Vanishing-concentration-compactness alternative for the Trudinger–Moser inequality in ℝN

Ó, João Marcos do; Sani, Federica; Tarsi, Cristina

doi:10.1142/s021919971650036x

Cited by 10 publications

(11 citation statements)

References 22 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%

“…Thus, in this case, we have

\underset{n \to \infty}{trueprefixlim} \int_{{double-struckR}^{2}} \frac{e^{4 π (1 - \frac{β}{2}) {(||, v_{n})}^{2}} - 1}{{|x|}^{β}} d x = 0 .

Case 2 :

\int_{B_{R}^{c}} {(||, \nabla, v_{n})}^{2} d x \to 0

for all

R > 0

. In this case,

((), v_{n})

is a maximizing (radial) normalized concentrating sequence (see ). Set

\begin{matrix} true \\ right r & left = e^{- t / 2}, \\ right w_{n} ((), t) & left = {(4 π)}^{1 / 2} v_{n} ((), r), \end{matrix}

we can easily check that

\begin{matrix} true \\ left \int_{R^{2}} {(||, v_{n})}^{2} d x = \frac{1}{4} \int_{- \infty}^{\infty} {(||, w_{n}, (t))}^{2} e^{- t} d t; \\ left \int_{R^{2}} {(||, \nabla, v_{n})}^{2} d x = \int_{- \infty}^{\infty} {(||, w_{n}^{'}, (t))}^{2} d t; \\ left \int_{R^{2}} exp () \end{matrix}

…”

Section: Proof Of Theoremmentioning

confidence: 94%

“…To prove Theorem B, the author in proposed the following new idea: the first observation is that

S T M_{β} (\cdot)

is a continuous function. Hence, the supremum in will become a maximum if we can control the two limits

\underset{s ↓ 0}{trueprefixlim} {((), \frac{1 - {((), \frac{s}{α})}^{\frac{N - 1}{N} a}}{{(\frac{s}{α})}^{\frac{N - 1}{N} b}})}^{\frac{N - β}{b}} S T M_{β} (s)

and

\underset{s ↑ α}{trueprefixlim} {((), \frac{1 - {((), \frac{s}{α})}^{\frac{N - 1}{N} a}}{{(\frac{s}{α})}^{\frac{N - 1}{N} b}})}^{\frac{N - β}{b}} S T M_{β} (s) .

Actually, these two limits are the vanishing and concentration phenomena of

T M_{a, b, β}

in the sense of , correspondingly. When the supremum in becomes a maximum, using a reasonable scaling argument, the maximizer for

T M_{a, b, β}

can be pointed out.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

show abstract

“…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%

“…Thus, in this case, we have

\underset{n \to \infty}{trueprefixlim} \int_{{double-struckR}^{2}} \frac{e^{4 π (1 - \frac{β}{2}) {(||, v_{n})}^{2}} - 1}{{|x|}^{β}} d x = 0 .

Case 2 :

\int_{B_{R}^{c}} {(||, \nabla, v_{n})}^{2} d x \to 0

for all

R > 0

. In this case,

((), v_{n})

is a maximizing (radial) normalized concentrating sequence (see ). Set

\begin{matrix} true \\ right r & left = e^{- t / 2}, \\ right w_{n} ((), t) & left = {(4 π)}^{1 / 2} v_{n} ((), r), \end{matrix}

we can easily check that

\begin{matrix} true \\ left \int_{R^{2}} {(||, v_{n})}^{2} d x = \frac{1}{4} \int_{- \infty}^{\infty} {(||, w_{n}, (t))}^{2} e^{- t} d t; \\ left \int_{R^{2}} {(||, \nabla, v_{n})}^{2} d x = \int_{- \infty}^{\infty} {(||, w_{n}^{'}, (t))}^{2} d t; \\ left \int_{R^{2}} exp () \end{matrix}

…”

Section: Proof Of Theoremmentioning

confidence: 94%

“…To prove Theorem B, the author in proposed the following new idea: the first observation is that

S T M_{β} (\cdot)

is a continuous function. Hence, the supremum in will become a maximum if we can control the two limits

\underset{s ↓ 0}{trueprefixlim} {((), \frac{1 - {((), \frac{s}{α})}^{\frac{N - 1}{N} a}}{{(\frac{s}{α})}^{\frac{N - 1}{N} b}})}^{\frac{N - β}{b}} S T M_{β} (s)

and

\underset{s ↑ α}{trueprefixlim} {((), \frac{1 - {((), \frac{s}{α})}^{\frac{N - 1}{N} a}}{{(\frac{s}{α})}^{\frac{N - 1}{N} b}})}^{\frac{N - β}{b}} S T M_{β} (s) .

Actually, these two limits are the vanishing and concentration phenomena of

T M_{a, b, β}

in the sense of , correspondingly. When the supremum in becomes a maximum, using a reasonable scaling argument, the maximizer for

T M_{a, b, β}

can be pointed out.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Lam

2018

Mathematische Nachrichten

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show abstract

“…For proving the existence of a maximizer of C N,α or D N,α (a, b) with α < α N , we need to avoid the lack of the compactness. In this case, concentration phenomena do not occur (see [10,Lemma 4.2]) and vanishing phenomena are issues due to the unboundedness of the domain. Concerning the maximizing problem D N,αN (a, b) with b ≤ N , we may also suffer from the lack of the compactness caused by the concentration.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Moreover, the author in [12] also proved the non-existence of a maximizer when N = 2 and 0 < α ≪ 1. For general a and b, the authors in [10] showed that D N,α (a, b) is attained when N ≥ 2, α = α N , a > N ′ and 0 < b < N .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Existence and non-existence of maximizers for the Moser–Trudinger type inequalities under inhomogeneous constraints

2018

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In this paper, we study the existence and non-existence of maximizers for the Moser-Trudinger type inequalities in R N of the formHereαN ] and ΦN (t) := e t − N−2 j=0 t j j! where αN := N ω 1/(N−1) N−1 and ωN−1 denotes the surface area of the unit ball in R N . We show the existence of the threshold α * = α * (a, b, N ) ∈ [0, αN ] such that DN,α(a, b) is not attained if α ∈ (0, α * ) and is attained if α ∈ (α * , αN ). We also provide the conditions on (a, b) in order that the inequality α * < αN holds.2010 Mathematics Subject Classification. 47J30; 46E35; 26D10.

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Maximizers for the variational problems associated with Sobolev type inequalities under constraints

Nguyen

2018

Math. Ann.

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We propose a new approach to study the existence and non-existence of maximizers for the variational problems associated with Sobolev type inequalities both in the subcritical case and critical case under the equivalent constraints. The method is based on an useful link between the attainability of the supremum in our variational problems and the attainablity of the supremum of some special functions defined on (0, ∞). Our approach can be applied to the same problems related to the fractional Laplacian operators. Our main results are new in the critical case and in the setting of the fractional Laplacian operator which was left open in the work of Ishiwata and Wadade [17,18]. In the subcritical case, our approach provides a new and elementary proof of the results of Ishiwata and Wadade [17,18]. p 1 γ , and S N,p,γ = {u ∈ W 1,p (R N ) : u W 1,p γ = 1}.

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Vanishing-concentration-compactness alternative for the Trudinger–Moser inequality in ℝN

Cited by 10 publications

References 22 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 non-existence of maximizers for the Moser–Trudinger type inequalities under inhomogeneous constraints

Maximizers for the variational problems associated with Sobolev type inequalities under constraints

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