Let n ≥ 2 and let be a Lipschitz wedge-like domain. We construct positive weak solutions of the problem

that vanish in a suitable trace sense on ∂Ω, but which are singular at a prescribed ‘edge’ of Ω if p is equal to or slightly above a certain exponent p0 > 1 that depends on Ω. Moreover, for the case in which Ω is unbounded, the solutions have fast decay at infinity.