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

We obtain an improved Sobolev inequality in $H^s$ 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 $H^s$ obtained in [P. Gérard, ESAIM 1998] using the abstract approach of dislocation spaces developed in [K. Tintarev & K. H. Fieseler, Imperial College Press 2007]. We also analyze directly the local defect of compactness of the Sobolev embedding in terms of measures in the spirit of [P. L. Lions, Rev. Mat. Iberoamericana 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 ([O. Rey, Manuscripta math. 1989; Z.-C. Han, Ann. Inst. H. Poincaré Anal. Non Linéaire 1991], [K. S. Chou & D. Geng, Differential Integral Equations 2000]).