For p∈ (1 , N) and Ω ⊆ RN open, the Beppo-Levi space D01,p(Ω) is the completion of Cc∞(Ω) with respect to the norm [∫Ω|∇u|pdx]1p. Using the p-capacity, we define a norm and then identify the Banach function space H(Ω) with the set of all g in Lloc1(Ω) that admits the following Hardy–Sobolev type inequality: ∫Ω|g||u|pdx≤C∫Ω|∇u|pdx,∀u∈D01,p(Ω),for some C> 0. Further, we characterize the set of all g in H(Ω) for which the map G(u)=∫Ωg|u|pdx is compact on D01,p(Ω). We use a variation of the concentration compactness lemma to give a sufficient condition on g∈ H(Ω) so that the best constant in the above inequality is attained in D01,p(Ω). © 2021, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature.