Critical exponents for the Pucci's extremal operators

Patricio L. Felmer; Alexander Quaas

doi:10.1016/S1631-073X(02)02605-5

Critical exponents for the Pucci's extremal operators
[Les exposants critiques pour l'opérateur extrémal de Pucci]

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

Comptes Rendus. Mathématique, Volume 335 (2002) no. 11, pp. 909-914.

Résumé
Abstract

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

ℳ_{λ, Λ}^{+} (D^{2} u) + u^{p} = 0, u ⩾ 0 dans ℝ^{N},

(∗)

où N⩾3, p>1 et

ℳ_{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^{*} < p ⩽ 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_{λ, Λ}^{-}

.

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

ℳ_{λ, Λ}^{+} (D^{2} u) + u^{p} = 0, u ⩾ 0 in ℝ^{N} .

(∗)

Here N⩾3, p>1 and

ℳ_{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^{*} < p ⩽ 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_{λ, Λ}^{-}

.

Reçu le : 2002-10-11
Publié le : 2002-11-16

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

Affiliations des auteurs :

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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     title = {Critical exponents for the {Pucci's} extremal operators},
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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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Guowei Dai Generalized limit theorem and bifurcation for problems with Pucci's operator, Topological Methods in Nonlinear Analysis, Volume 56 (2020) no. 1, pp. 229-261 | Zbl:1465.35044
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Patricio Felmer; Norihisa Ikoma Existence and nonexistence of positive solutions to some fully nonlinear equation in one dimension, Journal of Functional Analysis, Volume 275 (2018) no. 8, pp. 2162-2196 | DOI:10.1016/j.jfa.2018.07.009 | Zbl:1406.35020
Maria J. Esteban; Patricio L. Felmer; Alexander Quaas Large critical exponents for some second order uniformly elliptic operators, Communications in Partial Differential Equations, Volume 32 (2007) no. 4, pp. 543-556 | DOI:10.1080/03605300500394397 | Zbl:1187.35071
Patricio L. Felmer; Alexander Quaas; Moxun Tang On uniqueness for nonlinear elliptic equation involving the Pucci's extremal operator, Journal of Differential Equations, Volume 226 (2006) no. 1, pp. 80-98 | DOI:10.1016/j.jde.2005.12.008 | Zbl:1245.35038
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Patricio Felmer; Alexander Quaas Some Recent Results on Equations Involving the Pucci’s Extremal Operators, Contributions to Nonlinear Analysis, Volume 66 (2005), p. 263 | DOI:10.1007/3-7643-7401-2_18
Patricio L. Felmer; Alexander Quaas Positive radial solutions to a s̀emilinear equation involving the Pucci's operator, Journal of Differential Equations, Volume 199 (2004) no. 2, pp. 376-393 | DOI:10.1016/j.jde.2004.01.001 | Zbl:1070.34032

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