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

TL;DR

This paper proves a weighted Trudinger-Moser inequality in the entire space for a class of quasilinear elliptic operators that generalize well-known operators, involving fractional dimensions and new inequalities.

Contribution

It introduces a weighted Trudinger-Moser inequality for generalized elliptic operators in radial form, extending previous results to fractional dimensions and new weighted inequalities.

Findings

Established the inequality on weighted Sobolev spaces in f4f4f4 space.

Derived bounds for the optimal constant in a Gagliardo-Nirenberg inequality.

Developed a new weighted Pf3lya-Szegf6 principle.

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.