), 214–259.

    Article  Google Scholar 

  15. Chang, A. S. Y., and Yang, P., Conformal deformation of metric on S2, JDG 23 (1988), 259–296.

    Google Scholar 

  16. Chang, K. C., Variational methods for non-differentiate functionals and their applications to PDE, J. Math. Anal. Appl. 80 (1981), 102–129.

    Article  MathSciNet  Google Scholar 

  17. Chang, K. C., Solutions of asymptotically linear operator equations via Morse theory, Comm. Pure Appl. Math. 34 (1981), 693–712.

    Article  MathSciNet  Google Scholar 

  18. Chang, K. C., A variant mountain pass lemma, Sci. Sinica Ser. A, 26 (1983), 1241–1255.

    MathSciNet  MATH  Google Scholar 

  19. Chang, K. C., Infinite dimensional Morse theory and its applications, Univ. de Montreal 97, 1985.

    MATH  Google Scholar 

  20. Chang, K. C., Morse theory for harmonic maps, Variational methods, (Ed., H. Berestycki), PNLDE 4, Birkhäuser, Basel and Boston, (1990), 431–446.

    Google Scholar 

  21. Chang, K. C., Critical groups, Morse theory and applications to semilinear elliptic BVPS, Chinese Math, into the 21st Century, (Ed., Wu Wen-tsun and Cheng Min-de ), Peking Univ. Press. (1991), 41–65.

    Google Scholar 

  22. Chang, K. C., On the homology method in the critical point theory, Partial Diff. Eqs. and Related subjects, (Ed., M. Miranda), Pitman Res. Notes Math. 269 (1992), 59–77.

    Google Scholar 

  23. Chang, K. C., Infinite dimensional Morse theory and multiple solution problems, PNLDE 6, Birkhauser, Basel and Boston, 1993.

    Book  Google Scholar 

  24. Chang, K. C., H1 versus C1 isolated critical points, Peking Univ., preprint (1993).

    Google Scholar 

  25. Chang, K. C., and Eells, J., Unstable minimal surface coboundaries, Acta Math. Sinica 2, (1986), 233–247.

    Article  MathSciNet  Google Scholar 

  26. Chang, K. C., and Jiang, M. Y., The Lagrange intersection for (CPn, RPn), Manuscripta Math. 68 (1990), 89–100.

    Article  MathSciNet  Google Scholar 

  27. Chang K. C., and Liu, J. Q., A strong resonance problem, Chinese Ann. Math. 11 B. 2 (1990), 191–210.

    MathSciNet  MATH  Google Scholar 

  28. Chang K. C., Morse theory under general boundary conditions, J. Systems Sci Math. Sci. 4 (1991), 78–83.

    MathSciNet  Google Scholar 

  29. Chang K. C., On Nirenberg’s problem, Internat. J. Math. 4 (1993), 35–58.

    MATH  Google Scholar 

  30. Chang K. C., A prescribing geodesic curvature problem, ICTP preprint (1993).

    Google Scholar 

  31. Chang, K. C.; Liu, J. Q.; and Liu, M. J., Nontrivial periodic solutions for strong resonance Hamiltonian Systems, Peking Univ. preprint (1993).

    MATH  Google Scholar 

  32. Chen, W. X., and Ding W. Y., Scalar curvature on S2, TAMS, 303 (1987), 365–382.

    Google Scholar 

  33. Clapp, M., and Puppe, D., Critical point theory with symmetries, J. Reine An-gew. Math. 418 (1991), 1–29.

    MathSciNet  MATH  Google Scholar 

  34. Conley, C. C., and Zehnder, E., Morse type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math. 37 (1984), 207–253.

    Article  MathSciNet  Google Scholar 

  35. Corvellec, J. N.; De Giovanni, M.; and Marzocchi, M., Deformation properties for continuous functionals and critical point theory, Univ. di Pisa, preprint (1992).

    Google Scholar 

  36. Dancer, E. N., Degenerate critical points, homotopy indices and Morse inequalities, J. Reine Angew. Math. 350 (1984), 1–22.

    MathSciNet  MATH  Google Scholar 

  37. Fe, G. H., Ph.D. thesis, Nanjing Univ. (1994).

    Google Scholar 

  38. Fadell, E., and Hsseini, S., Infinite cup length in free loop spaces with an application to a problem of the N-body type, Analyse Non Lineaire 9 (1992), 305–320.

    MathSciNet  MATH  Google Scholar 

  39. Ghoussonb, N., Duality and perturbation methods in critical point theory, Cambridge Tracts in Math. 107, Cambridge Univ. Press, Cambridge and New York, 1993.

