Critical exponents for the Pucci's extremal operators

Felmer, Patricio L.; Quaas, Alexander

Comptes Rendus. Mathématique, Tome 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

$${ℳ}_{\lambda ,\Lambda }^{+}\left(D^{2}u\right)+u^p=0,u⩾0\mathrm{dans}{ℝ}^N,$$

(∗)

où N⩾3, p>1 et ${ℳ}_{\lambda ,\Lambda }^{+}$ 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 $1<{\mathrm{p}}_{+}^s<{\mathrm{p}}_{+}^{*}<{\mathrm{p}}_{+}^p$ tels que : (i) Si $1<\mathrm{p}<{\mathrm{p}}_{+}^{*}$, alors il n'existe pas de solution non triviale de ($*$). (ii) Si $\mathrm{p}={\mathrm{p}}_{+}^{*}$, il existe une unique solution de ($*$) à décroisssance rapide. (iii) Si $p^{*}<\mathrm{p}⩽{\mathrm{p}}_{+}^p$, il existe une unique solution de ($*$) à décroissance pseudo-lente. (iv) Si p^p₊<p, il existe une unique solution de ($*$) à décroissance lente. Un résultat similaire peut se démontrer pour ${ℳ}_{\lambda ,\Lambda }^{-}$.

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

$${ℳ}_{\lambda ,\Lambda }^{+}\left(D^{2}u\right)+u^p=0,u⩾0\text{in}{ℝ}^N.$$

(∗)

Here N⩾3, p>1 and ${ℳ}_{\lambda ,\Lambda }^{+}$ 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 $1<{\mathrm{p}}_{+}^s<{\mathrm{p}}_{+}^{*}<{\mathrm{p}}_{+}^p$, that satisfy: (i) If $1<\mathrm{p}<{\mathrm{p}}_{+}^{*}$ then there is no nontrivial solution of ($*$). (ii) If $\mathrm{p}={\mathrm{p}}_{+}^{*}$ then there is a unique fast decaying solution of ($*$). (iii) If $p^{*}<\mathrm{p}⩽{\mathrm{p}}_{+}^p$ then there is a unique pseudo-slow decaying solution to ($*$). (iv) If p^p₊<p then there is a unique slow decaying solution to ($*$). Similar results are obtained for the operator ${ℳ}_{\lambda ,\Lambda }^{-}$.

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