Fading absorption in non-linear elliptic equations

We study the equation -Delta u + h(x)vertical bar u vertical bar(q-1) u = 0, q > 1, in R-+(N) = RN-1 x R+ where h is an element of C(<(R-+(N))over bar>), h >= 0. Let (x(1), ... , x(N)) be a coordinate system such that R-+(N) = [x(N) > 0] and denote a point x is an element of R-N by (x', x(N)). Assume that h(x', x(N)) > 0 when x' not equal 0 but h(x', x(N)) -> 0 as vertical bar x'vertical bar -> 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. (C) 2012 Elsevier Masson SAS. All rights reserved.