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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 $(-\Delta)^s, 0<s<1$? 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!

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It seems so: www2.math.umd.edu/~mellet/publi/mrsNL-final.pdf –  Siminore Aug 2 '12 at 7:48
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1 Answer

To use Perron's method, you only need a comparison principle. I suggest you look at Luis Silvestre's thesis ( http://math.uchicago.edu/~luis/preprints/luisdissreadable.pdf ) for a nice proof of the maximum principle and other basic facts for the fractional Laplacian. The Schauder estimates for the fractional Laplacian also follow from the estimates presented by Luis.

The $W^{2,p}$ estimates for the fractional Laplacian follow from the Hardy-Littlewood-Sobolev inequality for Riesz potentials. So, e.g., Eli Stein's book on Singular Integrals for a nice study of how the Riesz potential maps from $L_p$ to $L_p$. The general question of $W^{2,p}$ style estimates for fractional-order operators is, so far as I know, still open.

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