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

We establish that every nonconstant bounded radial solution u of ?Δu=f(u) in all of $\text{R}{}^{\mathrm{n}}$ 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. To cite this article: X. Cabré, A. Capella, C. R. Acad. Sci. Paris, Ser. I 338 (2004).