Finsler $p$-Laplace equation with a potential: Maz'ya-type characterization and attainments of the Hardy constant

We study positive properties of the quasilinear elliptic equation $$-\mathrm{d}\mathrm{i}\mathrm{v}\mathcal{A}(x,\mathrm{\nabla }u)+V\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u=0(1<p<\mathrm{\infty })in\mathrm{\Omega },$$ where the function $\mathcal{A}(x,\xi )$ is induced by a family of norms on ${\mathbb{R}}^n$ ($n\ge 2$) parameterized by points in the domain $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$, and $V$ belongs to a certain local Morrey space. We first establish two-sided estimates for Bregman distances of $\mathrm{\mid }\xi {\mathrm{\mid }}_{s,a}^p$ ($1<s<\mathrm{\infty }$), where $a=(a_1,a_2,\dots ,a_n)$ and $a_1,a_2,\dots ,a_n$ are certain functions with positive local lower and upper bounds in $\mathrm{\Omega }$. 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 ${\tilde{W}}_0^{1,p}(\mathrm{\Omega })$.