Abstract.
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ahlfors, L.V.: Conformal invariants. Mc Graw Hill 1973
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)
Baernstein, A.: Integral means, univalent functions and circular symmetrization. Acta Math. 133, 139--169 (1974)
Baernstein, A., A unified approach to symmetrization. Symposia Matematica 35, 47--91 (1995)
Brezis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure and Appl. Math. 36, 437--477 (1983)
Brock, F., Solynin, A.: An approach to symmetrization via polarization. Trans. AMS 352, 1759--1796 (2000)
Byeon, J., Wang, Z.Q.
Caffarelli, L., Kohn, R., Nirenberg, L.: First order interpolation inequalities with weights. Compositio Math. 53, 259--275 (1984)
Cao, D., Peng, S.: The asymptotic behaviour of the ground state solutions for Hénon equation. To appear in J. Math. Anal. Appl.
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)
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)
Gidas, B., Ni, W.-M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68, 209--243 (1979)
Gidas, B., Spruck, J.: Global and local behavior of positive solutions of nonlinear elliptic equations. Comm. Pure Appl. Math. 34, 525--598 (1981)
Hénon, M.: Numerical experiments on the stability of spherical stellar systems. Astronomy and Astrophysics 24, 229--238 (1973)
Horiuchi, T.: Best constant in weighted Sobolev inequality with weights being powers of distance from the origin. J. Inequal. Appl. 1, 275--292 (1997)
Pacella, F.: Symmetry results for solutions of semilinear elliptic equations with convex nonlinearities. Preprint
Polya, G.: Sur la symetrisation circulaire. CRASP 230, 25--27 (1950)
Secchi, S., Smets, D., Willem, M.: In preparation
Smets, D., Su, J., Willem, M.: Non radial ground states for the Hénon equation. Comm. in Contemporary Math. 4, 467--480 (2002)
Willem, M.: A decomposition lemma and applications. Proceedings of Morningside Center of Mathematics, Beijing 1999
Author information
Authors and Affiliations
Corresponding author
Additional information
Received: 5 May 2002, Accepted: 3 September 2002, Published online: 17 December 2002
Mathematics Subject Classification (2000):
35B40 - 35J20
Rights and permissions
About this article
Cite this article
Smets, D., Willem, M. Partial symmetry and asymptotic behavior for some elliptic variational problems. Cal Var 18, 57–75 (2003). https://doi.org/10.1007/s00526-002-0180-y
Issue Date:
DOI: https://doi.org/10.1007/s00526-002-0180-y