Preface ........................................................ ix
Some Notations and Conventions ................................. xi
1. Introduction ................................................ 1
1. Five Illustrating Problems ............................... 1
2. Critical Points Via Minimization ............................ 7
1. Basic Results ............................................ 7
2. Application to a Dirichlet Problem ...................... 11
3. Exercises ............................................... 15
3. The Deformation Theorem .................................... 19
1. Preliminaries ........................................... 19
2. Some Versions of the Deformation Theorem ................ 20
3. A Minimum Principle and an Application .................. 24
4. Exercises ............................................... 27
4. The Mountain-Pass Theorem .................................. 29
1. Critical Points of Minimax Type ......................... 29
2. The Mountain-Pass Theorem ............................... 31
3. Two Basic Applications .................................. 32
4. Exercises ............................................... 37
5. The Saddle-Point Theorem ................................... 39
1. Preliminaries. The Topological Degree ................... 39
2. The Abstract Result ..................................... 41
3. Application to a Resonant Problem ....................... 42
4. Exercises ............................................... 46
6. Critical Points under Constraints .......................... 49
1. Introduction. The Basic Minimization Principle
Revisited ............................................... 49
2. Natural Constraints ..................................... 50
3. Applications ............................................ 52
4. Exercises ............................................... 62
7. A Duality Principle ........................................ 63
1. Convex Functions. The Legendre Fenchel Transform ........ 63
2. A Variational Formulation for a Class of Problems ....... 66
3. A Dual Variational Formulation .......................... 67
4. Applications ............................................ 70
8. Critical Points under Symmetries ........................... 75
1. Introduction ............................................ 75
2. The Lusternik - Schnirelman Theory ...................... 76
3. The Basic Abstract Multiplicity Result .................. 78
4. Application to a Problem with a Z2-Symmetry ............. 83
9. Problems with an S1-Symmetry ............................... 87
1. A Geometric S1-index .................................... 87
2. A Multiplicity Result ................................... 90
3. Application to a Class of Problems ...................... 92
4. A Dirichlet Problem on a Plane Disk ..................... 95
10. Problems with Lack of Compactness .......................... 99
1. Introduction ............................................ 99
2. Two Beautiful Lemmas ................................... 100
3. A Problem in  N ........................................ 103
11. Lack of Compactness for Bounded Ω ......................... 115
1. (PS)с for Strongly Resonant Problems ................... 115
2. A Class of Indefinite Problems ......................... 118
3. An Application ......................................... 121
12. Appendix .................................................. 125
1. Ekeland Variational Principle .......................... 125
References .................................................... 131
Index ......................................................... 135
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