Partial Differential Equations
On the stability of radial solutions of semilinear elliptic equations in all of n
Comptes Rendus. Mathématique, Volume 338 (2004) no. 10, pp. 769-774.

We establish that every nonconstant bounded radial solution u of −Δu=f(u) in all of  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.

On montre que toute solution u non constante, bornée et radiale de l'équation −Δu=f(u) dans tout  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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.03.013
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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Cabré, Xavier; Capella, Antonio. On the stability of radial solutions of semilinear elliptic equations in all of $ \mathbb{R}^{n}$. Comptes Rendus. Mathématique, Volume 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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