Concentration Compactness - Functional-Analytic Grounds and Applications

Cocompactness and minimizers for inequalities of Hardy–Sobolev type involving N-Laplacian

“…For N = 2 the theorem is an immediate application of Corollary 3.2 from [14], whose conditions are verified by (5.1). Interpretation of relation (3.9) from [14] by (5.2) gives (5.4).…”

“…Our starting point for this result is the following version of global compactness, see [14,Theorem 5.1]:…”

A geometric criterion for compactness of invariant subspaces

“…[7]. For the case of integer k, we refer the reader to [17]. For fractional s, the result follows from the continuity of Sobolev embeddings (see Strichartz [15]), the monotonicity of the Sobolev scale with respect to s, and the Hölder inequality.…”

“…In particular, the Sobolev spaces on Riemannian manifolds were studied by Hebey, Vaugon [5,6], and one of the authors [13]. A more abstract approach was developed by the second author and his collaborators [3,10,17] The purpose of this paper is to identify a general geometric condition that is necessary and sufficient for such a compactness phenomenon to occur in a setting when the original embedding is inherently non-compact. We consider an embedding of an abstract normed space A into the Lebesgue space of a non-compact complete Riemannian manifold M .…”

Concentration Compactness at the Mountain Pass Level in Semilinear Elliptic Problems

“…Decompositions of critical sequences similar to (1.4) have been introduced by Struwe [31] for the critical exponent case in bounded domains, followed by Brezis and Coron [7] and by Lions [26] for subcritical problems in R N and numerous authors afterward. We use the version of the "multibump" decomposition, Theorem 6.1 [34,Theorem 5.1] that is not restricted to critical sequences of a specific functional. In restriction to critical sequences this is close, up to elementary modifications, to the "splitting lemma" in Benci and Cerami [4].…”