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Topological Methods in Nonlinear Analysis
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The intrinsic mountain pass principle

Authors: Ribarska, Nadezhda; Tsachev, Tsvetomir; Krastanov, Mikhail;

The intrinsic mountain pass principle

Abstract

The celebrated mountain pass principle of Ambrosetti and Rabinowitz is one of the most important tools in nonlinear analysis for proving the existence of critical points of \(C^1\) real functionals. The authors establish in this paper a general mountain pass principle, dropping any smoothness or even continuity assumptions on the functional. The classical Fréchet differential is replaced by the weak slope, in the sense defined by Degiovanni, and the basic tool in the proof is a nonsmooth version of the deformation lemma. As a corollary of the main result the authors deduce the existence of a point with an arbitrarily small slope if the ``low'' part of the ``barrier'' is sufficiently far from the ``boundary'', obtaining in addition estimates for the location of the point.

Keywords

intrinsic, mountain pass principle, weak slope, nonsmooth analysis, variational methods, critical point, Abstract critical point theory (Morse theory, Lyusternik-Shnirel'man theory, etc.) in infinite-dimensional spaces, Variational principles in infinite-dimensional spaces, complete metric space, Variational problems in a geometric measure-theoretic setting, deformation lemma

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
2
Average
Average
Average
Green
bronze