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

TL;DR

This paper establishes sharp Liouville-type theorems for fully nonlinear elliptic equations with power nonlinearities, identifying critical exponents that determine the existence of solutions in exterior domains.

Contribution

It generalizes previous results by identifying a critical exponent for fully nonlinear equations using scaling and maximum principle techniques.

Findings

Existence of a critical exponent depending on the fundamental solution's homogeneity.

Characterization of solution existence in exterior domains based on this critical exponent.

Extension of classical results to a broader class of fully nonlinear operators.

Abstract

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.