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 $M_{\lambda ,\mathrm{\Lambda }}^{-}$ be the Pucci’s inf- operator, defined as the infimum of all linear uniformly elliptic operators with ellipticity constants $\mathrm{\Lambda }\ge \lambda >0$. Then, we prove that the inequality $M_{\lambda ,\mathrm{\Lambda }}^{-}(D^{2}u)+u^p\le 0$ does not have any positive viscosity solution in a halfspace provided that $-1\leq p \leq (\Lambda/\lambda n+1)/(\Lambda/\lambda n-1)$,whereaspositivesolutionsdoexistifeither$p < -1$or$p > (\Lambda/\lambda (n-1)+2)/(\Lambda/\lambda (n-1))$. This will be accomplished by constructing explicit subsolutions of the homogeneous equation $M_{\lambda ,\mathrm{\Lambda }}^{-}(D^{2}u)=0$ and by proving a nonlinear version in a halfspace of the classical Hadamard three-circles Theorem for entire superharmonic functions.