Existence and nonexistence of extremal functions for sharp Trudinger-Moser inequalities

Lam, Nguyen; Lu, Guozhen; Zhang, Lu

doi:10.1016/j.aim.2019.06.020

Cited by 35 publications

(8 citation statements)

References 40 publications

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“…Proof of Theorem When

β = 0

and

a \geq 2

, it was proved in (see also ) that

T M_{a, 2, 0} (4 π) > 4 π

. Hence in this case if

4 π \geq \frac{a}{2} π e

, that is

a \leq \frac{8}{e}

, then by combining this with Lemma and Lemma , we get that

\underset{s ↓ 0}{trueprefixlim} (\frac{1 - {(\frac{s}{4 π})}^{\frac{a}{2}}}{\frac{s}{4 π}}) S T M_{0} (s) < T M_{a, 2, 0} (4 π)

and

\underset{s ↑ 4 π}{trueprefixlim} (\frac{1 - {(\frac{s}{4 π})}^{\frac{a}{2}}}{\frac{s}{4 π}}) S T M_{0} (s) < T M_{a, 2, 0} (4 π) .

This, together with Lemma and the identity , give us that there exists

s \in (0, 4 π)

such that

(\frac{1 - {(\frac{s}{4 π})}^{\frac{a}{2}}}{\frac{s}{4 π}}) S T M_{0} (s) = T <...>

…”

Section: Proof Of Theoremmentioning

confidence: 75%

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Optimizers for the singular Trudinger–Moser inequalities in the critical case in

Lam

2018

Mathematische Nachrichten

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The main purpose of this paper is to study the existence of extremal functions for the singular Trudinger–Moser inequalities in the critical case in R2. More precisely, let 0≤β<2 and denote 0true81.60004pt74.0ptTMa,b,β()α=sup∇u2a+u2b≤1∫double-struckR2[]expα()1−β2false|ufalse|2−1dx||xβ,then we will prove in this article that there exists ε>0 such that TMa,2,β4π can be achieved for all 0≤β<ε, 2−ε

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“…Proof of Theorem When

β = 0

and

a \geq 2

, it was proved in (see also ) that

T M_{a, 2, 0} (4 π) > 4 π

. Hence in this case if

4 π \geq \frac{a}{2} π e

, that is

a \leq \frac{8}{e}

, then by combining this with Lemma and Lemma , we get that

\underset{s ↓ 0}{trueprefixlim} (\frac{1 - {(\frac{s}{4 π})}^{\frac{a}{2}}}{\frac{s}{4 π}}) S T M_{0} (s) < T M_{a, 2, 0} (4 π)

and

\underset{s ↑ 4 π}{trueprefixlim} (\frac{1 - {(\frac{s}{4 π})}^{\frac{a}{2}}}{\frac{s}{4 π}}) S T M_{0} (s) < T M_{a, 2, 0} (4 π) .

This, together with Lemma and the identity , give us that there exists

s \in (0, 4 π)

such that

(\frac{1 - {(\frac{s}{4 π})}^{\frac{a}{2}}}{\frac{s}{4 π}}) S T M_{0} (s) = T <...>

…”

Section: Proof Of Theoremmentioning

confidence: 75%

“…Proof of Theorem 1.1. When = 0 and ≥ 2, it was proved in [9,11,16] (see also [10]) that ,2,0 (4 ) > 4 . Hence in this case if 4 ≥ 2 , that is ≤ 8 , then by combining this with Lemma 3.2 and Lemma 3.4, we get that…”

Section: Applying Lemma 23 and Lemma 24 Withmentioning

confidence: 99%

Optimizers for the singular Trudinger–Moser inequalities in the critical case in

Lam

2018

Mathematische Nachrichten

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“…We finally remark that there is some recent development on the existence and nonexistence of extremal functions for subcritical Trudinger-Moser inequalities established by Lam, Lu and Zhang [21] using the equivalence and identities between the supremums for the critical and subcritical Trudinger-Moser inequalities in R n established by the same authors in [22]. For subcritical Adams inequalities on the entire space, the existence of extremal functions has been proved by Chen, Lu and Zhang [7].…”

Section: Introductionmentioning

confidence: 89%

Existence and nonexistence of extremals for critical Adams inequalities in R4 and Trudinger-Moser inequalities in R2
Chen
¹
,
Lu
²
,
Zhu
³

2020
Advances in Mathematics
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22
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4
0
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Though much work has been done with respect to the existence of extremals of the critical first order Trudinger-Moser inequalities in W 1,n (R n ) and higher order Adams inequalities on finite domain Ω ⊂ R n , whether there exists an extremal function for the critical higher order Adams inequalities on the entire space R n still remains open. The current paper represents the first attempt in this direction. The classical blowup procedure cannot apply to solving the existence of critical Adams type inequality because of the absence of the Pólya-Szegö type inequality. In this paper, we develop some new ideas and approaches based on a sharp Fourier rearrangement principle (see [24]), sharp constants of the higher-order Gagliardo-Nirenberg inequalities and optimal poly-harmonic truncations to study the existence and nonexistence of the maximizers for the Adams inequalities in R 4 of the formwhere α ∈ (−∞, 32π 2 ). We establish the existence of the threshold α * , where α * ≥ (32π 2 ) 2 B2 2 and B 2 ≥ 1 24π 2 , such that S (α) is attained if 32π 2 − α < α * , and is not attained if 32π 2 − α > α * . This phenomena has not been observed before even in the case of first order Trudinger-Moser inequality. Therefore, we also establish the existence and non-existence of an extremal function for the Trudinger-Moser inequality on R 2 . Furthermore, the symmetry of the extremal functions can also be deduced through the Fourier rearrangement principle.

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“…Concerning the existence and nonexistence of the best constants of the Trudinger-Moser type inequalities, the following results were proved in [11,15,16,17,23,26,31]:…”

Section: Nguyen Lammentioning

confidence: 99%

Equivalence of sharp Trudinger-Moser-Adams Inequalities

Lam¹

2017

Communications on Pure &Amp; Applied Analysis

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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.

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Existence and nonexistence of extremal functions for sharp Trudinger-Moser inequalities

Cited by 35 publications

References 40 publications

Optimizers for the singular Trudinger–Moser inequalities in the critical case in

Optimizers for the singular Trudinger–Moser inequalities in the critical case in

Existence and nonexistence of extremals for critical Adams inequalities in R4 and Trudinger-Moser inequalities in R2
Chen
¹
,
Lu
²
,
Zhu
³

2020
Advances in Mathematics
Self Cite
22
0
4
0
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Equivalence of sharp Trudinger-Moser-Adams Inequalities

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Existence and nonexistence of extremal functions for sharp Trudinger-Moser inequalities

Cited by 35 publications

References 40 publications

Optimizers for the singular Trudinger–Moser inequalities in the critical case in

Optimizers for the singular Trudinger–Moser inequalities in the critical case in

Existence and nonexistence of extremals for critical Adams inequalities in R4 and Trudinger-Moser inequalities in R2Chen1, Lu2, Zhu3 2020Advances in MathematicsSelf Cite22040View full textAdd to dashboardCite

Equivalence of sharp Trudinger-Moser-Adams Inequalities

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Existence and nonexistence of extremals for critical Adams inequalities in R4 and Trudinger-Moser inequalities in R2
Chen
¹
,
Lu
²
,
Zhu
³

2020
Advances in Mathematics
Self Cite
22
0
4
0
View full text Add to dashboard Cite