  We consider the monomial weight $x^{A}=\vert x_{1}\vert^{a_{1}}\ldots\vert
x_{N}\vert^{a_{N}}$, where $a_{i}$ is a nonnegative real number for each
$i\in\{1,\ldots,N\}$, and we establish the existence and nonexistence of
isoperimetric inequalities with different monomial weights. We study positive
minimizers of $\int_{\partial\Omega}x^{A}\mathcal{H}^{N-1}(x)$ among all smooth
bounded sets $\Omega$ in $\mathbb{R}^{N}$ with fixed Lebesgue measure with
monomial weight $\int_{\Omega}x^{B}dx$.
