Skip to main content
SpringerLink
Log in

Anisotropic Gagliardo–Nirenberg inequality with fractional derivatives

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

Studied here is anisotropic Gagliardo–Nirenberg inequality in n-dimensional case. A fractional version of this inequality will be proved. Its sharp constant is also elicited in terms of the ground states of the associated nonlocal equation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams R.A.: Reduced Sobolev inequalities. Can. Math. Bull. 31, 159–167 (1988)

    Article  MATH  Google Scholar 

  2. Adams R.A.: Anisotropic Sobolev inequalities. Čsopis Pěst. Mat. 113, 267–279 (1988)

    MATH  Google Scholar 

  3. Algervik, R.: Embedding theorems for mixed norm spaces and applications. PhD thesis, Karlstad University Studies (2010)

  4. Bellazzini J., Frank R., Visciglia N.: Maximizers for Gagliardo–Nirenberg inequalities and related non-local problems. Math. Ann. 360, 653–673 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Berestycki H., Cazenave T.: Instabilité des états stationnaires dans les équations de Schrödinger et de Klein–Gordon non linéaires. C. R. Acad. Sci. Paris 293, 489–492 (1981)

    MathSciNet  MATH  Google Scholar 

  6. Besov O.V., Il’in V.P., Nikolskiĭ S.M.: Integral representations of functions and imbedding theorems, vol. I. Wiley, New York (1978)

    MATH  Google Scholar 

  7. Brezis H., Mironescu P.: Gagliardo–Nirenberg, composition and products in fractional Sobolev spaces. J. Evol. Equ. 1, 387–404 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chang X., Wang Z.-Q.: Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity. Nonlinearity 26, 479–494 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen H., Bona J.L.: Existence and asymptotic properties of solitary-wave solutions of Benjamin-type equations. Adv. Differ. Equ. 3, 51–84 (1998)

    MathSciNet  MATH  Google Scholar 

  10. Chen Z.-Q., Kim P., Song R.: Dirichlet heat kernel estimates for \({\Delta^{\alpha/2}+\Delta^{\beta/2}}\). Illinois J. Math. 54, 1357–1392 (2010)

    MathSciNet  MATH  Google Scholar 

  11. Chen Z.-Q., Kim P., Song R., Vondraček Z.: Boundary Harnack principle for \({\Delta +\Delta^{\alpha/2}}\). Trans. Am. Math. Soc. 364, 4169–4205 (2012)

    Article  MATH  Google Scholar 

  12. Cheng M.: Bound state for the fractional Schrödinger equation with unbounded potential. J. Math. Phys. 53, 043507 (2012)

    Article  MathSciNet  Google Scholar 

  13. Cordero-Erausquin D., Nazaret B., Villani C.: A mass-transportation approach to sharp Sobolev and Gagliardo–Nirenberg inequalities. Advances in Mathematics 182, 307–332 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. de Bouard A., Saut J.-C.: Solitary waves of generalized Kadomtsev–Petviashvili equations. Ann. Inst. H. Poincaré, Anal. Non Linéaire 14, 211–236 (1997)

    Article  MATH  Google Scholar 

  15. Del Pino M., Dolbeault J.: Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions. J. Math. Pures Appl. 81, 847–875 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Del Pino M., Dolbeault J.: The optimal Euclidean L p-Sobolev logarithmic inequality. J. Func. Anal. 197, 151–161 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Di Nezza E., Palatucci G., Valdinoci E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dipierro S., Palatucci G., Valdinoci E.: Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian. Matematiche (Catania) 68, 201–216 (2013)

    MathSciNet  MATH  Google Scholar 

  19. Escher J., Scarpellini B.: Stability properties of parabolic equations in unbounded domains. Arch. Math. 71, 31–45 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  20. Escher J., Yin Z.: Stable equilibria to parabolic systems in unbounded domains. J. Nonlinear Math. Phys. 11, 243–255 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Esfahani A., Pastor A.: Instability of solitary wave solutions for the generalized BO-ZK equation. J. Differ. Equ. 247, 3181–3201 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Esfahani, A., Pastor, A., Bona, J.L.: Stability and decay properties of solitary wave solutions for the generalized BOZK equation. Adv. Differ. Equ. 20, 801–834 (2015)

  23. Felmer P., Quaas A., Tan J.: Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian. Proc. R. Soc. Edinburgh Sect. A 142, 1237–1262 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Feng, B.: Ground states for the fractional Schr¨odinger equation. Electron. J. Differ. Equ. 2013(127), 1–11 (2013)

