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 Ω ⊂ R N . We deal with harmonic functions associated to uniformly elliptic, fully nonlinear nonlocal operators, including the linear case ( − Δ ) s u = 0 in Ω , where ( − Δ ) s denotes the fractional Laplacian of order 2 s ∈ ( 0 , 2 ) . We use the viscosity solution’s theory and Perron’s method to construct harmonic functions with zero exterior condition in Ω c and a boundary blow-up profile lim x → x 0 , x ∈ Ω dist ( x , ∂ Ω ) 1 − s u ( x ) = h ( x 0 ) , for all x 0 ∈ ∂ Ω , for any given boundary data h ∈ C ( ∂ Ω ) . Our method allows us to provide a blow-up rate for the gradient of the solution.