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

Abstract

We study the boundary behavior of positive functions u satisfying (E) - Delta u-kappa/d(2)(x)u+g(u) = 0 in a bounded domain ohm of R-N, where 0 < kappa <= 1/4, g is a continuous nondecreasing function and d(.) is the distance function to delta ohm. We first construct the Martin kernel associated to the linear operator L-kappa = -Delta - kappa d(2)(x) and give a general condition for solving equation (E) with any Radon measure mu for boundary data. When g(u) = vertical bar u vertical bar(q-1)u we show the existence of a critical exponent q(c) = q(c) (N, kappa) > 1 with the following properties: when 0 < q < q(c) any measure is eligible for solving (E) with mu for boundary data; if q >= q(c), a necessary and sufficient condition is expressed in terms of the absolute continuity of mu with respect to some Besov capacity. The same capacity characterizes the removable compact boundary sets. At end any positive solution (F) - Delta u - kappa/d(2)(x)u + vertical bar u vertical bar(q-1)u = 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.