Book contents
- Frontmatter
- Contents
- Introduction
- 1 Retrospective
- I First Steps Toward the Mountains
- II Reaching the Mountain Pass Through Easy Climbs
- III A Deeper Insight in Mountains Topology
- 9 The Limiting Case in the MPT
- 10 Palais-Smale Condition versus Asymptotic Behavior
- 11 Symmetry and the MPT
- 12 The Structure of the Critical Set in the MPT
- 13 Weighted Palais-Smale Conditions
- IV The Landscape Becoming Less Smooth
- V Speculating about the Mountain Pass Geometry
- VI Technical Climbs
- A Background Material
- Bibliography
- Index
11 - Symmetry and the MPT
Published online by Cambridge University Press: 04 September 2009
- Frontmatter
- Contents
- Introduction
- 1 Retrospective
- I First Steps Toward the Mountains
- II Reaching the Mountain Pass Through Easy Climbs
- III A Deeper Insight in Mountains Topology
- 9 The Limiting Case in the MPT
- 10 Palais-Smale Condition versus Asymptotic Behavior
- 11 Symmetry and the MPT
- 12 The Structure of the Critical Set in the MPT
- 13 Weighted Palais-Smale Conditions
- IV The Landscape Becoming Less Smooth
- V Speculating about the Mountain Pass Geometry
- VI Technical Climbs
- A Background Material
- Bibliography
- Index
Summary
In the present section, we shall develop methods, employing ideas contained in some of L.A. Lyusternik's work, which allow us to establish the existence of a denumerable number of stable critical values of an even functional – they do not disappear under small perturbations by odd functionals.
M. A. Krasnosel'skii, Topological methods in the theory nonlinear integral equations, 1956.This chapter is devoted to the study of the symmetric MPT and its subsequent extensions. It is a multiplicity result asserting the existence of multiple critical points, when the functional is invariant under the action of a group of symmetries. It has been stated in the same time as the classical MPT by Ambrosetti and Rabinowitz [50]. This theorem can be seen as an extension of older multiplicity results of Ljusternik Schnirelman type. We will also review two other ways of obtaining multiplicity results; a procedure that inductively uses the (classical) MPT and does not pass by any Index theory, and a generalization of the symmetric MPT, the fountain theorem of Bartsch and its dual form by Bartsch and Willem.
Some basic references for the material presented here include [93, 734, 748, 882] and of course [50]. The lecture notes [93] by Bartsch discuss very nicely and exhaustively the role of symmetry in variational methods.
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- Information
- The Mountain Pass TheoremVariants, Generalizations and Some Applications, pp. 114 - 133Publisher: Cambridge University PressPrint publication year: 2003