In this paper, we study positive solutions of the quasilinear elliptic equation $$Q'_{ p,\mathcal{ A},V}[u]\triangleq-\mathrm{ div}{ \mathcal{ A}(x,\nabla u)}+V(x)|u|^{ p-2}u=0,$$ in a domain $\Omega\subseteq \mathbb{ R}^n$, 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.
Categories: math.AP