  25. Frank R., Lenzmann E., Silvestre L.: Uniqueness of radial solutions for the fractional Laplacian. Math. Ann. 360, 653–673 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Gagliardo E.: Proprieta di alcune classi di funzioni in pia variabili. Richerche Mat. 7, 102–137 (1958)

    MathSciNet  MATH  Google Scholar 

  27. Gagliardo E.: Proprieta di alcune classi di funzioni in pia variabili. Richerche Mat. 9, 24–51 (1959)

    MathSciNet  Google Scholar 

  28. Guo B., Huang D.: Existence and stability of standing waves for nonlinear fractional Schrödinger equations. J. Math. Phys. 53, 083702 (2012)

    Article  MathSciNet  Google Scholar 

  29. Hajaiej H., Yub X., Zhai Z.: Fractional Gagliardo–Nirenberg and hardy inequalities under Lorentz norms. J. Math. Anal. Appl. 396, 569–577 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kenig C.E., Martel Y., Robbiano L.: Local well-posedness and blow-up in the energy space for a class of L 2 critical dispersion generalized Benjamin–Ono equations. Ann. Inst. H. Poincaré 28, 853–887 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kolyada V.I.: Mixed norms and Sobolev type inequalities, vol. 72, pp. 141–160. Banach Center Publ., Warsaw (2006)

    Google Scholar 

  32. Kolyada, V.I.: On embedding theorems, nonlinear analysis, function spaces and applications. In Proceedings of the 8th Spring School held at the Czech University of Agriculture, Prague, May 30-June 6, vol. 8, pp. 35–94

  33. Kolyada V.I., Pérez Lázaro F.J.: Estimates of difference norms for functions in anisotropic Sobolev spaces. Math. Nachr. 267, 46–64 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kolyada V.I., Pérez Lázaro F.J.: On Gagliardo–Nirenberg type inequalities. J Fourier Anal. Appl. 20, 577–607 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Lehrer R., Maia L.A., Squassina M.: On fractional p-Laplacian problems with weight. Differ. Integr. Equ. 28, 15–28 (2015)

    MathSciNet  Google Scholar 

  36. Linares F., Pilod D., Saut J.-C.: Dispersive perturbations of Burgers and hyperbolic equations I: Local theory. SIAM J. Math. Anal. 46, 1505–1537 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. Lizorkin P.I.: Generalized Liouville differentiation and function spaces \({L_p^r(E_n)}\). Embedding theorems. Math. Sb. 60, 325–353 (1963)

    MathSciNet  Google Scholar 

  38. Nagy B.V. Sz.: Über Integralgleichungen zwischen einer Funktion und ihrer Ableitung. Acta Sci. Math. 10, 64–74 (1941)

    Google Scholar 

  39. Nikol’skiǐ, S.M.: Approximation of functions of several variables and imbedding theorems, Nauka, Moscow 1977; English translation of 1st ed. Springer, New York (1975)

  40. Nikol’skiǐ, S.M.: Inequalities for entire functions of finite degree and their application in the theory of differentiable functions of several variables, (Russian) Trudy Mat. Inst. Steklov. 38, 244–278, Trudy Mat. Inst. Steklov. 38, Izdat. Akad. Nauk SSSR, Moscow (1951)

  41. Nirenberg L.: On elliptic partial differential equations. Ann. Sc. Norm. Sup. Pisa, Ser. III 13, 115–162 (1959)

    MathSciNet  MATH  Google Scholar 

  42. Pérez Lázaro F.J.: Embeddings for anisotropic Besov spaces. Acta Math. Hung. 119, 25–40 (2008)

    Article  MATH  Google Scholar 

  43. Secchi S.: Ground state solutions for nonlinear fractional Schrödinger equations in \({\mathbb{R}^N}\). J. Math. Phys. 54, 031501 (2013)

    Article  MathSciNet  Google Scholar 

  44. Serrin J., Tang M.: Uniqueness of ground states for quasilinear elliptic equations. Indiana Univ. Math. J. 49, 897–923 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  45. Stein, E.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series 30, Princeton, NJ (1970)

  46. Weinstein M.I.: Nonlinear Schrödinger equations and sharp interpolation estimates. Commun. Math. Phys. 87, 567–576 (1983)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Computer Science, Damghan University, Damghan, 36715-364, Iran

    Amin Esfahani

Authors
  1. Amin Esfahani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Amin Esfahani.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Esfahani, A. Anisotropic Gagliardo–Nirenberg inequality with fractional derivatives. Z. Angew. Math. Phys. 66, 3345–3356 (2015). https://doi.org/10.1007/s00033-015-0586-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00033-015-0586-y

Mathematics Subject Classification

Keywords

Search

Navigation