Partial symmetry and asymptotic behavior for some elliptic variational problems

  • Didier SmetsEmail author
  • Michel Willem
Original Paper


A short elementary proof based on polarizations yields a useful (new) rearrangement inequality for symmetrically weighted Dirichlet type functionals. It is then used to answer some symmetry related open questions in the literature. The non symmetry of the Hénon equation ground states (previously proved in [19]) as well as their asymptotic behavior are analyzed more in depth. A special attention is also paid to the minimizers of the Caffarelli-Kohn-Nirenberg [8] inequalities.


Asymptotic Behavior Variational Problem Type Functional Elementary Proof Partial Symmetry 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ahlfors, L.V.: Conformal invariants. Mc Graw Hill 1973Google Scholar
  2. 2.
    Badiale, M., Tarentello, G.: A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics. To appear in Arch. Rational Mech. Anal. (2001)Google Scholar
  3. 3.
    Baernstein, A.: Integral means, univalent functions and circular symmetrization. Acta Math. 133, 139--169 (1974)zbMATHGoogle Scholar
  4. 4.
    Baernstein, A., A unified approach to symmetrization. Symposia Matematica 35, 47--91 (1995)Google Scholar
  5. 5.
    Brezis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure and Appl. Math. 36, 437--477 (1983)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Brock, F., Solynin, A.: An approach to symmetrization via polarization. Trans. AMS 352, 1759--1796 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Byeon, J., Wang, Z.Q.Google Scholar
  8. 8.
    Caffarelli, L., Kohn, R., Nirenberg, L.: First order interpolation inequalities with weights. Compositio Math. 53, 259--275 (1984)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Cao, D., Peng, S.: The asymptotic behaviour of the ground state solutions for Hénon equation. To appear in J. Math. Anal. Appl.Google Scholar
  10. 10.
    Catrina, F., Wang, Z.Q.: On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions. Comm. Pure Appl. Math. 54, 229--258 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Catrina, F., Wang, Z.Q.: Asymptotic uniqueness and exact symmetry of k-bump solutions for a class of degenerate elliptic problems. Proc. of the Kennesaw Conference, Disc. Cont. Dyn. Syst. 80--88 (2001)Google Scholar
  12. 12.
    Gidas, B., Ni, W.-M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68, 209--243 (1979)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Gidas, B., Spruck, J.: Global and local behavior of positive solutions of nonlinear elliptic equations. Comm. Pure Appl. Math. 34, 525--598 (1981)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Hénon, M.: Numerical experiments on the stability of spherical stellar systems. Astronomy and Astrophysics 24, 229--238 (1973)Google Scholar
  15. 15.
    Horiuchi, T.: Best constant in weighted Sobolev inequality with weights being powers of distance from the origin. J. Inequal. Appl. 1, 275--292 (1997)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Pacella, F.: Symmetry results for solutions of semilinear elliptic equations with convex nonlinearities. PreprintGoogle Scholar
  17. 17.
    Polya, G.: Sur la symetrisation circulaire. CRASP 230, 25--27 (1950)zbMATHGoogle Scholar
  18. 18.
    Secchi, S., Smets, D., Willem, M.: In preparationGoogle Scholar
  19. 19.
    Smets, D., Su, J., Willem, M.: Non radial ground states for the Hénon equation. Comm. in Contemporary Math. 4, 467--480 (2002)CrossRefzbMATHGoogle Scholar
  20. 20.
    Willem, M.: A decomposition lemma and applications. Proceedings of Morningside Center of Mathematics, Beijing 1999Google Scholar

Copyright information

© Springer-Verlag Berlin/Heidelberg 2003

Authors and Affiliations

  1. 1.Laboratoire Jacques-Louis LionsUniversité de Paris 6ParisFrance

Personalised recommendations