  We consider a class of isoperimetric problems on $\mathbb{R}^{N}_{+} $ where
the volume and the area element carry two different weights of the type
$|x|^lx_N^\alpha$. We solve them in a special case while a more detailed study
is contained in \cite{ABCMP2}. Our results imply a weighted Polya-Sz\"ego
principle and a priori estimates for weak solutions to a class of boundary
value problems for degenerate elliptic equations
