Relative Category and The Calculus of Variations

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


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 \( {\varphi ^c} = {\varphi ^{ - 1}}(] - \infty ,c]). \)


Morse Theory Critical Point Theory Relative Category Distinct Solution Finsler Structure 
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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

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