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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 asymmetric 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 assumethat 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 characterizethe space of all Hardy-weights for the functional $Q_{p,A,V}$, extendingMaz'ya's well known characterization of the space of Hardy-weights for the$p$-Laplacian. In addition, we provide various sufficient conditions on thepotential $V$ and the Hardy-weight $g$ such that the best constant of thecorresponding variational problem is attained in an appropriate Beppo-Levispace.

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