Large harmonic functions for fully nonlinear fractional operators

We study existence, uniqueness and boundary blow-up profile for fractional harmonic functions on a bounded smooth domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$. We deal with harmonic functions associated to uniformly elliptic, fully nonlinear nonlocal operators, including the linear case $$(-\mathrm{\Delta }{)}^{s}u=0in\mathrm{\Omega },$$ where $(-\mathrm{\Delta }{)}^s$ denotes the fractional Laplacian of order $2s\in (0,2)$. We use the viscosity solution’s theory and Perron’s method to construct harmonic functions with zero exterior condition in ${\bar{\mathrm{\Omega }}}^c$, and boundary blow-up profile $$\underset{x\to x_0,x\in \mathrm{\Omega }}{\lim }\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x,\mathrm{\partial }\mathrm{\Omega }{)}^{1-s}u(x)=h(x_0),forall{x}_0\in \mathrm{\partial }\mathrm{\Omega },$$ for any given boundary data $h\in C(\mathrm{\partial }\mathrm{\Omega })$. Our method allows us to provide blow-up rate for the solution and its gradient estimates. Results are new even in the linear case.