Large harmonic functions for fully nonlinear fractional operators

Abstract

We study existence, uniqueness and boundary blow-up profile for fractional harmonic functions on a bounded smooth domain $\Omega \subset {\mathbb{R}}^N$. We deal with harmonic functions associated to uniformly elliptic, fully nonlinear nonlocal operators, including the linear case $$ (-Δ)^s u = 0 \quad \mbox{in} \ Ω, $$ where $(-\Delta {)}^s$ 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 ${\bar{\Omega }}^c$, and boundary blow-up profile $$ \lim_{x\to x_0, x \in Ω}\mathrm{dist}(x, \partial Ω)^{1-s}u(x)=h(x_0), \quad \mbox{for all} \quad x_0\in \partial Ω, $$ for any given boundary data $h\in C(\partial \Omega )$. Our method allows us to provide blow-up rate for the solution and its gradient estimates. Results are new even in the linear case.