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

  • Giampiero Palatucci
  • Adriano PisanteEmail author


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).

Mathematics Subject Classification (2000)

35J60 35C20 35B33 49J45 



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].


  1. 1.
    Aubin, T.: Problèmes isopérimétriques et espaces de Sobolev. J. Differ. Geom. 11, 573–598 (announced in C. R. Acad. Sci. Paris 280, 279–282 (1975)) (1976)Google Scholar
  2. 2.
    Bahouri, H., Gérard, P.: High frequency approximation of solutions to critical nonlinear wave equations. Am. J. Math. 121, 131–175 (1999)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bianchi, G., Egnell, H.: A note on the Sobolev inequality. J. Funct. Anal. 100, 18–24 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Brezis, H.: How to recognize constant functions. A connection with Sobolev spaces. Russ. Math. Surv. 57, 639–708 (2002)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Brezis, H., Lieb, E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88, 486–490 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Brezis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents Comm. Pure Appl. Math. 36, 437–477 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32, 1245–1260 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Chang, A., Gonzalez, M.d.M.: Fractional Laplacian in conformal geometry. Adv. Math. 226, 1410–1432 (2011)Google Scholar
  9. 9.
    Chen, W., Li, C., Ou, B.: Classification of solutions for an integral equation. Commun. Pure Appl. Math. 59, 330–343 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Chou, K.-S., Geng, D.: Asymptotic of positive solutions for a biharmonic equation involving critical exponent. Differ. Integral Equ. 13, 921–940 (2000)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Cotsiolis, A., Tavoularis, N.K.: Best constants for Sobolev inequalities for higher order fractional derivatives. J. Math. Anal. Appl. 295, 225–236 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. math. 136, 521–573 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Edmunds, D.E., Fortunato, D., Jannelli, E.: Critical exponents, critical dimensions and the biharmonic operator. Arch. Ration. Mech. Anal. 112, 269–289 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Escobar, J.: Sharp constant in a Sobolev trace inequality. Indiana Univ. Math. J. 37, 687–698 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Flucher, M.: Variational problems with concentration. In: Progress in Nonlinear Differential Equations and their Applications, vol. 36. Birkhäuser Verlag, Basel (1999)Google Scholar
  16. 16.
    Fanelli, L., Vega, L., Visciglia, N.: Existence of maximizers for Sobolev-Strichartz inequalities. Adv. Math. 229, 1912–1923 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Frank, R., Seiringer, R.: Non-linear ground state representations and sharp Hardy inequalities. J. Funct. Anal. 255, 3407–3430 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Gallagher, I.: Profile decomposition for solutions of the Navier-Stokes equations. Bull. Soc. Math. France 129, 285–316 (2001)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Gérard, P.: Description du défaut de compacité de l’injection de Sobolev. ESAIM Control Optim. Calc. Var. 3, 213–233 (1998)Google Scholar
  20. 20.
    Gérard, P., Meyer, Y., Oru, F.: Inégalités de Sobolev précisées. Séminaire sur les Équations aux Dérivées Partielles 1996–1997, École Polytech., Palaiseau., Exp. no. IV (1997)Google Scholar
  21. 21.
    Gonzalez, M.D.M., Qing, J.: Fractional conformal Laplacians and fractional Yamabe problems. Anal. PDE, available at
  22. 22.
    Han, Z.-C.: Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent. Ann. Inst. Henri Poincaré Anal. Non Linéaire 8, 159–174 (1991)Google Scholar
  23. 23.
    Hebey, E., Robert, F.: Coercivity and Struwe’s compactness for Paneitz type operators with constant coefficients. Calc. Var. Partial Differ. Equ. 13, 491–517 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Jaffard, S.: Analysis of the lack of compactness in the critical Sobolev embeddings. J. Funct. Anal. 161, 384–396 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Kenig, C.E., Merle, F.: Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case. Invent. Math. 166, 645–675 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Kenig, C.E., Ponce, G., Vega, L.: On the concentration of blow up solutions for the generalized KdV equation critical in \(L^2\). In: Nonlinear Wave Equations (Providence, RI, 1998). Contemp. Math. 263, pp. 131–156. American Mathematical Society, Providence, RI (2000)Google Scholar
