Large harmonic functions for fully nonlinear fractional operators

• Published in: Communications in partial differential equations Vol. 49; no. 10-12; pp. 919 - 937
• Main Authors: Dávila, Gonzalo, Quaas, Alexander, Topp, Erwin
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 01.12.2024; Taylor & Francis Ltd
• Subjects: Dirichlet problem; Harmonic functions; large solutions; Nonlocal operator; Operators (mathematics); viscosity solutions
• ISSN: 0360-5302; 1532-4133
• DOI: 10.1080/03605302.2024.2405923

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.