Ordered subsolution and supersolution (${\phi }^{-}\le {\phi }^{+}$)

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.

For smooth non-increasing $V$ and an ordered subsolution and supersolution, begin with $u_0={\phi }^{-}$ and solve successive Poisson equations with forcing $V(u_k)$ and the fixed boundary data. The weak maximum principle for elliptic operators applied to the negatives of consecutive differences givesThe global Schauder estimate and the Hölder interpolation inequality bound the iterates uniformly in $C^{2,\alpha }$. Their bounded monotone sequence converges, and compact embedding of Hölder spaces permits passing to the equation. The uniform Hölder seminorm bounds retain $C^{2,\alpha }$ regularity of the limit.