Comptes Rendus
Critical exponents for the Pucci's extremal operators
[Les exposants critiques pour l'opérateur extrémal de Pucci]
Comptes Rendus. Mathématique, Volume 335 (2002) no. 11, pp. 909-914.

Dans cette Note nous présentons des résultats d'existence des solutions radiales pour l'équa- tion elliptique non linéare

λ,Λ + (D 2 u)+u p =0,u0 dans N ,(∗)
N⩾3, p>1 et λ,Λ + est l'opérateur extrémal de Pucci avec les paramètres 0<λ⩽Λ. L'objectif de cette Note est décrire l'ensemble des solutions en fonction de p. On trouve des exposants critiques 1<p + s <p + * <p + p tels que : (i) Si 1<p<p + * , alors il n'existe pas de solution non triviale de (*). (ii) Si p=p + * , il existe une unique solution de (*) à décroisssance rapide. (iii) Si p * <pp + p , il existe une unique solution de (*) à décroissance pseudo-lente. (iv) Si pp+<p, il existe une unique solution de (*) à décroissance lente. Un résultat similaire peut se démontrer pour λ,Λ - .

In this Note we present some results on the existence of radially symmetric solutions for the nonlinear elliptic equation

λ,Λ + (D 2 u)+u p =0,u0in N .(∗)
Here N⩾3, p>1 and λ,Λ + denotes the Pucci's extremal operators with parameters 0<λ⩽Λ. The goal is to describe the solution set as function of the parameter p. We find critical exponents 1<p + s <p + * <p + p , that satisfy: (i) If 1<p<p + * then there is no nontrivial solution of (*). (ii) If p=p + * then there is a unique fast decaying solution of (*). (iii) If p * <pp + p then there is a unique pseudo-slow decaying solution to (*). (iv) If pp+<p then there is a unique slow decaying solution to (*). Similar results are obtained for the operator λ,Λ - .

Reçu le :
Publié le :
DOI : 10.1016/S1631-073X(02)02605-5

Patricio L. Felmer 1 ; Alexander Quaas 1

1 Departamento de Ingenierı́a Matemática and Centro de Modelamiento Matemático, UMR2071 CNRS-UChile, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
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Patricio L. Felmer; Alexander Quaas. Critical exponents for the Pucci's extremal operators. Comptes Rendus. Mathématique, Volume 335 (2002) no. 11, pp. 909-914. doi : 10.1016/S1631-073X(02)02605-5. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02605-5/

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