Extremals for a Hardy–Trudinger–Moser Inequality with Remainder Terms

Let B ⊂ R2 be the unit disk , H be the completion of C∞ 0 (B) under the norm||u||H B |∇u|2dx − B u2 (1 − |x|2)2 dx 1/2 . In this paper, we consider a maximum problem concerning the Hardy–Trudinger– Moser inequalities containing lower order perturbation. Namely, there exists a positive constant ε0 such that if γ ≤ 4π + ε0, then sup u∈H, ||u||H≤1 B (e4πu2 − γ u2)dx can be achieved by some functions u0 ∈ H with ||u0||H = 1. This expands the results of Wang and Ye (Adv Math 230:294–320, 2012