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 Mλ,Λ− be the Pucciʼs inf-operator with ellipticity constants Λ⩾λ>0. We prove that the inequality Mλ,Λ−(D2u)+up⩽0 does not have any positive viscosity solution in a halfspace provided that −1⩽p⩽Λλn+1Λλn−1, whereas positive solutions do exist if either p<−1 or p>Λλ(n−1)+2Λλ(n−1). The proof relies on the construction of explicit subsolutions of the homogeneous equation Mλ,Λ−(D2u)=0 and on a nonlinear version in a halfspace of the classical Hadamard three-circles theorem for entire superharmonic functions.