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

الأصل · EN

We study fully nonlinear elliptic equations such as F(D²u) = uᵖ, p>1, in 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 B as well as Cutri and Leoni CL, who found critical exponents for supersolutions in the whole space 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.