  We solve a class of isoperimetric problems on $\mathbb{R}^2_+ :=\left\{
(x,y)\in \mathbb{R} ^2 : y>0 \right\}$ with respect to monomial weights. Let
$\alpha $ and $\beta $ be real numbers such that $0\le \alpha <\beta+1$,
$\beta\le 2 \alpha$. We show that, among all smooth sets $\Omega$ in
$\mathbb{R} ^2_+$ with fixed weighted measure $\iint_{\Omega } y^{\beta} dxdy$,
the weighted perimeter $\int_{\partial \Omega } y^\alpha \, ds$ achieves its
minimum for a smooth set which is symmetric w.r.t. to the $y$--axis, and is
explicitly given. Our results also imply an estimate of a weighted Cheeger
constant and a lower bound for the first eigenvalue of a class of nonlinear
problems.
