Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces
In the interesting paper under review, the authors derive an improved version of the Sobolev inequality in \(\dot H^s\)-spaces which involves Morrey norms. This refinement provides a direct proof of existence of optimizers and compactness up to symmetry for the optimizing sequences for what concerns the usual Sobolev embedding. Moreover, it allows to get a more transparent proof of the profile decomposition for \(\dot H^s\) given in [\textit{P. Gérard}, ESAIM, Control Optim. Calc. Var. 3, 213--233 (1998; Zbl 0907.46027)] which uses the abstract approach of dislocation spaces. The local defect of compactness of the Sobolev embedding is also analyzed directly in terms of measures in the spirit of \textit{P. L. Lions} [Rev. Mat. Iberoam. 1, No. 1, 145--201 (1985; Zbl 0704.49005); Rev. Mat. Iberoam. 1, No. 2, 45--121 (1985; Zbl 0704.49006)]. As a model application, the authors study the asymptotic limit of a family of subcritical problems, obtaining concentration results for the corresponding optimizers that are well known when \(s\) is integer [\textit{O. Rey}, Manuscr. Math. 65, No. 1, 19--37 (1989; Zbl 0708.35032); \textit{Z.-C. Han}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 8, No. 2, 159--174 (1991; Zbl 0729.35014); \textit{K.-S. Chou} and \textit{D. Geng}, Differ. Integral Equ. 13, No. 7--9, 921--940 (2000; Zbl 0977.35043)].
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