  We study fully nonlinear elliptic equations such as \[ F(D^2u) = u^p, \quad
p>1, \] 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-V\'eron \cite{B} as well as Cutri and Leoni \cite{CL}, 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.
