Abstract
In this chapter, we consider problems which are derived from a functional on a function space in variational problems. We will begin with an introduction of existence theory by using minimization techniques and some elementary facts about monotone operators. By using a compactness condition (condition (C)), we give a minimax principle in Banach spaces. This result will then be used to prove a theorem (mountain pass theorem) on the existence of critical points which are not necessary minimal points. An application to a semi-linear hyperbolic equation will be considered. In Sections 4.8 and 4.9, we present a summary of Ljusternik-Schnirelman theory on Banach manifolds with an application.
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© 1982 Springer-Verlag New York Inc.
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Chow, SN., Hale, J.K. (1982). Variational Method. In: Methods of Bifurcation Theory. Grundlehren der mathematischen Wissenschaften, vol 251. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8159-4_4
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DOI: https://doi.org/10.1007/978-1-4613-8159-4_4
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4613-8161-7
Online ISBN: 978-1-4613-8159-4
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