Extremals for Hardy-Sobolev type inequalities with monomial weights

Abstract

In this article we study the existence and non-existence of extremals for the following family of Hardy-Sobolev inequalities(∫(R+)k×RN−k|xBu|q)1q≤C(∫(R+)k×RN−k|xA∇u|p)1p, which holds for suitable values of A,B∈RN, q>p>1. Here the quantity xA (respectively xB) denotes the monomial weight defined asxA=|x1|a1⋅…⋅|xk|ak(respectively xB=|x1|b1⋅…⋅|xk|bk).