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Variational methods for strongly indefinite problems. (English) Zbl 1133.49001

Interdisciplinary Mathematical Sciences 7. Hackensack, NJ: World Scientific (ISBN 978-981-270-962-2/hbk). viii, 168 p. (2007).
This is a research monograph dealing with critical point theory for indefinite functionals without compactness and its applications. The author provides a unified approach to many problems of such kind. Main abstract results, presented in Chapter 4, are theorems of linking type that dive the existence of either Palais-Smale, or Cerami sequences. The key idea is to endow the underlying Banach space with certain mixed “strong-weak” topology. The rest of the book is devoted to a comprehensive discussion of several important applications. These are homoclinics in first order Hamiltonian systems, both finite dimensional (Chapter 5) and infinite dimensional (Chapter 8), spatially localized standing waves in nonlinear Schrödinger equations (Chapter 6), and finite energy solutions of nonlinear Dirac equations (Chapter 7). In all these cases the author presents existence results for both superlinear and asymptotically linear nonlinearities. If, in addition, the nonlinearity is odd, the author obtains the existence of infinitely many essentially (geometrically) distinct solutions.
The book contains a lot of material that can be useful for researchers in the field of critical point theory and its applications.

MSC:

49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
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