The Concentration-Compactness Principle in the Calculus of Variations. The Limit Case, Part 2

Abstract

This paper is the second part of a work devoted to the study of variational problems (with constraints) in functional spaces defined on domains presenting some (local) form of invariance by a non-compact group of transformations like the dilations in ${\mathbb{R}}^N$. This contains for example the class of problems associated with the determination of extremal functions in inequalities like Sovolev inequalities, convolution or trace inequalities... We show how the concentration-compactness principle and method introduced in the so-called locally compact case are to be modified in order to solve these problems and we present applications to Functional Analysis, Mathematical Physics, Differential Geometry and Harmonic Analysis.