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

Xavier Cabré; Antonio Capella

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

Xavier Cabré ¹ ; Antonio Capella ¹

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

Comptes Rendus. Mathématique, Volume 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 :

Xavier Cabré ¹ ; Antonio Capella ¹

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

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     author = {Xavier Cabr\'e and Antonio Capella},
     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},
     publisher = {Elsevier},
     volume = {338},
     number = {10},
     year = {2004},
     doi = {10.1016/j.crma.2004.03.013},
     language = {en},
}

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Xavier Cabré; Antonio Capella. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.03.013/

Bibliographie
Cité par

[1] G. Alberti; L. Ambrosio; X. Cabré On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property, Acta Appl. Math., Volume 65 (2001), pp. 9-33

[2] L. Ambrosio; X. Cabré Entire solutions of semilinear elliptic equations in $ℝ^{3}$ and a conjecture of De Giorgi, J. Amer. Math. Soc., Volume 13 (2000), pp. 725-739

[3] H. Berestycki; P.-L. Lions Nonlinear scalar field Equations. I. Existence of a ground state, Arch. Rational Mech. Anal., Volume 82 (1983), pp. 313-345

[4] H. Berestycki; P.-L. Lions; L.A. Peletier An ODE approach to the existence of positive solutions for semilinear problems in $ℝ^{N}$ , Indiana Univ. Math. J., Volume 30 (1981), pp. 141-157

[5] A. Capella, Ph.D. Thesis

[6] B. Franchi; E. Lanconelli; J. Serrin Existence and uniqueness of nonnegative solutions of quasilinear equations in $ℝ^{n}$ , Adv. Math., Volume 118 (1996), pp. 177-243

[7] E. Giusti Minimal Surfaces and Functions of Bounded Variation, Monographs Math., vol. 80, Birkhäuser, Basel, 1984

[8] C. Gui; W.-M. Ni; X. Wang On the stability and instability of positive steady states of a semilinear heat equation in $ℝ^{n}$ , Comm. Pure Appl. Math., Volume 45 (1992), pp. 1153-1181

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