The sub-supersolution method for the following semilinear equation \left\{ \begin{alignedat}{2}\tag{1} - \Delta u & = f(x,u) \quad && \text{in }\Omega\\ u & = 0 \quad && \text{on }\partial \Omega \end{alignedat} \right. states that:
Let $\underline{U}$ (resp., $\overline{U}$) be a subsolution (resp., a supersolution) to problem (1) such that $\underline{U} \leq \overline{U}$ in $\Omega$ . Then the following properties hold true:
(I) there exists a solution $u$ of (1) satisfying $\underline{U} \leq u \leq \overline{U}$;
(II) there exist minimal and maximal solutions $\underline{u}$ and $\overline{u}$ of problem (1) with respect to the interval $[ \underline{U}, \overline{U} ]$.
My question is: is there a sub-supersolution method for fractional Laplacian $(-\Delta)^s, 0? Or for what operators there are sub-supersolution methods? For the $(-\Delta)^s$ case,if yes, how to prove it?
If we follow the standard "monotone iterations method", it seems that we need a maximum principle for $(-\Delta)^s+c(x)$, a $W^{2,p}$ estimate and a Schauder estimate for $(-\Delta)^s$. But I don't know those results.
Can anyone help me? Thanks!