  27. 27.
    Koch, G.: Profile decompositions for critical Lebesgue and Besov space embeddings. Indiana Univ. Math. J. 59, 1801–1830 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Ledoux, M.: On improved Sobolev embedding theorems. Math. Res. Lett. 10, 659–669 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Lemarié-Rieusset, P.G.: Recent developments in the Navier-Stokes problem, Research Notes in Mathematics 431. Chapman & Hall/CRC, Boca Raton (2002)Google Scholar
  30. 30.
    Lieb, E.: Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Ann. Math. 118, 349–374 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Lions, P.L.: The concentration-compactness principle in the calculus of variations. The limit case, part 1. Rev. Mat. Iberoamericana 1, 145–201 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Lions, P.L.: The concentration-compactness principle in the calculus of variations. The limit case, part 2. Rev. Mat. Iberoamericana 1, 45–121 (1985)CrossRefzbMATHGoogle Scholar
  33. 33.
    Maz’ya, V., Shaposhnikova, T.: Theory of Sobolev multipliers. With applications to differential and integral operators, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 337, Springer, Berlin (2009)Google Scholar
  34. 34.
    Merle, F., Vega, L.: Compactness at blow-up time for \(L^2\) solutions of the critical nonlinear Schrödinger equation in 2D. Internat. Math. Res. Notices 1998, 399–425 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Mironescu, P., Pisante, A.: A variational problem with lack of compactness for \(H^{1/2}(S^1;S^1)\) maps of prescribed degree. J. Funct. Anal. 217, 249–279 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Palatucci, G.: Subcritical approximation of the Sobolev quotient and a related concentration result. Rend. Sem. Mat. Univ. Padova 125, 1–14 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Palatucci, G.: \(p\)-Laplacian problems with critical Sobolev exponent. Asymptot. Anal. 73, 37–52 (2011)zbMATHMathSciNetGoogle Scholar
  38. 38.
    Palatucci, G., Pisante, A., Sire, Y.: Subcritical approximation of a Yamabe type non local equation: a Gamma-convergence approach. Ann. Sci. Norm. Super. Pisa Cl. Sci. (5)Google Scholar
  39. 39.
    Pucci, P., Serrin, J.: Critical exponents and critical dimensions for polyharmonic operators. J. Math. Pures Appl. 69, 55–83 (1990)zbMATHMathSciNetGoogle Scholar
  40. 40.
    Rey, O.: Proof of the conjecture of H. Brezis and L. A. Peletier. Manuscr. Math. 65, 19–37 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    Sawyer, E., Wheeden, R.L.: Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces. Am. J. Math. 114, 813–874 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    Sawano, Y., Sugano, S., Tanaka, H.: Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces. Trans. Am. Math. Soc. 363, 6481–6503 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  43. 43.
    Servadei, R., Valdinoci, E.: The Brezis-Nirenberg result for the fractional Laplacian. Trans. Am. Math. Soc. (to appear)Google Scholar
  44. 44.
    Solimini, S.: A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sobolev space. Ann. Inst. H. Poincaré Anal. Non Linéaire 12, 319–337 (1995)Google Scholar
  45. 45.
    Struwe, M.: A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math. Z. 187, 511–517 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  46. 46.
    Swanson, C.: The best Sobolev constant. Appl. Anal. 47, 227–239 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  47. 47.
    Talenti, G.: Best constants in Sobolev inequality. Ann. Mat. Pura Appl. 110(4), 353–372 (1976)Google Scholar
  48. 48.
    Tao, T.: Concentration compactness and the profile decomposition. Terence Tao Blog: What’s new., 5 Nov (2008)
  49. 49.
    Tao, T.: Concentration compactness via nonstandard analysis. Terence Tao Blog: What’s new., 10 Nov (2010)
  50. 50.
    Taylor, M.: Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations. Commun. Partial Differ. Equ. 17, 1407–1456 (1992)CrossRefzbMATHGoogle Scholar
  51. 51.
    Taylor, M.: Commutator estimates. Proc. Am. Math. Soc. 131, 1501–1507 (2003)CrossRefzbMATHGoogle Scholar
  52. 52.
    Tintarev, K., Fieseler, K.-H.: Concentration Compactness. Functional-Analytic Grounds and Applications. Imperial College Press, London (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità degli Studi di ParmaParmaItaly
  2. 2.Dipartimento di MatematicaSapienza Università di RomaRomeItaly

Personalised recommendations