Infinite Dimensional Morse Theory and Multiple Solution Problems

The book is based on my lecture notes "Infinite dimensional Morse theory and its applications", 1985, Montreal, and one semester of graduate lectures delivered at the University of Wisconsin, Madison, 1987. Since the aim of this monograph is to give a unified account of the topics in critical point theory, a considerable amount of new materials has been added. Some of them have never been published previously. The book is of interest both to researchers following the development of new results, and to people seeking an introduction into this theory. The main results are designed to be as self-contained as possible. And for the reader's convenience, some preliminary background information has been organized. The following people deserve special thanks for their direct roles in help­ ing to prepare this book. Prof. L. Nirenberg, who first introduced me to this field ten years ago, when I visited the Courant Institute of Math Sciences. Prof. A. Granas, who invited me to give a series of lectures at SMS, 1983, Montreal, and then the above notes, as the primary version of a part of the manuscript, which were published in the SMS collection. Prof. P. Rabinowitz, who provided much needed encouragement during the academic semester, and invited me to teach a semester graduate course after which the lecture notes became the second version of parts of this book. Professors A. Bahri and H. Brezis who suggested the publication of the book in the Birkhiiuser series.

Inhalt

I: Infinite Dimensional Morse Theory.- 1. A Review of Algebraic Topology.- 2. A Review of the Banach-Finsler Manifold.- 3. Pseudo Gradient Vector Field and the Deformation Theorems.- 4. Critical Groups and Morse Type Numbers.- 5. Gromoll-Meyer Theory.- 6. Extensions of Morse Theory.- 6.1. Morse Theory Under General Boundary Conditions.- 6.2. Morse Theory on a Locally Convex Closed Set.- 7. Equivariant Morse Theory.- 7.1. Preliminaries.- 7.2. Equivariant Deformation.- 7.3. The Splitting Theorem and the Handle Body Theorem for Critical Manifolds.- 7.4. G-Cohomology and G-Critical Groups.- II: Critical Point Theory.- 1. Topological Link.- 2. Morse Indices of Minimax Critical Points.- 2.1. Link.- 2.2. Genus and Cogenus.- 3. Connections with Other Theories.- 3.1. Degree theory.- 3.2. Ljusternik-Schnirelman Theory.- 3.3. Relative Category.- 4. Invariant Functional.- 5. Some Abstract Critical Point Theorems.- 6. Perturbation Theory.- 6.1. Perturbation on Critical Manifolds.- 6.2. Uhlenbeck's Perturbation Method.- III: Applications to Semilinear Elliptic Boundary Value Problems.- 1. Preliminaries.- 2. Superlinear Problems.- 3. Asymptotically Linear Problems.- 3.1. Nonresonance and Resonance with the Landesman-Lazer Condition.- 3.2. Strong Resonance.- 3.3. A Bifurcation Problem.- 3.4. Jumping Nonlinearities.- 3.5. Other Examples.- 4. Bounded Nonlinearities.- 4.1. Functional Bounded From Below.- 4.2. Oscillating Nonlinearity.- 4.3. Even Functional.- 4.4. Variational Inequalities.- IV: Multiple Periodic Solutions of Hamiltonian Systems.- 1. Asymptotically Linear Systems.- 2. Reductions and Periodic Nonlinearities.- 2.1. Saddle Point Reduction.- 2.2. A Multiple Solution Theorem.- 2.3. Periodic Nonlinearity.- 3. Singular Potentials.- 4. The Multiple Pendulum Equation.- 5. Some Results on Arnold Conjectures.- 5.1. Conjectures.- 5.2. The Fixed Point Conjecture on (T2n, ?0).- 5.3. Lagrange Intersections for (?Pn, ?Pn).- V: Applications to Harmonic Maps and Minimal Surfaces.- 1. Harmonic Maps and the Heat Flow.- 2. The Morse Inequalities.- 3. Morse Decomposition.- 4. The Existence and Multiplicity for Harmonic Maps.- 5. The Plateau Problem for Minimal Surfaces.- Appendix: Witten's Proof of the Morse Inequalities.- 1. A Review of Hodge Theory.- 2. The Witten Complex.- 3. Weak Morse Inequalities.- 4. Morse Inequalities.- References.- Index of Notation.