Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces

Abstract

We obtain an improved Sobolev inequality in \(\dot{H}^s\) spaces involving Morrey norms. This refinement yields a direct proof of the existence of optimizers and the compactness up to symmetry of optimizing sequences for the usual Sobolev embedding. More generally, it allows to derive an alternative, more transparent proof of the profile decomposition in \(\dot{H}^s\) obtained in Gérard (ESAIM Control Optim Calc Var 3:213–233, 1998) using the abstract approach of dislocation spaces developed in Tintarev and Fieseler (Concentration compactness. Functional-analytic grounds and applications. Imperial College Press, London, 2007). We also analyze directly the local defect of compactness of the Sobolev embedding in terms of measures in the spirit of Lions (Rev Mat Iberoamericana 1:145–201, 1985, Rev Mat Iberoamericana 1:45–121, 1985). As a model application, we study the asymptotic limit of a family of subcritical problems, obtaining concentration results for the corresponding optimizers which are well known when \(s\) is an integer (Rey in Manuscr Math 65:19–37, 1989, Han in Ann Inst Henri Poincaré Anal Non Linéaire 8:159–174, 1991, Chou and Geng in Differ Integral Equ 13:921–940, 2000).