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

Abstract

We obtain an improved Sobolev inequality in 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 obtained in G,rard (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 is an integer (Rey in Manuscr Math 65:19-37, 1989, Han in Ann Inst Henri Poincar, Anal Non Lin,aire 8:159-174, 1991, Chou and Geng in Differ Integral Equ 13:921-940, 2000)

Local behavior of fractional p-minimizers

“…For this, we will have to handle not only the usual nonlocal character of such fractional operators, but also the difficulties given by the corresponding nonlinear behavior. As a consequence, we can make use neither of the powerful framework provided by the Caffarelli-Silvestre s-harmonic extension ( [4]) nor of various tools as, e. g., the sharp 3-commutators estimates introduced in [5] to deduce the regularity of weak fractional harmonic maps, the strong barriers and density estimates in [26,28,29], the commutator and energy estimates in [25,27], and so on. Indeed, the aforementioned tools seem not to be trivially adaptable to a nonlinear framework; also, increasing difficulties are due to the non-Hilbertian structure of the involved fractional Sobolev spaces W s,p when p is different than 2.…”

Concentration analysis in Banach spaces

“…The existence of profile decompositions involving the usual rescalings and shifts was established by Kyriasis [18] Koch [17], Bahouri, Cohen and Koch [5] for imbeddings involving Besov, Triebel-Lizorkin and BMO spaces, although, like all similar work based on the use of wavelet bases, it provided only a weak form of remainder. Related results involving Morrey spaces were obtained recently in [27], using, like here, more classical decompositions of spaces instead of wavelets. We refer the reader for details to the recent survey of profile decompositions, [36].…”

On Some Nonlinear Fractional Equations Involving the Bessel Operator

“…Proof Since L α,2 (R N ) is equivalent to H s (R N ), the proof of [29,Lemma 5] carries over with only minor modifications.…”

“…Proof It suffices to remark that the proof of [29,Lemma 6] actually contains a proof of our statement.…”