On a Weighted Adachi–Tanaka Type Trudinger–Moser Inequality in Nonradial Sobolev Spaces

In this article we establish a Trudinger–Moser inequality of Adachi–Tanaka type in nonradial weighted Sobolev spaces for functions defined on the whole $\\mathbb{R}^{N}$. The main tools are the Besecovitch covering lemma and a Trudinger–Moser inequality on the whole space established by S. Adachi and K. Tanaka.