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 $\widetilde{ W}^{ 1,p}_{ 0}(\Omega)$.
Categories: math.AP math.FA
