  We solve a class of weighted isoperimetric problems of the form $$
  \min\left\{\int\_{\partial E}w e^V\,dx:\int\_E e^V\,dx={\rm constant}\right\}
$$ where $w$ and $V$ are suitable functions on $\R^d$. As a consequence, we
prove a comparison result for the solutions of degenerate elliptic equations.
