We classify all the possible asymptotic behavior at the origin for positive solutions of quasilinear elliptic equations of the form div(∇|u|ᵖ⁻²∇u)=b(x)h(u) in Ω∖{0}, where 1∠p≤N and Ω is an open subset of ℝᴺ with 0∈Ω. Our main result provides a sharp extension of a well-known theorem of Friedman and Véron for h(u)=uq and b(x)≡1, and a recent result of the authors for p=2 and b(x)≡1. We assume that the function h is regularly varying at ∞ with index q (that is, limt→∞h(λt)/h(t)=λq for every λ>0) and the weight function b(x) behaves near the origin as a function b0(|x|) varying regularly at zero with index θ greater than −p. This condition includes b(x)=|x|θ and some of its perturbations, for instance, b(x)=|x|θ(−log|x|)ͫ for any m ∈ ℝ. Our approach makes use of the theory of regular variation and a new perturbation method for constructing sub- and super-solutions.