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 $\Lambda \ge \lambda >0$. Then, we prove that the inequality $M_{\lambda ,\Lambda }^{-}(D^{2}u)+u^p\le 0$ does not have any positive viscosity solution in a halfspace provided that $-1\le p\le (\Lambda /\lambda n+1)/(\Lambda /\lambda n-1)$, 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 $M_{\lambda ,\Lambda }^{-}(D^{2}u)=0$ and by proving a nonlinear version in a halfspace of the classical Hadamard three-circles Theorem for entire superharmonic functions.

18 pages

References in corpus (1)

• Singular solutions of fully nonlinear elliptic equations and applications