    Book  Google Scholar 

  40. De Giorgi, E.; Marino, A.; and Tosques, M., Problemi di evoluzione in spazi metrici e curve di massima pendenza, Atti Accad. Naz. Lincei Rend. CI. Sci. Fis. Mat. Natur. (8) 68 (1980), 180–187.

    MATH  Google Scholar 

  41. Hingston, N., Equivariant Morse theory and closed geodesies, JDG 19 (1984), 85–116.

    MathSciNet  MATH  Google Scholar 

  42. Jost, J., and Struwe, M., Morse-Conley theory for minimal surfaces of varying topological type, Invent. Math. 102, (1990), 465–499.

    Article  MathSciNet  Google Scholar 

  43. Liu, J. Q., A Morse index of a saddle point.

    Google Scholar 

  44. Liu, J. Q., and Li, P. L., Nirenberg’s problem on the 2-dimensional hemisphere, Internat. J. of Math. (1994)

    Google Scholar 

  45. Li, S. J., and Liu, J. Q., Morse theory and asymptotically linear Hamiltonian systems, JDE, 78 (1989), 53–73.

    Article  Google Scholar 

  46. Li, Y., Prescribing scalar curvature on Sn and related problems, Part 1 and Part 2, preprint (1994).

    Google Scholar 

  47. Long, Y. M., Maslov index, degenerate critical points and asymptotically linear Hamiltonian systems, Science in China, Ser. A 33 (1990), 1409–1419.

    MathSciNet  MATH  Google Scholar 

  48. Long, Y. M., and Zehnder, E., Morse theory for forced oscillations of asymptotically linear Hamiltonian systems, Stochastic Process, Physics, and Geometry, World Sci. Press, (1990), 528–563.

    Google Scholar 

  49. Majer, P., Variational methods on manifolds with boundary, SISSA preprint (1991).

    MATH  Google Scholar 

  50. Majer, P., and Terracini, S., Periodic solutions to some problems of n-body type, Arch. Rational Mech. Anal. 124 (1993), 381–404.

    Article  MathSciNet  Google Scholar 

  51. Morse, M., The calculus of variations in the large, Amer. Math. Soc. Colloq. Publ. no. 18, Providence, RI, 1934.

    Google Scholar 

  52. Parker, T. H., A Morse theory for equivariant Yang-Mills, Duke Math. J. 66 (1992), 337–356.

    Article  MathSciNet  Google Scholar 

  53. Problems of Mathematics, Princeton University Bicentennial Conferences, Princeton, NJ 1947.

    Google Scholar 

  54. Riahi, H., Periodic orbits of N-body type problems, Ph.D. thesis, Rutgers Univ. (1993).

    Google Scholar 

  55. Rothe, E., Critical point theory in Hilbert space under general boundary conditions, J. Math. Anal. Appl. 2 (1965), 357–409.

    Article  MathSciNet  Google Scholar 

  56. Salamon, D., and Zehnder, E., Morse theory for periodic solutions of Hamiltonian systems and the Morse index, Comm. Pure Appl. Math. 45 (1992), 1303–1360.

    Article  MathSciNet  Google Scholar 

  57. Struwe, M., Plateau’s problem and the calculus of variations, Princeton, Tokyo, 1989.

    Book  Google Scholar 

  58. Szulkin, A., Minimax principles for lower semi continuous functions and applications to nonlinear BVPS, Analyse Non Linéaire 3 (1986), 77–109.

    MATH  Google Scholar 

  59. Szulkin, A., Cohomology and Morse theory for strong indefinite functionals, Math. Z. 209 (1992), 375–418.

    Article  MathSciNet  Google Scholar 

  60. Wang, Z. Q., Equivariant Morse theory for isolated critical orbits and its applications to nonlinear problems, Lecture Notes in Math. 1306, Springer-Verlag, Berlin and New York (1988), 202–221.

    Google Scholar 

  61. Wang, Z. Q., On a superlinear elliptic equation, Analyse Non Linéaire 8 (1991), 43–58.

    MathSciNet  MATH  Google Scholar 

  62. Zhang, D., Ph.D. thesis, Stanford Univ. (1990).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Peking University, Beijing, 100871, China

    Kung-Ching Chang

Authors
  1. Kung-Ching Chang
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Département de Mathématiques, EPFL, 1015, Laussane, Switzerland

    S. D. Chatterji

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Birkhäuser Verlag, Basel, Switzerland

About this paper

Cite this paper

Chang, KC. (1995). Morse Theory in Differential Equations. In: Chatterji, S.D. (eds) Proceedings of the International Congress of Mathematicians. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-9078-6_99

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-9078-6_99

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9897-3

  • Online ISBN: 978-3-0348-9078-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation