We discuss the existence and the asymptotic behavior of positive radial solutions for the following equation: Δpu(x) + f (u, x) = 0, where Δpu = div(Dup-2Du), x ℝn, n > p > 1, and we assume that f ≥ 0 is subcritical for u large and x small and supercritical for u small and x large, with respect to the Sobolev critical exponent. We give sufficient conditions for the existence of ground states with fast decay. As a corollary we also prove the existence of ground states with slow decay and of singular ground states with fast and slow decays. For the proofs we use a Fowler transformation which enables us to use dynamical arguments. This approach allows to unify the study of different types of non-linearities and to complete the results already appeared in literature with the analysis of singular solutions.