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

We study the boundary behavior of positive functions u satisfying(E) - [Delta]u - [kappa]/d super(2)(x)u+g(u) = 0 in a bounded domain [Omega] of [dbl-struck R] super(N), where 0 < [kappa] [< or =] [1/4], g is a continuous nondecreasing function and d(.) is the distance function to [partialdifferential][Omega]. We first construct the Martin kernel associated to the linear operator L sub([kappa]) = -[Delta]- [kappa]/d super(2)(x) and give a general condition for solving equation (E) with any Radon measure [mu] for boundary data. When g(u) = u super(q-1)u we show the existence of a critical exponent q sub(c) = q sub(c)(N, [kappa]) > 1 with the following properties: when 0 < q < q sub(c) any measure is eligible for solving (E) with [mu] for boundary data; if q > or = q sub(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 super(2)(x)u + u super(q-1)u = 0 with q > 1 admits a boundary trace which is a positive outer regular Borel measure. When 1 < q < q sub(c) we prove that to any positive outer regular Borel measure we can associate a positive solutions of (F) with this boundary trace.