On a semilinear fractional Dirichlet problem on a bounded domain

Let D   be a bounded C1,1C1,1-domain in RdRd (d⩾2d⩾2), 0<α<20<α<2 and σ<1σ<1. We are concerned with the existence, the uniqueness and the asymptotic behavior of positive continuous solution for the following semilinear fractional differential equation(-Δ|D)α2u=a(x)uσ(x)inD,with the boundary Dirichlet conditionlimx⟶∂Dδ(x)2-αu(x)=0.Here (-Δ|D)α2 is the fractional Laplacian associated to the subordinate killed Brownian motion process in D   and δ(x)=d(x,∂D)δ(x)=d(x,∂D) denotes the Euclidean distance between x   and ∂D∂D. Our arguments are based on potential theory associated to the fractional Laplacian and some Karamata regular variation theory tools.