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

TL;DR

This paper investigates a Finsler $p$-Laplace equation with a potential, establishing estimates and characterizations related to Hardy inequalities and conditions for attaining the Hardy constant in a specific functional space.

Contribution

It provides a Maz'ya-type characterization for Hardy-weights and sufficient conditions for Hardy constant attainability in a Finsler $p$-Laplace setting.

Findings

Two-sided Bregman distance estimates for $| abla u|^p$

Maz'ya-type characterization for Hardy-weights

Conditions for Hardy constant attainment

Abstract

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 $\widetilde{W}^{1,p}_{0}(\Omega)$.