H s versus C 0-weighted minimizers

  • Antonio IannizzottoEmail author
  • Sunra Mosconi
  • Marco Squassina


We study a class of semi-linear problems involving the fractional Laplacian under subcritical or critical growth assumptions. We prove that, for the corresponding functional, local minimizers with respect to a C 0-topology weighted with a suitable power of the distance from the boundary are actually local minimizers in the natural H s -topology.

Mathematics Subject Classification

35P15 35P30 35R11 


Fractional Laplacian Fractional Sobolev spaces Local minimizers 


  1. 1.
    Aikawa H., Kipleläinen T., Shanmugalingam N., Zhong X.: Boundary Harnack principle for p-harmonic functions in smooth euclidean spaces. Potential Anal. 26, 281–301 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Barrios, B.; Colorado, E.; Servadei, R.; Soria, F.: A critical fractional equation with concave-convex power nonlinearities. Ann. Inst. H. Poincaré Anal. Non Linéaire (to appear, 2014)Google Scholar
  3. 3.
    Brasco, L.; Franzina, G.: Convexity properties of Dirichlet integrals and Picone-type inequalities. Kodai Math. J. (2014a, to appear)Google Scholar
  4. 4.
    Brasco, L., Lindgren, E., Parini, E.: The fractional Cheeger problem (2014b, preprint)Google Scholar
  5. 5.
    Brezis H., Nirenberg L.: H 1 versus C 1 minimizers. C. R. Acad. Sci. Paris. 317, 465–472 (1993)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Cabré X., Sire Y.: Nonlinear equations for fractional Laplacians I: regularity, maximum principles, and Hamiltonian estimates. Ann. Inst. Henri Poincaré (C) Nonlinear Anal. 31, 23–53 (2014)CrossRefzbMATHGoogle Scholar
  7. 7.
    Cabré, X.; Sire, Y.: Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions. Trans. Am. Math. Soc. (2014, to appear)Google Scholar
  8. 8.
    Caffarelli L.A.: Nonlocal equations, drifts and games. Nonlinear Partial Differ. Equ. Abel Symposia. 7, 37–52 (2012)MathSciNetGoogle Scholar
  9. 9.
    Caffarelli L., Silvestre L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32, 1245–1260 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Caffarelli L., Roquejoffre J.-M., Sire Y.: Variational problems with free boundaries for the fractional Laplacian. J. Eur. Math. Soc. 12, 1151–1179 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Chen W., Li C., Ou B.: Classification of solutions for an integral equation. Commun. Pure Appl. Math. 59, 330–343 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Di Castro, A., Kuusi, T., Palatucci, G.: Local behavior of fractional p-minimizers (2014, preprint)Google Scholar
  13. 13.
    Di Nezza E., Palatucci G., Valdinoci E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Fiscella, A., Servadei, R., Valdinoci, E.: Density properties for fractional Sobolev spaces (2014, preprint)Google Scholar
  15. 15.
    Franzina, G., Palatucci, G.: Fractional p-eigenvalues. Riv. Mat. Univ. Parma. 5 (2014)Google Scholar
  16. 16.
    Garcìa Azorero J.P., Peral Alonso I., Manfredi J.J.: Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations. Commun. Contemp. Math. 2, 385–404 (2000)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Iannizzotto, A., Liu, S., Perera, K., Squassina, M.: Existence results for fractional p-Laplacian problems via Morse theory (2014, preprint)Google Scholar
  18. 18.
    Li S., Perera K., Su J.: Computation of critical groups in elliptic boundary-value problems where the asymptotic limits may not exist. Proc. R. Soc. Edinburgh Sec. A. 131, 721–732 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Lindgren E., Lindqvist P.: Fractional eigenvalues. Calc. Var. PDE. 49, 795–826 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Liu J., Liu S.: The existence of multiple solutions to quasilinear elliptic equations. Bull. London Math. Soc. 37, 592–600 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Mingione G.: Gradient potential estimates. J. Eur. Math. Soc. 13, 459–486 (2011)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Ros-Oton X., Serra J.: The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J. Math. Pures Appl. 101, 275–302 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Ros-Oton X., Serra J.: The Pohožaev identity for the fractional laplacian. Arch. Rat. Mech. Anal. 213, 587–628 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Ros-Oton, X., Serra, J.: Nonexistence results for nonlocal equations with critical and supercritical nonlinearities. Commun. Partial Differ. Equ. (2014, to appear)Google Scholar
  25. 25.
    Servadei R., Valdinoci E.: Mountain pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389, 887–898 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Servadei R., Valdinoci E.: Variational methods for non-local operators of elliptic type. Discrete Contin. Dyn. Syst. 33, 2105–2137 (2013)zbMATHMathSciNetGoogle Scholar
  27. 27.
    Servadei R., Valdinoci E.: Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators. Rev. Mat. Iberoam. 29, 1091–1126 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Servadei, R., Valdinoci, E.: The Brezis–Nirenberg result for the fractional Laplacian. Trans. Am. Math. Soc. (2014, to appear)Google Scholar
  29. 29.
    Servadei R., Valdinoci E.: Weak and viscosity solutions of the fractional Laplace equation. Publ. Mat. 58, 1–261 (2014)CrossRefMathSciNetGoogle Scholar
  30. 30.
    Silvestre L.: Regularity of the obstacle problem for a fractional power of the laplace operator. Commun. Pure Appl. Math. 60, 67–112 (2007)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Antonio Iannizzotto
    • 1
    Email author
  • Sunra Mosconi
    • 2
  • Marco Squassina
    • 1
  1. 1.Dipartimento di InformaticaUniversità degli Studi di VeronaVeronaItaly
  2. 2.Dipartimento di Matematica e InformaticaUniversità degli Studi di CataniaCataniaItaly

Personalised recommendations