We study positive solutions of half-linear second-order elliptic equations of the form QA,V(u)≔−div(|∇u|Ap−2A(x)∇u)+V(x)|u|p−2u=0in Ω, where 1<p<∞, Ω is a domain in Rn, n≥2, V∈Lloc∞(Ω), A=(aij)∈Lloc∞(Ω,Rn2) is a symmetric and locally uniformly positive definite matrix in Ω, and |ξ|A2:=〈A(x)ξ,ξ〉=∑i,j=1naij(x)ξiξjx∈Ω,ξ=(ξ1,…,ξn)∈Rn. We extend criticality theory which has been established for linear operators and for half-linear operators involving the p-Laplacian, to the operator QA,V. We prove Liouville-type theorems, and study the behavior of positive solutions of the equation QA,V(u)=0 near an isolated singularity and near infinity in Ω, and obtain some perturbations results.