Variational Principles and Critical Point Theory

Abstract

This chapter addresses variational principles and critical point theory that will be applied later in the book for setting up variational methods in the case of nonlinear elliptic boundary value problems. The first section of the chapter illustrates the connection between the variational principles of Ekeland and Zhong and compactness-type conditions such as the Palais–Smale and Cerami conditions. The second section contains the deformation theorems that form the basis of the critical point and Morse theories. These results are proved in the setting of Banach spaces relying on the construction of a pseudogradient vector field and by using the Cerami condition. The third section focuses on important minimax theorems encompassing various linking situations: mountain pass, saddle point, generalized mountain pass, and local linking. The fourth section studies critical points for functionals with symmetries providing minimax values corresponding to index theories whose prototype is the Krasnosel’skiĭ genus. The fifth section is devoted to generalizations: critical point theory on Banach manifolds and nonsmooth critical point theories. Comments and related references are available in a remarks section.