Sharp Liouville results for fully nonlinear equations with power-growth nonlinearities

Armstrong, Scott N.; Sirakov, Boyan

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 3, pp. 729-746.

We study fully nonlinear elliptic equations such as

$$F\left(D^{2}u\right)=u^p,p>1,$$

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.

Classification : 35B53, 35J60

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