We start this chapter by stating and proving Ekeland's well known variational principle since it will be used frequently throughout this monograph. We also give some of its lesser known applications to constrained minimization problems that eventually yield global critical points for the functional in question. We introduce the Palais-Smale condition around a set and we present the first of many examples which show its relevance. We give an existence result for nonhomogeneous elliptic equations involving the critical Sobolev exponent, due to Tarantello. We then establish the more recent smooth variational principle of Borwein-Preiss and we apply it to the study of Hartree-Fock equations for Coulomb systems as was done by P.L. Lions. Finally, we follow the ideas of Ghoussoub and Maurey to deal with the more general problem of identifying those classes of functions that can serve as perturbation spaces in an appropriate minimization principle. As an application, we give a result of Deville et al, stating that the perturbations can be taken to be as smooth as the norm of the Banach space involved. We then apply this result to the problem of existence and uniqueness of viscosity solutions for first order Hamilton-Jacobi equations on general Banach spaces.

Ekeland's variational principle

The following theorem will be of constant use throughout this monograph. The applications of this principle to non-linear analysis are numerous and well documented in several books ([A-E], [Ek 1], [De]).