Critical and Subcritical Anisotropic Trudinger–Moser Inequalities on the Entire Euclidean Spaces

Abstract

We investigate the subcritical anisotropic Trudinger–Moser inequality in the entire space ${ℝ}^N$ , obtain the asymptotic behavior of the supremum for the subcritical anisotropic Trudinger–Moser inequalities on the entire Euclidean spaces, and provide a precise relationship between the supremums for the critical and subcritical anisotropic Trudinger–Moser inequalities. Furthermore, we can prove critical anisotropic Trudinger–Moser inequalities under the nonhomogenous norm restriction and obtain a similar relationship with the supremums of subcritical anisotropic Trudinger–Moser inequalities.