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

Abstract

We study the boundary behavior of positive functions u satisfying (E) −Δu−κd2(x)u+g(u)=0 in a bounded domain Ω of RN, where 0<κ≤14, g is a continuous nondecreasing function and d(.) is the distance function to ∂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 μ for boundary data. When g(u)=|u|q−1u we show the existence of a critical exponent qc=qc(N,κ)>1 with the following properties: when 0<q<qc any measure is eligible for solving (E) with μ for boundary data; if q≥qc, a necessary and sufficient condition is expressed in terms of the absolute continuity of μ with respect to some Besov capacity. The same capacity characterizes the removable compact boundary sets. At end any positive solution (F) −Δu−κd2(x)u+|u|q−1u=0 with q>1 admits a boundary trace which is a positive outer regular Borel measure. When 1<q<qc we prove that to any positive outer regular Borel measure we can associate a positive solutions of (F) with this boundary trace.