Equivalence of sharp Trudinger-Moser-Adams Inequalities

Abstract

Improved Trudinger-Moser-Adams type inequalities in the spirit of Lions were recently studied in [21]. The main purpose of this paper is to prove the equivalence of these versions of the Trudinger-Moser-Adams type inequalities and to set up the relations of these Trudinger-Moser-Adams best constants. Moreover, using these identities, we will investigate the existence and nonexistence of the optimizers for some Trudinger-Moser-Adams type inequalities.

On attainability of Moser-Trudinger inequality with logarithmic weights in higher dimensions

“…Blow up analysis for Moser-Trudinger functional is dealt in [19,25,26,36]. We list a few other related works in this direction [1,2,3,10,12,13,18,16,20,22,30,32,34] and the references there in for the available literatures in this direction.…”

“…In Lemma 2.1, we show that a maximizing sequence can loose compactness only if it concentrates at the point x = 0. Then in Lemma 3.3 the concentration level J δ β,w0 (0) (see (12) for the formal definition) is explicitly calculated. It is shown that for all β ∈ [0, 1)…”

Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc

“…Later, Carleson and Chang's results were extended by Flucher [9] to arbitrary bounded domain in R 2 and by Lin [22] to general bounded domain in R n (n ≥ 2). Li [17,19] and [18] developed a blow-up method to establish the existence of extremal functions for the Trudinger-Moser inequality on Riemannian manifolds (see also [10,19,20,28,29,30]).…”