Critical point theory in infinite dimensional spaces using the Leray-Schauder index (Q1982268)

The article deals with variational approaches to equations of the form \(G'(u)=0\) for continuously differentiable functionals \(G:E\to\mathbb{R}\) on infinite-dimensional Banach spaces \(E\). Introducing the concept of ``sandwich sets'' for such a given functional, the author proves criteria for the existence of Palais-Smale sequences at some energy level which is typically different from the zero level. So the existence of a nontrivial critical point follows once the convergence of this sequence is proved. The proofs are based on a variant of the Deformation Lemma (Lemma 17) where the Leray-Schauder degree is used. As an application, the author obtains some generalized Palais-Smale sequence (Lemma 18) in a standard way. An application to a coupled indefinite system of equations is given in Section 6 (Theorem 14). The concept of sandwiching (Chapter 2) appears to be closely related to Linking and the whole chapter is very close to Schechter's earlier book on ``Linking methods in critical point theory''. Unfortunately, the author does not relate any of his results to contributions by others: a comparison to his own book, to Willem's book on Minimax Theorems or to Rabinowitz' contributions would have been desirable. Further background explanations would have been helpful: Why do other approaches fail? Aren't Saddle Point Theorems applicable? What are typical choices for the sets \(N,A,B\)? The application in Section 6 is not convincing given that it is very technical with strong assumptions on the nonlinearity, see for instance eq. (39). For the entire collection see [Zbl 1471.34005].