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 $M^-_{λ, Λ}$ be the Pucci's inf- operator, defined as the infimum of all linear uniformly elliptic operators with ellipticity constants $Λ\geq λ>0$. Then, we prove that the inequality $M^-_{λ, Λ}(D^2u) +u^p \leq 0$ does not have any positive viscosity solution in a halfspace provided that $-1\leq p \leq (Λ/λn+1)/(Λ/λn-1)$, whereas positive solutions do exist if either $p < -1$ or $p > (Λ/λ(n-1)+2)/(Λ/λ(n-1))$. This will be accomplished by constructing explicit subsolutions of the homogeneous equation $M^-_{λ, Λ}(D^2u)=0$ and by proving a nonlinear version in a halfspace of the classical Hadamard three-circles Theorem for entire superharmonic functions.

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