= Ordered subsolution and supersolution
{title2=$\varphi^-\leq\varphi^+$}
For $Qu=\Delta u-V(u)$, a subsolution and supersolution satisfy $Q\varphi^-\geq0$ and $Q\varphi^+\leq0$, respectively. An ordered pair additionally satisfies $\varphi^-\leq\varphi^+$ 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 $V(s)=-\lambda_1s$, a positive first <Dirichlet Laplacian eigenfunction> $e_1$ and $-e_1$ both solve the homogeneous equation and vanish on the boundary, but taking $\varphi^-=e_1$ and $\varphi^+=-e_1$ produces reversed interior barriers.
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