Trudinger-Moser inequality with remainder terms

摘要

The paper gives the following improvement of the Trudinger-Moser inequality: related to the Hardy-Sobolev-Mazya inequality in higher dimensions. We show (0.1) with ψ(u) = ∫_Ω V(x)u~2 dx for a class of V > 0 that includes which refines two previously known cases of (0.1) proved by Adimurthi and Druet [2] and by Wang and Ye [23]. In addition, we verify (0.1) for ψ(u) = λ||u||_p~2, as well as give an analogous improvement for the Onofri-Beckner inequality for the unit disk (Beckner [6]).