The Concentration-Compactness Principle in the Calculus of Variations. The limit case, Part 1

After the study made in the locally compact case for variational problems with some translation invariance, we investigate here the variational problems (with constraints) for example in \mathbb R^N where the invariance of \mathbb R^N by the group of dilatations creates some possible loss of compactness. This is for example the case for all the problems associated with the determination of extremal functions in functional inequalities (like for example the Sobolev inequalities). We show here how the concentration-compactness principle has to be modified in order to be able to treat this class of problems and we present applications to Functional Analysis, Mathematical Physics, Differential Geometry and Harmonic Analysis.