The concentration-compactness principle in the calculus of variations. The limit case. II. (English) Zbl 0704.49006

Lions, P.-L.

Rev. Mat. Iberoam. 1, No. 2, 45-121 (1985).

Summary: [For part I see the author, ibid. No.1, 145-201 (1985; Zbl 0704.49005).]
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 noncompact 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 such as Sobolev 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.

MSC:

49J10	Existence theories for free problems in two or more independent variables
49J27	Existence theories for problems in abstract spaces
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

Sobolev inequalities; convolution or trace inequalities; concentration- compactness principle

Citations:

Zbl 0704.49005