Variational Methods for Strongly Indefinite Problems

✍ Scribed by Ding Yangeng

• Publisher: World Scientific Pub Co (
• Year: 2007
• Tongue: English
• Leaves: 177
• Series: Interdisciplinary Mathematical Sciences
• Category: Library

✦ Synopsis

This unique book focuses on critical point theory for strongly indefinite functionals in order to deal with nonlinear variational problems in areas such as physics, mechanics and economics. With the original ingredients of Lipschitz partitions of unity of gage spaces (nonmetrizable spaces), Lipschitz normality, and sufficient conditions for the normality, as well as existence-uniqueness of flow of ODE on gage spaces, the book presents for the first time a deformation theory in locally convex topological vector spaces. It also offers satisfying variational settings for homoclinic-type solutions to Hamiltonian systems, Schrödinger equations, Dirac equations and diffusion systems, and describes recent developments in studying these problems. The concepts and methods used open up new topics worthy of in-depth exploration, and link the subject with other branches of mathematics, such as topology and geometry, providing a perspective for further studies in these areas. The analytical framework can be used to handle more infinite-dimensional Hamiltonian systems.

✦ Table of Contents

Contents......Page 8
Preface......Page 6
1. Introduction......Page 10
2. Lipschitz partitions of unity......Page 14
Appendix......Page 21
3. Deformations on locally convex topological vector spaces......Page 24
4. Critical point theorems......Page 34
5.1 Existence and multiplicity results for periodic Hamiltonians......Page 44
5.2 Spectrum of the Hamiltonian operator......Page 48
5.3 Variational setting......Page 50
5.4 Linking structure......Page 51
5.5 The (C) sequences......Page 54
5.6 Proofs of the main results......Page 62
5.7 Non periodic Hamiltonians......Page 63
5.7.1 Variational setting......Page 65
5.7.2 Linking structure......Page 69
5.7.3 The (C)-condition......Page 71
5.7.4 Proof of Theorem 5.3......Page 74
6.1 Introduction and results......Page 76
6.2 Preliminaries......Page 80
6.3 The linking structure......Page 81
6.4 The (C) sequences......Page 83
6.5 Proofs of the existence and multiplicity......Page 90
6.6 Semiclassical states of a system of Sch odinger equations......Page 91
6.6.1 An equivalent variational problem......Page 93
6.6.2 Proofs of Theorem 6.5......Page 97
6.6.3 Proof of Theorem 6.6......Page 102
7.1 Relative studies......Page 106
7.2 Existence results for scalar potentials......Page 109
7.3 Variational setting......Page 112
7.4 The asymptotically quadratic case......Page 115
7.5 Super-quadratic case......Page 125
7.6 More general external fields......Page 130
7.6.1 Main results......Page 131
7.6.2 Variational arguments......Page 132
7.6.4 Proofs of Theorems 7.6 and 7.7......Page 140
7.7 Semiclassical solutions......Page 142
8.1 Reviews......Page 148
8.2 Main results......Page 151
8.3 Linear preliminaries......Page 152
8.4 Functional setting......Page 155
8.5 Solutions to (FS)......Page 160
8.6.1 0 is a boundary point of (S)......Page 163
8.6.3 More general nonlinearities......Page 164
8.6.4 More general systems......Page 165
Acknowledgments......Page 168
Bibliography......Page 170
Index......Page 176