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

Abstract

Let $p\in (1,\infty )$ and $\Omega \subset {\mathbb{R}}^N$ be a domain. Let $A:=(a_{ij})\in L_{\text{loc}}^{\infty }(\Omega ;{\mathbb{R}}^{N\times N})$ be a symmetric and locally uniformly positive definite matrix. Set $|\xi {|}_A^2:=\sum _{i,j=1}^N{a}_{ij}(x){\xi }_i{\xi }_j$, $\xi \in {\mathbb{R}}^N$, and let $V$ be a given potential in a certain local Morrey space. We assume that the energy functional $$Q_{p,A,V}(\varphi ):={\int }_{\Omega }[|\nabla \varphi {|}_A^p+V|\varphi {|}^p]\mathrm{dx}$$ 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.

30 pages