Minimax Methods

Abstract

In the preceding chapter we have seen that (weak sequential) lower semicontinuity and (weak sequential) compactness of the sub-level sets of a functional E on a Banach space V suffice to guarantee the existence of a minimizer of E.

To prove the existence of saddle points we will now strengthen the regularity hypothesis on E and in general require E to be of class C ¹(V), that is continuously Fréchet differentiable. In this case, the notion of critical point is defined and it makes sense to classify such points as relative minima or saddle points as we did in the introduction to Chapter I.

Moreover, we will impose a certain compactness assumption on E, to be stated in Section 2. First, however, we recall a classical result in finite dimensions.