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

Abstract

We study positive properties of the quasilinear elliptic equation $$-\mathrm{div}\mathcal{A}(x,\nabla u)+V|u|^{p-2}u=0\quad (1<p<\infty)\qquad \mbox{ in } Ω,$$ 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 $\Omega \subseteq {\mathbb{R}}^n$, and $V$ belongs to a certain local Morrey space. We first establish two-sided estimates for Bregman distances of $|\xi {|}_{s,a}^p$ ($1<s<\infty$), 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 $\Omega$. 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 ${\tilde{W}}_0^{1,p}(\Omega )$.

34 pages; the introduction part, Definition 2.39, and Theorem 2.40 modified, References 3 and 17 added, and some other minor changes made