On compactness in the Trüdinger-Moser inequality

Abstract

We show that the Moser functional J (u) = R (e 4⇡u 2 1) dx on the set B = {u 2 H 1 0 () : kruk 2  1}, where ⇢ R 2 is a bounded domain, fails to be weakly continuous only in the following exceptional case. Defineup to translations and up to a remainder vanishing in the Sobolev norm. In other words, the weak continuity fails only on translations of concentrating Moser functions. The proof is based on a profile decomposition similar to that of Solimini [16], but with different concentration operators, pertinent to the two-dimensional case.

Extremal functions for Trudinger–Moser inequalities of Adimurthi–Druet type in dimension two

“…Theorem 1 obviously implies C. Tintarev's inequality (9) in the case V(x) ≡ α, and whence leads to Adimurthi and O. Druet's original inequality (3). It should be remarked that Theorem 1 does not imply that the supremum in (3) is attained for all α, 0 ≤ α < λ 1 (Ω).…”

“…Before ending this section, we remark that for results in this paper, there is a possibility of another proof, which is based on the explicit structure of putative weakly vanishing maximizing sequences as concentrating Moser functions. For details about this new method, we refer the reader to Adimurthi and C. Tintarev [3].…”

Singular Moser–Trudinger inequality on simply connected domains

“…where F sup B1 < ∞ is the constant given by the singular Moser-Trudinger embedding, see (2). Since ǫ was arbitrary, this proves the theorem.…”

“…This generalizes the result of Flucher [6], who has proven the case β = 0. On the history of Flucher's result and other recent developments on the subject we refer to [2], [3], [6], [10] , [11] and [13]. Flucher gives two different proofs, one for simply connected domains and one for general domains.…”

A profile decomposition for the limiting Sobolev embedding

“…[2]), or when one only has a semigroup of isometries (e.g. [1]), or when the profile decomposition can be expressed without the explicit use of a group (e.g. Struwe [12]) and so when [11,Theorem 2.10] does not apply.…”