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

TL;DR

This paper establishes a Liouville theorem for positive viscosity supersolutions of certain fully nonlinear elliptic inequalities in halfspaces, identifying conditions under which no solutions exist and constructing explicit subsolutions.

Contribution

It introduces explicit subsolutions and a halfspace version of Hadamard's three-circles theorem to analyze nonlinear elliptic inequalities.

Findings

No positive solutions for certain p ranges in the inequality

Existence of solutions when p is outside the specified range

Construction of explicit subsolutions for the homogeneous equation

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.