This book is an account of critical point theory focusing on new and powerful methods for solving difficult problems that arise in applications. The topics covered have been collected from hundreds of papers and books; it is pleasant to see them covered in a single volume. Although there are other excellent books on these theories, this book mainly covers recent developments, which do not seem to be available elsewhere in this form.
Starting with Morse theory in Banach spaces, the book covers linking, the Fučík spectrum, critical groups, the minimax principle, jumping nonlinearities, sandwich pairs, and the cohomological index. For Morse theory and linking the book gives applications to semilinear elliptic boundary value problems. The Fučík spectrum is presented in an abstract setting, so that many problems in applications become special cases. A very general class of sandwich pairs is constructed and its wide applicability is implied. A special case is used in p-Laplacian problems. The presentation is elegant and concise, and highlights most important aspects of every topic included.
The authors have presented extremely powerful methods in critical point theory. It can be presumed that researchers in these subjects had been awaiting such an excellent source and here they have it. The book may be used as a graduate level text for highly motivated individuals with strong interests in the theory. It is undoubtedly an excellent reference for research scientists in mathematics, physics and engineering.
Dhruba Adhikari is an assistant professor of mathematics at Southern Polytechnic State University, Marietta, Georgia.
1. Morse theory
3. Applications to semilinear problems
4. Fučík spectrum
5. Jumping nonlinearities
6. Sandwich pairs
Appendix: Sobolev spaces