In this paper we consider the problem of finding a positive solution of the equation Δ u + | x | ν u ( n + 2 + 2 ν ) / ( n − 2 ) = 0 \Delta u + |x{|^\nu }{u^{(n + 2 + 2\nu )/(n - 2)}} = 0 in a cone C \mathcal {C} , with zero boundary data. We are only interested in solutions that are regular at infinity (i.e. such that u ( x ) = o ( | x | 2 − n ) u(x) = o(|x{|^{2 - n}}) , as C ∋ x → ∞ \mathcal {C} \ni x \to \infty ). We will always assume that ν > − 2 \nu > - 2 . We show that the existence of a solution depends on the sign of ν \nu and also on the shape of the cone C \mathcal {C} .