# Revista Matemática Iberoamericana

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**Volume 1, Issue 2, 1985, pp. 45–121**

**DOI: 10.4171/RMI/12**

Published online: 1985-06-30

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

Pierre-Louis Lions^{[1]}(1) Université de Paris-Dauphine, Paris, France

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.

*Keywords: *Concentration-compactness principle, minimization problems, unbounded domains, dilation invariance, concentration function, nonlinear field equations, Dirac masses, Morse theory, Sobolev inequalities, convolution, weak $L^p$ spaces, trace theorems, Yamabe problem, scalar curvature, conformal invariance.

Lions Pierre-Louis: The Concentration-Compactness Principle in the Calculus of Variations. The Limit Case, Part 2. *Rev. Mat. Iberoam.* 1 (1985), 45-121. doi: 10.4171/RMI/12