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We study existence, uniqueness and boundary blow-up profile for fractionalharmonic functions on a bounded smooth domain $\Omega \subset \mathbb R^N$. Wedeal with harmonic functions associated to uniformly elliptic, fully nonlinearnonlocal operators, including the linear case $$ (-\Delta)^s u = 0 \quad\mbox{in} \ \Omega, $$ where $(-\Delta)^s$ denotes the fractional Laplacian oforder $2s \in (0,2)$. We use the viscosity solution's theory and Perron'smethod to construct harmonic functions with zero exterior condition in $\bar\Omega^c$, and boundary blow-up profile $$ \lim_{x\to x_0, x \in\Omega}\mathrm{dist}(x, \partial \Omega)^{1-s}u(x)=h(x_0), \quad \mbox{for all}\quad x_0\in \partial \Omega, $$ for any given boundary data $h \in C(\partial\Omega)$. Our method allows us to provide blow-up rate for the solution and itsgradient estimates. Results are new even in the linear case.

Large harmonic functions for fully nonlinear fractional operators