Relative Category and The Calculus of Variations

  • G. Fournier
  • M. Willem
Chapter
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 4)

Abstract

Contrary to Morse theory, Lusternik-Schnirelman theory is not applicable to functions which are unbounded from below. In order to overcome this difficulty, a notion of relative category was introduced in [6]. Under some assumptions, the following estimate is true:
$$ \# \{ u \in {\varphi ^{ - 1}}([a,b]):\varphi \prime (u) = 0\} \geqslant ca{t_{{\varphi ^b},{\varphi ^a}}}({\varphi ^b}) $$
where \( {\varphi ^c} = {\varphi ^{ - 1}}(] - \infty ,c]). \)

Keywords

Morse Theory Critical Point Theory Relative Category Distinct Solution Finsler Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    K.C. Chang, Infinite dimensional Morse theory and its applications, Séminaire de Mathématiques supérieures, Presses de l’Université de Montréal, Montréal, 1985.Google Scholar
  2. [2]
    K.C. Chang, On the periodic nonlinearity and the multiplicity of solutions, Nonlinear Analysis, TMA 13 (1987), 527–537.Google Scholar
  3. [3]
    C. Conley and E. Zehnder, The Birkhoff—Lewis fixed point theorem and a conjecture of V. Arnold, Invent. Math. 73 (1983), 33–45.MathSciNetMATHGoogle Scholar
  4. [4]
    E. Fadell, Cohomological methods in non-free G-spaces with applications to general Borsuk—Ulam theorems and critical point theorems for invariant functionals, in Nonlinear functional analysis and its applications, D. Reidel, 1986, 1–45.Google Scholar
  5. [5]
    A. Fonda and J. Mawhin, Multiple periodic solutions of conservative systems with periodic nonlinearity, preprint.Google Scholar
  6. [6]
    G. Fournier and M. Willem, Multiple solutions of the forced double pendulum equation, Analyse Non Linéaire, Gauthier-Villars, Paris (1989), 259–281.Google Scholar
  7. [7]
    L. Lusternik and L. Schnirelman, Méthodes topologiques dans les problèmes variationnels, Hermann, Paris, 1934.MATHGoogle Scholar
  8. [8]
    J. Mawhin and M. Willem, Critical point theory and Hamiltonian systems, Springer-Verlag, New York, 1989.MATHGoogle Scholar
  9. [9]
    E. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • G. Fournier
    • 1
  • M. Willem
    • 2
  1. 1.Département de Math. InfoUniversité de SherbrookeSherbrooke, QuébecCanada
  2. 2.Dept. MathUniversité Cath. de LouvainLouvain-la-NeuveBelgium

Personalised recommendations