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 (Formula presented.). We deal with harmonic functions associated to uniformly elliptic, fully nonlinear nonlocal operators, including the linear case (Formula presented.) where (Formula presented.) denotes the fractional Laplacian of order (Formula presented.). We use the viscosity solution’s theory and Perron’s method to construct harmonic functions with zero exterior condition in (Formula presented.) and a boundary blow-up profile (Formula presented.) for any given boundary data (Formula presented.). Our method allows us to provide a blow-up rate for the gradient of the solution. © 2024 Taylor & Francis Group, LLC.