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

Abstract

Let p ∈ (1, ∞) and Ω ⊂ ℝ^N be a domain. Let

$$A: = ({a_{ij}}) \in L_{{\rm{loc}}}^\infty (\Omega;{\mathbb{R}^{N \times N}})$$

be a symmetric and locally uniformly positive definite matrix. Set

$$|\xi |_A^2:\sum\limits_{i,j = 1}^N {{a_{ij}}(x){\xi _i}{\xi _j}},$$

ξ ∈ ℝ^N, and let V be a given potential in a certain local Morrey space. We assume that the energy functional

$${Q_{p,A,V}}(\phi ): = \int_\Omega {[|\nabla \phi |_A^p + V|\phi {|^p}]{\rm{d}}x} $$

is nonnegative in W^1,p(Ω) ∩ C_c(Ω).

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.