  We study positive properties of the quasilinear elliptic equation
  $$-\mathrm{div}\mathcal{A}(x,\nabla u)+V|u|^{p-2}u=0\quad (1<p<\infty)\qquad
\mbox{ in } \Omega,$$ where the function $\mathcal{A}(x,\xi)$ is induced by a
family of norms on $\mathbb{R}^{n}$ ($n\geq 2$) parameterized by points in the
domain $\Omega\subseteq\mathbb{R}^{n}$, and $V$ belongs to a certain local
Morrey space. We first establish two-sided estimates for Bregman distances of
$|\xi|^{p}_{s,a}$ ($1<s<\infty$), where $a=(a_{1},a_{2},\ldots,a_{n})$ and
$a_{1},a_{2},\ldots,a_{n}$ are certain functions with positive local lower and
upper bounds in $\Omega$. These estimates lead to a Maz'ya-type
characterization for Hardy-weights of the corresponding functionals. Then we
prove three types of sufficient conditions for the attainment of the Hardy
constant in a certain space $\widetilde{W}^{1,p}_{0}(\Omega)$.
