On existence and nonexistence of isoperimetric inequality with differents monomial weights

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