On existence and nonexistence of isoperimetric inequality with differents monomial weights

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