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

Abstract

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