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, p. 729-746

We study fully nonlinear elliptic equations such as

$F\left({D}^{2}u\right)={u}^{p},\phantom{\rule{1em}{0ex}}p>1,$

in ${ℝ}^{n}$ 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 ${ℝ}^{n}$, 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 : 2018-06-21
Classification:  35B53,  35J60
@article{ASNSP_2011_5_10_3_729_0,
author = {Armstrong, Scott N. and Sirakov, Boyan},
title = {Sharp Liouville results for fully nonlinear equations with power-growth nonlinearities},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 10},
number = {3},
year = {2011},
pages = {729-746},
zbl = {1250.35050},
mrnumber = {2905384},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2011_5_10_3_729_0}
}

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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