Positive solutions of $Delta u+u^p = 0$ whose singular set is a manifold with boundary

The aim of this paper is to prove the existence of weak solutions to the equation , with , which are positive in a domain and which are singular along a k-dimensional submanifold with smooth boundary. Here, the exponent p is required to lie in the interval , where is the dimension of the singular set. In the particular case where and , solutions correspond to solutions of the singular Yamabe problem. Received: 7 October 2001 / Accepted: 7 March 2002 / Published online: 6 August 2002