The aim of this paper is to prove the existence of weak solutions to the equation Au+ up= 0 which are positive in a domain Ω CRN, vanish at the boundary, and have prescribed isolated singularities. The exponent p is required to lie in the interval (N/(N-2),(JV+ 2)/(N-2)). We also prove the existence of solutions to the equation Δu+ up= 0 which are positive in a domain Ω c K n and which are singular along arbitrary smooth Λ-dimensional submanifolds in the interior of these domains provided p lies in the interval ((nk)/(nk-2),(nk+ 2)/(nk-2)). A particular case is when p=(n+ 2)/(n—2), in which case solutions correspond to solutions of the singular Yamabe problem. The method used here is a mixture of different ingredients used by both authors in their separate constructions of solutions to the singular Yamabe problem, along with a new set of scaling techniques.