On the stability of radial solutions of semilinear elliptic equations in all of R
نویسندگان
چکیده
We establish that every nonconstant bounded radial solution u of −∆u = f(u) in all of R is unstable if n ≤ 10. The result applies to every C 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. Sur la stabilité des solutions radiales des équations elliptiques semilinéaires dans tout R Résumé. On montre que toute solution u non constante, bornée et radiale de l'équation −∆u = f(u) dans tout R est instable si n ≤ 10. Ce résultat s'applique à toute nonlinéarité f de classe C 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. Version française abrégée On étudie les propriétés de stabilité des solutions bornées de l'équation elliptique −∆u = f(u) dans R, (1) où f ∈ C(R). La forme quadratique associée au problème linéarisé de (1) en u est donnée par
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