Comptes Rendus
Partial Differential Equations
Remarks on a Hardy–Sobolev inequality
Comptes Rendus. Mathématique, Volume 336 (2003) no. 10, pp. 811-815.

We compute the optimal constant for a generalized Hardy–Sobolev inequality, and using the product of two symmetrizations we present an elementary proof of the symmetries of some optimal functions. This inequality was motivated by a nonlinear elliptic equation arising in astrophysics.

Nous calculons la meilleure constante dans une inégalité de Hardy–Sobolev généralisée, et en utilisant le produit de deux symétrisations, nous montrons de manière élémentaire la symétrie de certaines fonctions optimales. Cette inégalité est motivée par une équation elliptique non-linéaire en astrophysique.

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Published online:
DOI: 10.1016/S1631-073X(03)00202-4

Simone Secchi 1; Didier Smets 2; Michel Willem 3

1 Dipartimento di Matematica, Università di Pisa, via F. Buonarroti, 2, 56127 Pisa, Italy
2 Laboratoire J.-L. Lions, Université Paris 6, 4, pl. Jussieu, BC 187, 75252 Paris cedex 05, France
3 Université Catholique de Louvain, 2, chemin du cyclotron, 1348 Louvain-la-Neuve, Belgium
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Simone Secchi; Didier Smets; Michel Willem. Remarks on a Hardy–Sobolev inequality. Comptes Rendus. Mathématique, Volume 336 (2003) no. 10, pp. 811-815. doi : 10.1016/S1631-073X(03)00202-4. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00202-4/

[1] M. Badiale; G. Tarantello A Sobolev–Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics, Arch. Rational Mech. Anal., Volume 163 (2002), pp. 259-293

[2] F. Brock; A.Y. Solynin An approach to symmetrization via polarization, Trans. Amer. Math. Soc., Volume 352 (2000) no. 4, pp. 1759-1796

[3] O. Kavian Introduction à la théorie des points critiques, Springer-Verlag, Paris, 1993

[4] G. Mancini, K. Sandeep, Cylindrical symmetry of extremals of a Hardy–Sobolev inequality, Preprint, Dipartimento di Matematica, Università di Roma III, Roma, Italy

[5] E. Mitidieri A simple approach to Hardy inequalities, Math. Notes, Volume 67 (2000) no. 4

[6] M. Willem Analyse fonctionnelle élémentaire, Cassini, Paris, 2003

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