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

Abstract We study positivity properties of the quasilinear elliptic equation $$\begin{aligned} -\textrm{div}\mathcal {A}(x,\nabla u)+V|u|^{p-2}u=0\quad (1 - div A ( x , ∇ u ) + V | u | p - 2 u = 0 ( 1 < p < ∞ ) in Ω , where the function $$\mathcal {A}(x,\xi )$$ A ( x , ξ ) is induced by a family of norms on $$\mathbb {R}^{n}$$ R n ( $$n\ge 2$$ n ≥ 2 ) parameterized by points in the domain $$\Omega \subseteq \mathbb {R}^{n}$$ Ω ⊆ R n , and V belongs to a certain local Morrey space. We first establish some two-sided estimates for the Bregman distances of $$|\xi |^{p}_{s,a}$$ | ξ | s , a p ( $$1 1 < s < ∞ ), where $$a=(a_{1},a_{2},\ldots ,a_{n})$$ a = ( a 1 , a 2 , … , a n ) and $$a_{1},a_{2},\ldots ,a_{n}$$ a