Nonlinear Analysis and Semilinear Elliptic Problems

Graduate text explaining how methods of nonlinear analysis can be used to tackle nonlinear differential equations.

Antonio Ambrosetti (Author), Andrea Malchiodi (Author)

9780521863209, Cambridge University Press

Hardback, published 4 January 2007

328 pages, 55 b/w illus. 28 exercises
23.5 x 15.8 x 2.1 cm, 0.653 kg

'In the reviewer's opinion, this book can serve very well as a textbook in topological and variational methods in nonlinear analysis. Even researchers working in this field might find some interesting material (at least the reviewer did).' Zentralblatt MATH

Many problems in science and engineering are described by nonlinear differential equations, which can be notoriously difficult to solve. Through the interplay of topological and variational ideas, methods of nonlinear analysis are able to tackle such fundamental problems. This graduate text explains some of the key techniques in a way that will be appreciated by mathematicians, physicists and engineers. Starting from elementary tools of bifurcation theory and analysis, the authors cover a number of more modern topics from critical point theory to elliptic partial differential equations. A series of Appendices give convenient accounts of a variety of advanced topics that will introduce the reader to areas of current research. The book is amply illustrated and many chapters are rounded off with a set of exercises.

Preface
1. Preliminaries
Part I. Topological Methods: 2. A primer on bifurcation theory
3. Topological degree, I
4. Topological degree, II: global properties
Part II. Variational Methods, I: 5. Critical points: extrema
6. Constrained critical points
7. Deformations and the Palais-Smale condition
8. Saddle points and min-max methods
Part III. Variational Methods, II: 9. Lusternik-Schnirelman theory
10. Critical points of even functionals on symmetric manifolds
11. Further results on Elliptic Dirichlet problems
12. Morse theory
Part IV. Appendices: Appendix 1. Qualitative results
Appendix 2. The concentration compactness principle
Appendix 3. Bifurcation for problems on Rn
Appendix 4. Vortex rings in an ideal fluid
Appendix 5. Perturbation methods
Appendix 6. Some problems arising in differential geometry
Bibliography
Index.

Subject Areas: Mathematical modelling [PBWH], Differential calculus & equations [PBKJ], Calculus & mathematical analysis [PBK]