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. Gerard, 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. Poincare Anal. Non Lineaire 1991], [K. S. Chou & D. Geng, Differential
Integral Equations 2000]).Comment: 33 page
