Critical point theory has become a very powerful tool for solving many problems. The theory has enjoyed significant development over the past several years. The impetus for this development is the fact that many new problems could not be solved by the older theory.

There have been several excellent books written on critical point theory from various points of view; see, e.g., Berger [19], Zeidler [161], Rabinowitz [129], Mawhin and Willem [91], Chang [29, 30], Ghoussoub [56], Ambrosetti and Prodi [8], Willem [158], Chabrowski [26], Dacorogna [36], and Struwe [153] (see also Schechter [143, 144, 147], Zou and Schechter [163], and Perera et al. [113]). In this book we present more recent developments in the subject that do not seem to be covered elsewhere, including some results of the authors dealing with nonstandard linking geometries and sandwich pairs.

Chapter 1 is a brief review of Morse theory in Banach spaces. We prove the first and second deformation lemmas under the Cerami compactness condition. As the variational functionals associated with applications given later in the book will only be C1, we discuss critical groups of C1-functionals. We include discussions on minimizers, nontrivial critical points, mountain pass points, and the three critical points theorem. We also give a generalized notion of local linking that yields a nontrivial critical group, which will be applied to problems with jumping nonlinearities in Chapter 5. We close the chapter with a recent result of Perera [110] on nontrivial critical groups in p-Laplacian problems.