Sharp Liouville results for fully nonlinear equations with power-growth nonlinearities
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 3, pp. 729-746.

We study fully nonlinear elliptic equations such as

F ( D 2 u ) = u p , p > 1 ,

in n or in exterior domains, where F is any uniformly elliptic, positively homogeneous operator. We show that there exists a critical exponent, depending on the homogeneity of the fundamental solution of F, that sharply characterizes the range of p>1 for which there exist positive supersolutions or solutions in any exterior domain. Our result generalizes theorems of Bidaut-Véron [6] as well as Cutri and Leoni [11], who found critical exponents for supersolutions in the whole space n , in case -F is Laplace’s operator and Pucci’s operator, respectively. The arguments we present are new and rely only on the scaling properties of the equation and the maximum principle.

Published online:
Classification: 35B53, 35J60
Armstrong, Scott N. 1; Sirakov, Boyan 2

1 Department of Mathematics       Louisiana State University Baton Rouge, LA 70803
2 UFR SEGMI Université Paris 10 92001 Nanterre Cedex, France and CAMS, EHESS 54 bd Raspail 75270 Paris Cedex 06, France
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Armstrong, Scott N.; Sirakov, Boyan. Sharp Liouville results for fully nonlinear equations with power-growth nonlinearities. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 3, pp. 729-746. http://www.numdam.org/item/ASNSP_2011_5_10_3_729_0/

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