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

Abstract

We study fully nonlinear elliptic equations such as \[ F(D^2u) = u^p, \quad p>1, \] 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-Véron \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.

16 pages, new existence results added