Fading absorption in non-linear elliptic equations

We study the equation −Δu+h(x)|u|q−1u=0, q>1, in where , h⩾0. Let (x1,…,xN) be a coordinate system such that and denote a point x∈RN by (x′,xN). Assume that h(x′,xN)>0 when x′≠0 but h(x′,xN)→0 as |x′|→0. For this class of equations we obtain sharp necessary and sufficient conditions in order that singularities on the boundary do not propagate in the interior.