  In this paper, we study positive solutions of the quasilinear elliptic
equation $$ Q'_{p,\mathcal{A},V}[u]\triangleq -\dive{\mathcal{A}(x,\nabla
u)}+V(x)|u|^{p-2}u=0, $$ in a domain $\Gw\subseteq \R^n$, where $n\geq 2$,
$1<p<\infty$, the divergence of~$\mathcal{A}$ is the well known
$\mathcal{A}$-Laplace operator considered in the influential book of Heinonen,
Kilpel\"{a}inen, and Martio, and the potential $V$ belongs to a certain local
Morrey space. The main aim of the paper is to extend criticality theory to the
operator $Q'_{p,\mathcal{A},V}$. In particular, we prove an
Agmon-Allegretto-Piepenbrink (AAP) type theorem, establish the uniqueness and
simplicity of the principal eigenvalue of $Q'_{p,\mathcal{A},V}$ in a domain
$\omega\Subset\Omega$, and give various characterizations of criticality.
Furthermore, we also study positive solutions of the equation
$Q'_{p,\mathcal{A},V}[u]=0$ of minimal growth at infinity in $\Omega$, the
existence of a minimal positive Green function, and the minimal decay at
infinity of Hardy-weights.
