Singular Type Trudinger–Moser Inequalities with Logarithmic Weights and the Existence of Extremals.

In this paper, we study the existence of extremals for the following singular critical Trudinger–Moser inequality with logarithmic weights: sup u ∈ W 0 , r 1 , n (B , ω β) , u ω β ≤ 1 ∫ B exp (α n , β , σ u n n - 1 1 - β ) x σ d x < ∞ , <graphic href="9_2023_2582_Article_Equ20.gif"></graphic> where B is the unit ball in R n , β ∈ [ 0 , 1) , σ ∈ [ 0 , n) , α n , β , σ = n - σ [ ω n - 1 1 n - 1 1 - β ] 1 1 - β , W 0 , r 1 , n (B , ω β) denotes the radial weighted Sobolev space with the norm u ω β = ∫ B ∇ u n ω β x d x 1 n , ω β (x) = (log e x ) β (n - 1) . Moreover, for m > 0 , we establish a singular supercritical Trudinger–Moser inequality with logarithmic weights sup u ∈ W 0 , r 1 , n (B , ω β) , u ω β ≤ 1 ∫ B exp ((α n , β , σ + x m) u n n - 1 1 - β ) x σ d x < ∞ , <graphic href="9_2023_2582_Article_Equ21.gif"></graphic> and prove the existence of its extremal functions. [ABSTRACT FROM AUTHOR]