The isoperimetric problem for a class of non-radial weights and applications

We study a class of isoperimetric problems on ${\mathbb{R}}_{+}^N$ where the densities of the weighted volume and weighted perimeter are given by two different non-radial functions of the type $\mathrm{\mid }x{\mathrm{\mid }}^k{x}_N^{\alpha }$. Our results imply some sharp functional inequalities, like for instance, Caffarelli-Kohn-Nirenberg type inequalities.