Sobolev embeddings and distance functions

On a general open set of the euclidean space, we study the relation between the embedding of the homogeneous Sobolev space D 0 1 , p \mathcal{D}^{{1,p}}_{0} into L q L^{q} and the summability properties of the distance function. We prove that, in the superconformal case (i.e. when 𝑝 is larger than the dimension), these two facts are equivalent, while in the subconformal and conformal cases (i.e. when 𝑝 is less than or equal to the dimension), we construct counterexamples to this equivalence. In turn, our analysis permits to study the asymptotic behavior of the positive solution of the Lane–Emden equation for the 𝑝-Laplacian with sub-homogeneous right-hand side, as the exponent 𝑝 diverges to ∞. The case of first eigenfunctions of the 𝑝-Laplacian is included, as well. As particular cases of our analysis, we retrieve some well-known convergence results, under optimal assumptions on the open sets. We also give some new geometric estimates for generalized principal frequencies.