Advertisement

Mathematische Zeitschrift

, Volume 273, Issue 3–4, pp 883–905 | Cite as

On the regularity of positive solutions of a class of Choquard type equations

  • Yutian Lei
Article

Abstract

This paper is concerned with positive solutions of a class of Choquard type equations. Such equations are equivalent to integral systems involving the Bessel potential and the Riesz potential. By using two regularity lifting lemmas introduced by Chen and Li [2], we study the regularity for integrable solutions u. We first use the Hardy–Littlewood–Sobolev inequality to obtain an integrability result. Then, it is improved to \({u \in L^s(R^n)}\) for all \({s \in [1, \infty]}\) by an iteration. Next, we use the properties of the contraction map and the shrinking map to prove that u is Lipschitz continuous. Finally, we establish the smoothness of u by a bootstrap argument. Our technique can also be used to handle other integral systems involving the Riesz potential or the Bessel potential, such as the Hartree type equations.

Keywords

Choquard equation Integral equations Integrability intervals Hardy–Littlewood–Sobolev inequality Bessel potential Regularity lifting lemmas 

Mathematics Subject Classification

35J15 45E10 45G05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ackermann N.: On a periodic Schrödinger equation with nonlocal superlinear part. Math. Z. 248, 423–443 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Chen W., Li C.: Methods on Nonlinear Elliptic Equations. AIMS Book Ser. Differ. Equ. Dyn. Syst. 4 (2010)Google Scholar
  3. 3.
    Chen W., Li C.: Regularity of solutions for a system of integral equations. Commun. Pure Appl. Anal. 4, 1–8 (2005)MathSciNetGoogle Scholar
  4. 4.
    Chen W., Li C., Ou B.: Classification of solutions for an integral equation. Commun. Pure Appl. Math. 59, 330–343 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Chen W., Li C., Ou B.: Classification of solutions for a system of integral equations. Commun. Partial Differ. Equ. 30, 59–65 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Gilbarg D., Trudinger N.: Elliptic Partial Differential Equations of Second Order. Springer-Verlag, New York (1977)zbMATHCrossRefGoogle Scholar
  7. 7.
    Ginibret J., Velo G.: On a class of non linear Schrödinger equations with non local interaction. Math. Z. 170, 109–136 (1980)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Han X., Lu G.: Regularity of solutions to an integral equation associated with Bessel potential. Commun. Pure Appl. Anal. 10, 1111–1119 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Hang F.: On the integral systems related to Hardy–Littlewood–Sobolev inequality. Math. Res. Lett. 14, 373–383 (2007)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Huang X., Li D., Wang L.: Existence and symmetry of positive solutions of an integral equation system. Math. Comput. Model. 52, 892–901 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Huang X., Li D., Wang L.: Symmetry and monotonicity of integral equation systems. Nonlinear Anal. Real World Appl. 12, 3515–3530 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Jin C., Li C.: Qualitative analysis of some systems of integral equations. Calc. Var. Partial Differ. Equ. 26, 447–457 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Lei, Y., Li, C., Ma, C.: Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy–Littlewood–Sobolev system. Calc. Var. Partial Differ. Equ. doi: 10.1007/s00526-011-0450-7
  14. 14.
    Li Y.: Remark on some conformally invariant integral equations: the method of moving spheres. J. Eur. Math. Soc. 6, 153–180 (2004)zbMATHCrossRefGoogle Scholar
  15. 15.
    Lieb E.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57, 93–105 (1976/77)MathSciNetGoogle Scholar
  16. 16.
    Lieb E.: Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities. Ann. Math. 118, 349–374 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Lieb E., Simon B.: The Hartree–Fock theory for Coulomb systems. Commun. Math. Phys. 53, 185–194 (1977)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lions P.L.: The Choquard equation and related questions. Nonlinear Anal. 4, 1063–1072 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Lions, P.L.: The concentration-compactness principle in the calculus of variations, The locally compact case, I and II. Ann. Inst. H. Poincar., Anal. Nonlinaire. 1, 109–145, 223–283 (1984)Google Scholar
  20. 20.
    Liu S.: Regularity, symmetry, and uniqueness of some integral type quasilinear equations. Nonlinear Anal. 71, 1796–1806 (2009)zbMATHCrossRefGoogle Scholar
  21. 21.
    Ma C., Chen W., Li C.: Regularity of solutions for an integral system of Wolff type. Adv. Math. 226, 2676–2699 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Ma F., Huang X., Wang L.: A classification of positive solutions of some integral systems. Integr. Equ. Oper. Theory 69, 393–404 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Ma L., Chen D.: Radial symmetry and uniqueness for positive solutions of a Schrödinger type system. Math. Comput. Model. 49, 379–385 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Ma L., Zhao L.: Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Ration. Mech. Anal. 195, 455–467 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Smoller J.: Shock Waves and Reaction–Diffusion Equations, Grundlehren der Mathematischen Wissenschaften, vol. 258. Springer-Verlag, New York (1983)Google Scholar
  26. 26.
    Stein E.M.: Singular Integrals and Differentiability Properties of Function, Princetion Math. Series, vol. 30. Princeton University Press, Princeton, NJ (1970)Google Scholar
  27. 27.
    Ziemer W.: Weakly Differentiable Functions, Graduate Texts in Math., vol. 120. Springer-Verlag, New York (1989)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Institute of Mathematics, School of Mathematical SciencesNanjing Normal UniversityNanjingChina

Personalised recommendations