On the stability of radial solutions of semilinear elliptic equations in all of $ \mathbb{R}^{n}$

Cabré, Xavier; Capella, Antonio

doi:10.1016/j.crma.2004.03.013

Partial Differential Equations

On the stability of radial solutions of semilinear elliptic equations in all of

ℝ^{n}

[Sur la stabilité des solutions radiales des équations elliptiques semi-linéaires dans tout

ℝ^{n}

]

Présenté par : Haı̈m Brezis

Cabré, Xavier ¹ ; Capella, Antonio ¹

¹ Universitat Politècnica de Catalunya, Departament de Matemàtica Aplicada I, Diagonal 647, 08028 Barcelona, Spain

Comptes Rendus. Mathématique, Tome 338 (2004) no. 10, pp. 769-774.

Résumé
Abstract

On montre que toute solution u non constante, bornée et radiale de l'équation −Δu=f(u) dans tout $ℝ^{n}$ est instable si n⩽10. Ce résultat s'applique à toute nonlinéarité f de classe C¹ qui satisfait une condition générique de non dégénérescence. Il s'applique, en particulier, à toute nonlinéarité analytique et à toute nonlinéarité de type puissance. On donne aussi un exemple de solution u non constante, bornée et radiale qui est stable pour tout n⩾11, et où f est un polynôme.

We establish that every nonconstant bounded radial solution u of −Δu=f(u) in all of $ℝ^{n}$ is unstable if n⩽10. The result applies to every C¹ nonlinearity f satisfying a generic nondegeneracy condition. In particular, it applies to every analytic and every power-like nonlinearity. We also give an example of a nonconstant bounded radial solution u which is stable for every n⩾11, and where f is a polynomial.

Reçu le : 2004-03-16
Accepté le : 2004-03-16
Publié le : 2004-04-14

DOI : 10.1016/j.crma.2004.03.013

Affiliations des auteurs :

Cabré, Xavier ¹ ; Capella, Antonio ¹

¹ Universitat Politècnica de Catalunya, Departament de Matemàtica Aplicada I, Diagonal 647, 08028 Barcelona, Spain

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     title = {On the stability of radial solutions of semilinear elliptic equations in all of $ \mathbb{R}^{n}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {769--774},
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Cabré, Xavier; Capella, Antonio. On the stability of radial solutions of semilinear elliptic equations in all of $ \mathbb{R}^{n}$. Comptes Rendus. Mathématique, Tome 338 (2004) no. 10, pp. 769-774. doi : 10.1016/j.crma.2004.03.013. http://www.numdam.org/articles/10.1016/j.crma.2004.03.013/

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