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 $int_{Omega}x^{B}dx$.

On a weighted Trudinger-Moser inequality in $mathbb{R}^N$

We establish the Trudinger-Moser inequality on weighted Sobolev spaces in the whole space, and for a class of quasilinear elliptic operators in radial form of the type $displaystyle Lu:=-r^{-theta}(r^{alpha}vert u(r)vert^{beta}u(r))$, where $theta, betageq 0$ and $alpha>0$, are constants satisfying some existence conditions. It worth emphasizing that these operators generalize the $p$- Laplacian and $k$-Hessian operators in the radial case. Our results involve fractional dimensions, a new weighted Polya-Szeg{o} principle, and a boundness value for the optimal constant in a Gagliardo-Nirenberg type inequality.