An abstract version of the concentration compactness principle

Abstract

We prove an abstract version of concentration compactness principle in Hilbert space and show its applications to a range of elliptic problems on unbounded domains.

On interpolation of cocompact imbeddings

“…the above-mentioned result in [31]) of this refinement of the Banach-Alaoglu theorem states that, in the presence of a suitably chosen group of operators D acting on a Hilbert space A, every bounded subsequence in A has a subsequence of the following special structure: Each term in the subsequence is the sum of a principal term and a remainder term. The remainder terms form a sequence which converges D-weakly, and each principal term is a (possibly infinite) sum of "dislocated profiles", i.e.…”

“…For example, an abstract version of such a result, in the context of Hilbert spaces due to Schindler and the second author of this paper is given in [31] and also appears as Theorem 3.1 of [40] pp. 62-63.…”

Logarithmic Littlewood-Paley Decomposition and Applications to Orlicz Spaces

“…whose lack of compactness has been investigated by several authors (for further details, we refer to [7,8,12,21,22,26]). Since the works of P. -L. Lions ( [21,22]), it is well understood that the defect of compactness of the Sobolev embedding (1.3) in 2D is due to two reasons.…”

Description of the lack of compactness of some critical Sobolev embedding

“…Other studies have been conducted in various work ( [6], [7], [22], [25], [23],...) supplying us with a large amount of informations about solutions of nonlinear partial differential equations, both in the elliptic frame or the evolution frame. (Among other applications, one can mention [2], [15], [13], [16], [26],...).…”