Fractional Infinity Laplacian with Obstacle

This paper deals with the obstacle problem for the fractional infinity Laplacian with nonhomogeneous term $f(u)$, where $f:\mathbb{R}^+ \mapsto \mathbb{R}^+$: $$\begin{cases} L[u]=f(u) &\qquad in \{u>0\}\\ u \geq 0 &\qquad in\, \Omega\\ u=g &\qquad on\, \partial \Omega\end{cases},$$ with $$L[u](x)=\sup_{y\in \Omega,\,y\neq x}\dfrac{u(y)-u(x)}{|y-x|^{\alpha}}+\inf_{y\in \Omega,\,y\neq x} \dfrac{u(y)-u(x)}{|y-x|^\alpha},\qquad 0<\alpha<1.$$ Under the assumptions that $f$ is a continuous and monotone function and that the boundary datum $g$ is in $C^{0,\beta}(\partial\Omega)$ for some $0<\beta<\alpha$, we prove existence of a solution $u$ to this problem. Moreover, this solution $u$ is $\beta-$H\"olderian on $\overline{\Omega}$. Our proof is based on an approximation of $f$ by an appropriate sequence of functions $f_\varepsilon$ where we prove using Perron's method the existence of solutions $u_\varepsilon$, for every $\varepsilon>0$. Then, we show some uniform H\"older estimates on $u_\varepsilon$ that guarantee that $u_\varepsilon \rightarrow u$ where this limit function $u$ turns out to be a solution to our obstacle problem.