Abstract:

On a general open set of the euclidean space, we study the relation between the embedding of the homogeneous Sobolev space $D^{1,p}_0$  into $L^q$ 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 behaviour 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 $+\infty$. 

(a joint work with L. Brasco  and A.C. Zagati).