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 $\Omega \subset \ren$ is a bounded regular domain ($\mathcal{ C}^2$ is sufficient), $s\in (\frac 12, 1)$, $1<p$ 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, $p>2s$.
Categories: math.AP