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We prove a Liouville type theorem for arbitrarily growing positive viscositysupersolutions of fully nonlinear uniformly elliptic equations in halfspaces.Precisely, let $M^-_{\lambda, \Lambda}$ be the Pucci's inf- operator, definedas the infimum of all linear uniformly elliptic operators with ellipticityconstants $\Lambda \geq \lambda >0$. Then, we prove that the inequality$M^-_{\lambda, \Lambda}(D^2u) +u^p \leq 0$ does not have any positive viscositysolution in a halfspace provided that $-1\leq p \leq (\Lambda/\lambdan+1)/(\Lambda/\lambda n-1)$, whereas positive solutions do exist if either $p <-1$ or $p > (\Lambda/\lambda (n-1)+2)/(\Lambda/\lambda (n-1))$. This will beaccomplished by constructing explicit subsolutions of the homogeneous equation$M^-_{\lambda, \Lambda}(D^2u)=0$ and by proving a nonlinear version in ahalfspace of the classical Hadamard three-circles Theorem for entiresuperharmonic functions.

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