 Ordered subsolution and supersolution

For $Qu=\Delta u-V(u)$, a subsolution and supersolution satisfy $Q{\phi }^{-}\ge 0$ and $Q{\phi }^{+}\le 0$, respectively. An ordered pair additionally satisfies ${\phi }^{-}\le {\phi }^{+}$ throughout the domain, with boundary data $\psi$ between their boundary traces. This interior condition is separate from boundary ordering when $V$ is non-increasing: for $V(s)=-{\lambda }_{1}s$, a positive first Dirichlet Laplacian eigenfunction $e_1$ and $-e_1$ both solve the homogeneous equation and vanish on the boundary, but taking ${\phi }^{-}=e_1$ and ${\phi }^{+}=-e_1$ produces reversed interior barriers.