Skip to main content


Log in

Symmetry and regularity of extremals of an integral equation related to the Hardy–Sobolev inequality

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript


Let α be a real number satisfying 0 < α < n, \({0\leq t<\alpha, \alpha{^\ast}(t)=\frac{2(n-t)}{n-\alpha}}\). We consider the integral equation


which is closely related to the HardySobolev inequality. In this paper, we prove that every positive solution u(x) is radially symmetric and strictly decreasing about the origin by the method of moving plane in integral forms. Moreover, we obtain the regularity of solutions to the following integral equation

$$u(x)=\int\limits_{{\mathbb{R}^n}}\frac{|u(y)|^{p}u(y)}{|y|^t|x-y|^{n-\alpha}}\, dy\quad\quad\quad\quad\quad\quad\quad(2)$$

that corresponds to a large class of PDEs by regularity lifting method.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others


  1. Agmon S., Douglis A., Nirenberg L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary condition I. Commun. Pure Appl. Math. 12, 623–727 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aubin T.: Best constants in the Sobolev imbedding theorem: the Yamabe problem. In: Yau, S.-T. (eds) Seminar on Differential Geometry, pp. 173–184. Princeton University, Princeton (1982)

    Google Scholar 

  3. Badiale M., Tarantello G.: A Hardy–Sobolev inequality with applications to a nonlinear elliptic equation arising in astrophysics. Arch. Ration. Mech. Anal. 163, 259–293 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Caffarelli L., Gidas B., Spruck J.: Asymptotic symmetry and local behavior of semilinear elliptic equation with critical Sobolev growth. Commun. Pure Appl. Math. 42, 271–297 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  5. Caffarelli L., Kohn R., Nirenberg L.: First order interpolation inequalities with weights. Compos. Math. 53(3), 259–275 (1984)

    MathSciNet  MATH  Google Scholar 

  6. Chou K.S., Chu C.W.: On the best constant for a weighted Sobolev–Hardy inequality. J. Lond. Math. Soc. 2, 137–151 (1993)

    Article  MathSciNet  Google Scholar 

  7. Catrina F., Wang Z.Q.: On the Caffarelli–Kohn–Nirenberg inequalities: sharp constant, existence (and nonexistence) and symmetry of extremal functions. Commun. Pure Appl. Math. 54(2), 229–258 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen W., Li C.: Classification of solutions of some nonlinear elliptic equations. Duke Math. J. 63, 615–622 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, W., Li, C.: Methods on Nonlinear Elliptic Equations, AIMS book series, vol. 4 (2010)

  10. Chen W., Li C., Ou B.: Classification of solutions for an integral equation. Commun. Pure Appl. Math. 59, 330–343 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen W., Li C., Ou B.: Qualitative properties of solutions for an integral equation. Discret. Contin. Dyn. Syst. 12, 347–354 (2005)

    MathSciNet  MATH  Google Scholar 

  12. Chen, W., Zhu, J.: Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems. J. Math. Anal. Appl. (to appear)

  13. Cao D., Li Y.: Results on positive solutions of elliptic equations with a critical Hardy–Sobolev operator. Methods Appl. Anal. 15(1), 81–96 (2008)

    MathSciNet  MATH  Google Scholar 

  14. Chang S.-Y.A., Yang P.C.: On uniqueness of solutions of nth order differential equations in conformal geometry. Math. Res. Lett. 4(1), 91–102 (1997)

    MathSciNet  MATH  Google Scholar 

  15. Gidas, B., Ni, W.M., Nirenberg, L.: Symmetry of positive solutions of nonlinear elliptic equation in R n. Mathematical analysis and application. Part A, 369–402. Advances in Mathematics, Supplementary studies, 7a. Academic press, New York–London (1981)

  16. Jin C., Li C.: Symmetry of solutions to some systems of integral equations. Proc. AMS 134, 1661–1670 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li Y.Y.: Remark on some conformally invariant integral equations: the method of moving spheres. J. Eur. Math. Soc. 6, 153–180 (2004)

    Article  MATH  Google Scholar 

  18. Lieb E., Loss M.: Analysis, 2nd edn. American Mathematical Society, Rhode Island (2001)

    MATH  Google Scholar 

  19. Lieb E.H.: Sharp constants in the Hardy–Littlewood–Sobolev and related inequalites. Ann. Math. 118, 349–374 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mancini G., Fabbri I., Sandeep K.: Classification of soultions of a critical Hardy–Sobolev operater. J. Differ. Eqn. 224, 258–276 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Serrin J.: A symmetry problem in potential theory. Arch. Ration. Mech. Anal. 43, 304–318 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  22. Stein E.M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30, pp. xiv+290. Princeton University Press, N.J. Princeton (1970)

    Google Scholar 

  23. Talenti G.: Best constant in Sobolev inequality. Ann. Mat. Pura. Appl. 110, 353–372 (1976)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Jiuyi Zhu.

Additional information

Communcated by L. Ambrosio.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lu, G., Zhu, J. Symmetry and regularity of extremals of an integral equation related to the Hardy–Sobolev inequality. Calc. Var. 42, 563–577 (2011).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

Mathematics Subject Classification (2000)