Book contents
- Frontmatter
- Contents
- Introduction
- 1 Retrospective
- I First Steps Toward the Mountains
- II Reaching the Mountain Pass Through Easy Climbs
- 5 The Finite Dimensional MPT
- 6 The Topological MPT
- 7 The Classical MPT
- 8 The Multidimensional MPT
- III A Deeper Insight in Mountains Topology
- IV The Landscape Becoming Less Smooth
- V Speculating about the Mountain Pass Geometry
- VI Technical Climbs
- A Background Material
- Bibliography
- Index
7 - The Classical 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
- 5 The Finite Dimensional MPT
- 6 The Topological MPT
- 7 The Classical MPT
- 8 The Multidimensional MPT
- III A Deeper Insight in Mountains Topology
- IV The Landscape Becoming Less Smooth
- V Speculating about the Mountain Pass Geometry
- VI Technical Climbs
- A Background Material
- Bibliography
- Index
Summary
The mountain pass lemma of Ambrosetti and Rabinowitz is a result of great intuitive appeal as well as practical importance in the determination of critical points of functionals, particularly those which occur in the theory of ordinary and differential equations.
P. Pucci and J. Serrin, Extensions of the mountain pass theorem, J. Funct. Anal., 59, 185–210 (1984)One would expect that it is virtually impossible to find critical points which are not extrema. The first to show that this is not the case are Ambrosetti and Rabinowitz.
M. Schechter, Linking methods in critical point theory, Birkhäuser, 1999This chapter is devoted to the MPT, the source of inspiration for all the results that constitute the material of this monograph. We will also see two of its direct applications to boundary value problems: a superlinear Dirichlet problem and a problem of Ambrosetti-Prodi type.
A very interesting situation that occurs when treating nonlinear problems by variational methods is the following. The “energy” associated with the problem, whose critical points are the weak solutions, is indefinite, in the sense that it is bounded neither from above nor from below. In such cases, there are of course no absolute extrema, and the direct method of the calculus of variations that looks for absolute minimizers fails to apply.
- Type
- Chapter
- Information
- The Mountain Pass TheoremVariants, Generalizations and Some Applications, pp. 65 - 80Publisher: Cambridge University PressPrint publication year: 2003