This survey paper deals with the central role of Nehari manifold theory in the qualitative analysis of nonlinear problems with a variational structure. The central part of the paper under review is concerned with elliptic boundary value problems in bounded domains of Euclidean space. The authors make natural assumptions that ensure the existence of a local minimum for the associated energy functional \(\Phi\). The existence of sign-changing solutions and of infinitely many solutions in the case where \(\Phi\) is even is also discussed. In the next chapter, indefinite elliptic problems are considered, and the authors obtain solutions by minimizing \(\Phi\) over the generalized Nehari manifold \({\mathcal M}\). The arguments of the proofs combine various techniques in critical point theory and variational tools.
The reviewer would like to point out that some of the results contained in the present paper are new, such as Theorems 19, 41 and 42. The novelty consists in the fact that the authors admit general nonlinearities with superlinear growth at infinity, therefore replacing the more restrictive Ambrosetti-Rabinowitz technical assumption.
For the entire collection see [Zbl 1217.49001].