Improved Adams-type inequalities and their extremals in dimension 2m

Abstract

In this paper, we prove the existence of an extremal function for the Adams–Moser–Trudinger inequality on the Sobolev space [Formula: see text], where [Formula: see text] is any bounded, smooth, open subset of [Formula: see text], [Formula: see text]. Moreover, we extend this result to improved versions of Adams’ inequality of Adimurthi-Druet type. Our strategy is based on blow-up analysis for sequences of subcritical extremals and introduces several new techniques and constructions. The most important one is a new procedure for obtaining capacity-type estimates on annular regions.

Existence and nonexistence of extremals for critical Adams inequalities in R4 and Trudinger-Moser inequalities in R2Chen1, Lu2, Zhu3 2020Advances in Mathematics22040View full textAdd to dashboardCite

“…Bi-harmonic truncations. In the following, we will need some bi-harmonic truncations u M k which was studied in [9]. Roughly speaking, the value of truncations u M k is close to c k M in a small ball centered at x k , and coincides with u k outside the same ball.…”

“…In [34], the authors applied the capacity-type estimates and the Pohozaev identity to obtain the existence of extremals for bounded domains in the case n = 4 and m = 2. Recently, DelaTorre and Mancini [9] extended the results of [34] to arbitrary even dimension.…”

“…But as we know, in the second order Sobolev space H 2 (R 4 ), one cannot truncate u k linearly. To overcome this difficulty, we will apply the bi-harmonic type of truncation as used in [9]. We remark that this kind of truncations have many nice properties: on the one hand, they preserve the high order regularity, on the other hand, their behaviors are very similar to the constant in a ball.…”

Existence and Non-existence of Extremals for Critical Adams Inequality in any Even Dimension

“…After that, the existence of extremals was proved for any bounded domains in R n (see [24], [39], [3]). One can also see [34,35,37,57] for existence of extremals for the Trudinger-Moser inequalities on compact Riemannian manifold, unbounded domains, and see [42,20,13] for the existence of extremals for Adams' inequalities in bounded and unbounded domains. We note that the Trudinger-Moser-Adams inequalities on the Sobolev spaces W m, n m (Ω) without the Dirichlet boundary condition have also been established, the interested readers can refer to the work [11,30,16,43,51], and the references therein.…”

Estimate for concentration level of the Adams functional and extremals for Adams-type inequality

“…More precisely, it was proved that sup Hence, Corollary 1.2 sheds some new light on the existence of extremal functions for the inequality (1.1) in the critical case β = β 0 . Actually, for m = 1 the existence of extremals it was proved in the series of papers [5,14,18,29]; however concerning the higher order case m > 1, as far as we know, there are not so many results and we can only mention Lu and Yang [20], which proved extremals in the case m = 2 with Ω ⊂ R 4 and more recently DelaTorre and Mancini [8], where the existence of extremals for the case H m 0 (Ω) with Ω ⊂ R 2m is proved. We believe that the Corollary 1.2 is a significant contribution to solve completely this question.…”