Remarks on the Moser–Trudinger type inequality with logarithmic weights in dimension 𝑁

We provide a simpler proof of the Moser–Trudinger type inequality with logarithmic weight w β = ( − ln ⁡ | x | ) β ( N − 1 ) w_\beta = (-\ln |x|)^{\beta (N-1)} , β ∈ [ 0 , 1 ) \beta \in [0,1) in dimension N ≥ 2 N\geq 2 recently established by Calanchi and Ruf. Our proof is based on a suitable change of functions on B B and the classical Moser–Trudinger inequality on B B . We also prove the existence of maximizers for this inequality when β ≥ 0 \beta \geq 0 sufficiently small.