In this chapter, we present two results of Ghoussoub and Maurey: the first is concerned with linear perturbations and is applicable in reflexive Banach spaces. It will be used in Theorems 2.12 and 2.13 below to get generic minimization results in the case of critical exponents and non-zero data. The second one deals with the possibility of finding plurisubharmonic perturbations and can be applied to spaces as “bad” as L1. The proofs essentially boil down to showing that these new classes of functions are, in the terminology of Chapter 1, admissible cones of perturbations. However, the methods here are slightly more involved than the ones used earlier. Moreover, the minima for the perturbed functionals are not necessarily close to any given minimizing sequence of the original function, and the perturbations we are seeking here, can never be bounded on the whole Banach space.

Actually, our main goal for this chapter, is to introduce the reader to new and different methods for proving perturbed variational principles and especially, to the martingales techniques used in Theorem 2.17.

A minimization principle with linear perturbations

We shall first consider a situation where the perturbations can be taken to be linear.