  We study qualitative positivity properties of quasilinear equations of the
form \[
  Q'_{A,p,V}[v] := -\mathrm{div}(|\nabla v|_A^{p-2}A(x)\nabla v) +
V(x)|v|^{p-2}v =0 \qquad x\in\Omega, \] where $\Omega$ is a domain in
$\mathbb{R}^n$, $1<p<\infty$, $A=(a_{ij})\in L^\infty_{\rm
loc}(\Omega;\mathbb{R}^{n\times n})$ is a symmetric and locally uniformly
positive definite matrix, $V$ is a real potential in a certain local Morrey
space (depending on $p$), and \[
  |\xi|_{A}^{2}:=A(x)\xi\cdot\xi=\sum_{i,j=1}^n a_{ij}(x)\xi_i\xi_j \qquad
x\in\Omega ,~\xi=(\xi_1,\ldots,\xi_n)\in \mathbb{R}^n. \] Our assumptions on
the coefficients of the operator for $p\geq 2$ are the minimal (in the Morrey
scale) that ensure the validity of the local Harnack inequality and hence the
H\"older continuity of the solutions. For some of the results of the paper we
need slightly stronger assumptions when $p<2$.
  We prove an Allegretto-Piepenbrink-type theorem for the operator
$Q'_{A,p,V}$, and extend criticality theory to our setting. Moreover, we
establish a Liouville-type theorem and obtain some perturbation results. Also,
in the case $1<p\leq n,$ we examine the behavior of a positive solution near a
nonremovable isolated singularity and characterize the existence of the
positive minimal Green function for the operator $Q'_{A,p,V}[u]$ in $\Omega$.
