The Mountain Pass Theorem
Variants, Generalizations and Some Applications

This 2003 book presents min-max methods through a study of the different faces of the Mountain Pass Theorem of Ambrosetti and Rabinowitz.

Youssef Jabri (Author)

9780521827218, Cambridge University Press

Hardback, published 15 September 2003

382 pages
23.4 x 15.6 x 2.2 cm, 0.71 kg

Review of the hardback: 'This impressive research monograph provides an excellent basis for an advanced course or a seminar on problems of modern nonlinear analysis.' Zentralblatt MATH

This 2003 book presents min-max methods through a study of the different faces of the celebrated Mountain Pass Theorem (MPT) of Ambrosetti and Rabinowitz. The reader is led from the most accessible results to the forefront of the theory, and at each step in this walk between the hills, the author presents the extensions and variants of the MPT in a complete and unified way. Coverage includes standard topics, but it also covers other topics covered nowhere else in book form: the non-smooth MPT; the geometrically constrained MPT; numerical approaches to the MPT; and even more exotic variants. Each chapter has a section with supplementary comments and bibliographical notes, and there is a rich bibliography and a detailed index to aid the reader. The book is suitable for researchers and graduate students. Nevertheless, the style and the choice of the material make it accessible to all newcomers to the field.

1. Retrospective
Part I. First Steps toward the Mountains: 2. Palais-Smale condition. Definitions and examples
3. Variational principle
4. Deformation lemma
Part 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
Part III. A Deeper Insight in Mountain 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
Part IV. The Landscape Becoming Less Smooth: 14. The semismooth MPT
15. The nonsmooth MPT
16. The metric MPT
Part V. Speculating about the Mountain Pass Geometry: 17. The MPT on convex domains
18. A MPT in order intervals
19. The linking principle
20. The intrinsic MPT
21. Geometrically contrained MPT
Part VI. Technical Climbs: 22. Numerical MPT implementations
23. Perturbation from symmetry and the MPT
24. Applying the MPT in bifurcation problems
25. More climbs
Appendix A. Background material.

Subject Areas: Calculus & mathematical analysis [PBK]