  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 $|x|^k x_N^\alpha$. Our results
imply some sharp functional inequalities, like for instance,
Caffarelli-Kohn-Nirenberg type inequalities.
