Regularity for a class of degenerate fully nonlinear nonlocal elliptic equations

We consider a wide class of fully nonlinear integro-differential equations that degenerate when the gradient of the solution vanishes. By using compactness and perturbation arguments, we give a complete characterization of the regularity of viscosity solutions according to different diffusion orders. More precisely, when the order of the fractional diffusion is sufficiently close to 2, we obtain Hölder continuity for the gradient of any viscosity solutions and further derive an improved gradient regularity estimate at the origin. For the order of the fractional diffusion in the interval $(1,2)$, we prove that there is at least one solution of class $C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,\alpha }$. Additionally, if the order of the fractional diffusion is in the interval $(0,1]$, the local Hölder continuity of solutions is inferred.