Explicit subsolutions and a Liouville theorem for fully nonlinear uniformly elliptic inequalities in halfspaces

We prove a Liouville type theorem for arbitrarily growing positive viscosity supersolutions of fully nonlinear uniformly elliptic equations in halfspaces. Precisely, let M-lambda.Lambda(-) be the Pucci's inf-operator with ellipticity constants Lambda >= lambda > 0. We prove that the inequality M-lambda.Lambda(-)(D(2)u) + u(P) <= 0 does not have any positive viscosity solution in a halfspace provided that -1 <= p <= Lambda/lambda n+1/Lambda/lambda n-1, 2 whereas positive solutions do exist if either p < -1 or p > Lambda/lambda(n-1)+2/Lambda/lambda(n-1). The proof relies on the construction of explicit subsolutions of the homogeneous equation M-lambda.Lambda(-) (D(2)u) = 0 and on a nonlinear version in a halfspace of the classical Hadamard three-circles theorem for entire superharmonic functions. (C) 2012 Elsevier Masson SAS. All rights reserved.