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

Abstract

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 $Lu:=-r^{-\theta }(r^{\alpha }|u^{\prime }(r){|}^{\beta }{u}^{\prime }(r){)}^{\prime }$, where $\theta ,\beta \ge 0$ 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 Pólya-Szeg{ö} principle, and a boundness value for the optimal constant in a Gagliardo-Nirenberg type inequality.