Towards a deterministic KPZ equation with fractional diffusion: The stationary problem

In this work we analyze the existence of solution to the fractional quasilinear problem, \begin{equation*} \left{ \begin{array}{rcll} (-\Delta)^s u &= & |\nabla u|^{p}+ \l f & $\text{ in }$\Omega , u &=& 0 &\hbox{ in } $\mathbb{R}$^N\setminus\Omega, u&>&0 &\hbox{ in }\Omega, \end{array}% \right. \end{equation*}% where $\mathrm{\Omega }\subset \text{ren}$ is a bounded regular domain (${\mathcal{C}}^2$ is sufficient), $s\in (\frac{1}{2},1)$, $1<p$ and $f$ is a measurable nonnegative function with suitable hypotheses. The analysis is done separately in three cases, subcritical, $1<p<2s$, critical, $p=2s$, and supercritical, $p>2s$.