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 $\displaystyle Lu:=-r^{-\theta}(r^{\alpha}\vert u'(r)\vert^{\beta}u'(r))'$, where $\theta, \beta\geq 0$ 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 P\'olya-Szeg{\"o} principle, and a boundness value for the optimal constant in a Gagliardo-Nirenberg type inequality.