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

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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).

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Notes

  1. We immediately refer to Sect. 2 for the basic definitions and some properties of the relevant spaces we deal with in the paper.

  2. We defer to [52, Chapter 3], and to Sect. 5 below for precise definitions.

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Acknowledgments

We are indebted with Luis Vega for useful discussions about Sobolev inequalities and weighted estimates for the Riesz potentials, and for having drawn our attention on [26]. We would like to thank Piero D’Ancona for having pointed out to us the paper [42].

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università degli Studi di Parma, Parco Area delle Scienze, 53/A, 43124 , Parma, Italy

    Giampiero Palatucci

  2. Dipartimento di Matematica, Sapienza Università di Roma, P.le Aldo Moro, 5, 00185 , Rome, Italy

    Adriano Pisante

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  1. Giampiero Palatucci
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Correspondence to Adriano Pisante.

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Communicated by O.Savin.

G. Palatucci has been supported by the ERC grant 207573 “Vectorial Problems”.

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Palatucci, G., Pisante, A. Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces. Calc. Var. 50, 799–829 (2014). https://doi.org/10.1007/s00526-013-0656-y

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