The space of Hardy-weights for quasilinear equations: Maz'ya-type characterization and sufficient conditions for existence of minimizers

TL;DR

This paper extends Maz'ya's characterization of Hardy-weights to quasilinear equations with variable coefficients and potentials, providing conditions for the existence of minimizers in associated variational problems.

Contribution

It introduces a generalized notion of capacity for quasilinear functionals and characterizes Hardy-weights, extending classical results to more complex operators and potentials.

Findings

Characterization of Hardy-weights for quasilinear equations

Conditions for the attainability of the best variational constant

Extension of Maz'ya's classical results to variable coefficient operators

Abstract

Let $p \in (1,\infty)$ and $\Omega \subset \mathbb{R}^N$ be a domain. Let $ A: =(a_{ij}) \in L^{\infty}_{\text{loc}}(\Omega; \mathbb{R}^{N\times N})$ be a symmetric and locally uniformly positive definite matrix. Set $|\xi|_A^2:= \displaystyle \sum_{i,j=1}^N a_{ij}(x) \xi_i \xi_j$, $\xi \in \mathbb{R}^N$, and let $V$ be a given potential in a certain local Morrey space. We assume that the energy functional $$Q_{p,A,V}(\phi):=\displaystyle \int_{\Omega} [|\nabla \phi|_A^p + V|\phi|^p] {\rm dx} $$ is nonnegative in $W^{1,p}(\Omega)\cap C_c(\Omega)$. We introduce a generalized notion of $Q_{p,A,V}$-capacity and characterize the space of all Hardy-weights for the functional $Q_{p,A,V}$, extending Maz'ya's well known characterization of the space of Hardy-weights for the $p$-Laplacian. In addition, we provide various sufficient conditions on the potential $V$ and the Hardy-weight $g$ such that the best constant of the corresponding variational problem is attained in an appropriate Beppo-Levi space.