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

We study fully nonlinear elliptic equations such as $$F(D^{2}u)=u^p,p>1,$$ in ${\mathbb{R}}^n$ 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-V'eron \cite{B} as well as Cutri and Leoni \cite{CL}, who found critical exponents for supersolutions in the whole space ${\mathbb{R}}^n$, 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.