The Mountain Pass Theorem

This 2003 book presents min-max methods through a study of the different faces of the Mountain Pass Theorem of Ambrosetti and Rabinowitz. The reader is led from accessible results to the forefront of the theory, and at each step, the author presents the extensions and variants of the MPT in a complete and unified way.

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.