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

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

2010-01-25

arxiv.org/abs/1001.4489v3

Mathematics

Analysis of PDEs

16 pages, new existence results added

Scientific paper

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.

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