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Extremals for fractional order Hardy–Sobolev–Maz’ya inequality

  • Arka MallickEmail author
Article
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Abstract

In this article, we derive the existence of positive solution of a semi-linear, non-local elliptic PDE, involving a singular perturbation of the fractional laplacian, coming from the fractional Hardy–Sobolev–Maz’ya inequality, derived in this paper. We also derive symmetry properties and a precise asymptotic behaviour of solutions.

Mathematics Subject Classification

46E35 35J70 35J10 35S05 35B25 

Notes

Acknowledgements

I would like to thank my Ph.D. supervisor Prof. K. Sandeep for countless valuable discussions and suggestions. Also, I would like to thank the anonymous referee for many useful suggestions.

References

  1. 1.
    Abdellaoui, B., Medina, M., Peral, I., Primo, A.: The effect of the Hardy potential in some Calderón–Zygmund properties for the fractional Laplacian. J. Differ. Equ. 260(11), 8160–8206 (2016)CrossRefGoogle Scholar
  2. 2.
    Badiale, M., Tarantello, G.: A Sobolev–Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics. Arch. Ration. Mech. Anal. 163(4), 259–293 (2002)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Baernstein, A. II.: A unified approach to symmetrization. In: Partial Differential Equations of Elliptic Type (Cortona, 1992), Sympos. Math., XXXV, pp. 47–91. Cambridge Univ. Press, Cambridge (1994)Google Scholar
  4. 4.
    Berestycki, H., Nirenberg, L.: On the method of moving planes and the sliding method. Bol. Soc. Brasil. Mat. (N.S.) 22(1), 1–37 (1991)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bogdan, K., Dyda, B.: The best constant in a fractional Hardy inequality. Math. Nachr. 284(5–6), 629–638 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Brézis, H., Lieb, E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88(3), 486–490 (1983)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(7–9), 1245–1260 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Castorina, D., Fabbri, I., Mancini, G., Sandeep, K.: Hardy–Sobolev inequalities and hyperbolic symmetry. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 19(3), 189–197 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chen, W., Li, C., Biao, O.: Classification of solutions for an integral equation. Commun. Pure Appl. Math. 59(3), 330–343 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dipierro, S., Montoro, L., Peral, I., Sciunzi, B.: Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy–Leray potential. Calc. Var. Partial Differ. Equ. 55(4), Art. 99, 29 (2016)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Dipierro, S., Valdinoci, E.: A density property for fractional weighted Sobolev spaces. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26(4), 397–422 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Dyda, B., Lehrbäck, J., Vähäkangas, A.: Fractional Hardy–Sobolev type inequalities for half space and John domain (2017). arXiv: 1709.03296v1
  14. 14.
    Fall, M.M., Weth, T.: Nonexistence results for a class of fractional elliptic boundary value problems. J. Funct. Anal. 263(8), 2205–2227 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Felmer, P., Wang, Y.: Radial symmetry of positive solutions to equations involving the fractional laplacian. Commun. Contemp. Math. 16(01), 1350023 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Frank, R.L., Lieb, E.H., Seiringer, R.: Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators. J. Am. Math. Soc. 21(4), 925–950 (2008)CrossRefGoogle Scholar
  17. 17.
    Frank, R.L., Jin, T., Xiong, J.: Minimizers for the fractional Sobolev inequality on domains. J. Calc. Var. 57(3), 43 (2018)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Frank, R.L., Seiringer, R.: Non-linear ground state representations and sharp Hardy inequalities. J. Funct. Anal. 255(12), 3407–3430 (2008)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Frank, R.L., Seiringer, R..: Sharp fractional Hardy inequalities in half-spaces. In: Around the Research of Vladimir Maz’ya. I, vol 11 of Int. Math. Ser. (N. Y.), pp. 161–167. Springer, New York (2010)Google Scholar
  20. 20.
    Gazzini, M., Musina, R.: Hardy–Sobolev–Maz’ya inequalities: symmetry and breaking symmetry of extremal functions. Commun. Contemp. Math. 11(6), 993–1007 (2009)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Gazzini, M., Musina, R.: On a Sobolev-type inequality related to the weighted \(p\)-Laplace operator. J. Math. Anal. Appl. 352(1), 99–111 (2009)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ghoussoub, N., Shakerian, S.: Borderline variational problems involving fractional Laplacians and critical singularities. Adv. Nonlinear Stud. 15(3), 527–555 (2015)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Herbst, I.W.: Spectral theory of the operator \((p^2 +m^2)^{1/2} -Ze^2/r\). Commun. Math. Phys. 53(3), 285–294 (1977)CrossRefGoogle Scholar
  24. 24.
    Il’in, V.P.: Some integral inequalities and their applications in the theory of differentiable functions of several variables. Mat. Sb. (N.S.) 54(96), 331–380 (1961)MathSciNetGoogle Scholar
  25. 25.
    Jin, T., Li, Y.Y., Xiong, J.: On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions. J. Eur. Math. Soc. (JEMS) 16(6), 1111–1171 (2014)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kassmann, M.: A priori estimates for integro-differential operators with measurable kernels. Calc. Var. Partial Differ. Equ. 34(1), 1–21 (2009)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Lieb, E.H.: Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities. Ann. Math. (2) 118(2), 349–374 (1983)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The locally compact case. I. Ann. Inst. H. Poincaré Anal. Non Linéaire 1(2), 109–145 (1984)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The locally compact case. II. Ann. Inst. H. Poincaré Anal. Non Linéaire 1(4), 223–283 (1984)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Li, Y.Y.: Remark on some conformally invariant integral equations: the method of moving spheres. J. Eur. Math. Soc. (JEMS) 6(2), 153–180 (2004)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Mancini, G., Fabbri, I., Sandeep, K.: Classification of solutions of a critical Hardy–Sobolev operator. J. Differ. Equ. 224(2), 258–276 (2006)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Mancini, G., Sandeep, K.: On a semilinear elliptic equation in \({\mathbb{H}}^n\). Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 7(4), 635–671 (2008)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Maz’ja, V.G.: Sobolev spaces. Springer Series in Soviet Mathematics (Translated from the Russian by T. O. Shaposhnikova). Springer, Berlin (1985)Google Scholar
  34. 34.
    Musina, R.: Ground state solutions of a critical problem involving cylindrical weights. Nonlinear Anal. 68(12), 3972–3986 (2008)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Musina, R., Nazarov, A.I.: Fractional Hardy–Sobolev inequalities on half spaces (2017). ArXiv e-prints Google Scholar
  36. 36.
    Palatucci, G., Pisante, A.: Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces. Calc. Var. Partial Differ. Equ. 50(3–4), 799–829 (2014)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Silvestre, L.: Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60(1), 67–112 (2007)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Tertikas, A., Tintarev, K.: On existence of minimizers for the Hardy–Sobolev–Maz’ya inequality. Ann. Mat. Pura Appl. (4) 186(4), 645–662 (2007)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Tzirakis, K.: Sharp trace Hardy–Sobolev inequalities and fractional Hardy–Sobolev inequalities. J. Funct. Anal. 270(12), 4513–4539 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.TIFR Centre for Applicable MathematicsBangaloreIndia

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