Sharp Trudinger-Moser inequalities with monomial weights

Abstract

In this paper, we will study the Trudinger-Moser inequalities with the monomial weight \(\left| x_{1}\right| ^{A_{1}}...\left| x_{N}\right| ^{A_{N}}\) in \( \mathbb {R} ^{N}\) with \(A_{1}\ge 0,..., A_{N}\ge 0\). Moreover, we investigate the Trudinger-Moser inequalities on both domains of finite and infinite volume. More importantly, we will exhibit the best constants for our results. In the particular case \(A_{1}=\cdots =\) \(A_{N}=0\), we recover many results about the Trudinger-Moser inequalities without weight established in the literature.