Boundary singularities of solutions of semilinear elliptic equations with critical Hardy potentials

We study the boundary behavior of positive functions uu satisfying (E) −Δu−κd2(x)u+g(u)=0 in a bounded domain ΩΩ of RNRN, where 0<κ≤14, gg is a continuous nondecreasing function and d(.)d(.) is the distance function to ∂Ω∂Ω. We first construct the Martin kernel associated to the linear operator Lκ=−Δ−κd2(x) and give a general condition for solving equation (E) with any Radon measure μμ for boundary data. When g(u)=|u|q−1ug(u)=|u|q−1u we show the existence of a critical exponent qc=qc(N,κ)>1qc=qc(N,κ)>1 with the following properties: when 01q>1 admits a boundary trace which is a positive outer regular Borel measure. When 1