Revista Matemática Iberoamericana
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The Concentration-Compactness Principle in the Calculus of Variations. The Limit Case, Part 2
Pierre-Louis Lions (1)(1) CEREMADE, Univ de Paris IX Dauphine, Place du Marechal de Lattre de Tassigny, 75775, PARIS CEDEX 05, 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.