Variational Methods pp 95-104 | Cite as
Relative Category and The Calculus of Variations
Chapter
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: where \( {\varphi ^c} = {\varphi ^{ - 1}}(] - \infty ,c]). \)
$$ \# \{ u \in {\varphi ^{ - 1}}([a,b]):\varphi \prime (u) = 0\} \geqslant ca{t_{{\varphi ^b},{\varphi ^a}}}({\varphi ^b}) $$
Keywords
Morse Theory Critical Point Theory Relative Category Distinct Solution Finsler Structure
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