The thresholds of the existence of maximizers for the critical sharp singular Moser–Trudinger inequality under constraints
This paper is addressed to study the existence of maximizers for the singular Moser–Trudinger supremum under constraints in the critical case M T N ( a , β ) = sup u ∈ W 1 , N ( R N ) , ‖ ∇ u ‖ N a + ‖ u ‖ N N = 1 ∫ R N Φ N ( 1 - β / N ) α N | u | N N - 1 | x | - β d x , where a > 0 , β ∈ [ 0 , N...
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Veröffentlicht in: | Mathematische annalen 2021-08, Vol.380 (3-4), p.1933-1958 |
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container_end_page | 1958 |
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container_issue | 3-4 |
container_start_page | 1933 |
container_title | Mathematische annalen |
container_volume | 380 |
creator | Nguyen, Van Hoang |
description | This paper is addressed to study the existence of maximizers for the singular Moser–Trudinger supremum under constraints in the critical case
M
T
N
(
a
,
β
)
=
sup
u
∈
W
1
,
N
(
R
N
)
,
‖
∇
u
‖
N
a
+
‖
u
‖
N
N
=
1
∫
R
N
Φ
N
(
1
-
β
/
N
)
α
N
|
u
|
N
N
-
1
|
x
|
-
β
d
x
,
where
a
>
0
,
β
∈
[
0
,
N
)
,
Φ
N
(
t
)
=
e
t
-
∑
k
=
0
N
-
2
t
k
k
!
,
α
N
=
N
ω
N
-
1
1
/
(
N
-
1
)
, and
ω
N
-
1
denotes the surface area of the unit sphere in
R
N
. More precisely, we study the effect of the parameter
a
to the attainability of
M
T
N
(
a
,
β
)
. We will prove that for each
β
∈
[
0
,
N
)
there exist the thresholds
a
∗
(
β
)
and
a
∗
(
β
)
such that
M
T
N
(
a
,
β
)
is attained for any
a
∈
(
a
∗
(
β
)
,
a
∗
(
β
)
)
and is not attained for
a
<
a
∗
(
β
)
or
a
>
a
∗
(
β
)
. We also give some qualitative estimates for
a
∗
(
β
)
and
a
∗
(
β
)
. Our results complete the recent studies on the sharp Moser–Trudinger type inequality under constraints due to do Ó, Sani and Tarsi (Commun Contemp Math 19:27, 2016), Lam (Proc Am Math Soc 145:4885–4892, 2017; Math Nachr 291(14–15):2272–2287, 2018) and Ikoma, Ishiwata and Wadade (Math Ann 373(1–2):831–851, 2019). |
doi_str_mv | 10.1007/s00208-020-02010-8 |
format | Article |
fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2559301310</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2559301310</sourcerecordid><originalsourceid>FETCH-LOGICAL-c319t-afd66011c0240b7a4c59fa84ef9bd5a99bced3b8405593fffce4b2fdeb465c113</originalsourceid><addsrcrecordid>eNp9UE1PwzAMjRBIjMEf4BSJc8Fp2q09ookvaYjLOEdp6tBMXbvFrbQhDvwH_iG_hHRD4sbBtvz8ni0_xi4FXAuA6Q0BxJBFIQ0hIMqO2EgkMo5EBtNjNgrzNEozKU7ZGdESACRAOmIfiwp5V3mkqq1L4q0NHXLcOuqwMTgAK711K_eOnrht_X5uvOuc0TWnSvs1J9e89bX2_Lkl9N-fXwvflwFDz12Dm17XrtvxvikDYNqGOq9d09E5O7G6Jrz4rWP2en-3mD1G85eHp9ntPDJS5F2kbTmZgBAG4gSKqU5MmludJWjzokx1nhcGS1lkCaRpLq21BpMitiUWySQ1QsgxuzrsXft20yN1atn2vgknVTxIQEgBgRUfWMa3RB6tWnu30n6nBKjBZXVwWYWk9i6rLIjkQUSBPDz8t_of1Q-EWoQu</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2559301310</pqid></control><display><type>article</type><title>The thresholds of the existence of maximizers for the critical sharp singular Moser–Trudinger inequality under constraints</title><source>SpringerNature Journals</source><creator>Nguyen, Van Hoang</creator><creatorcontrib>Nguyen, Van Hoang</creatorcontrib><description>This paper is addressed to study the existence of maximizers for the singular Moser–Trudinger supremum under constraints in the critical case
M
T
N
(
a
,
β
)
=
sup
u
∈
W
1
,
N
(
R
N
)
,
‖
∇
u
‖
N
a
+
‖
u
‖
N
N
=
1
∫
R
N
Φ
N
(
1
-
β
/
N
)
α
N
|
u
|
N
N
-
1
|
x
|
-
β
d
x
,
where
a
>
0
,
β
∈
[
0
,
N
)
,
Φ
N
(
t
)
=
e
t
-
∑
k
=
0
N
-
2
t
k
k
!
,
α
N
=
N
ω
N
-
1
1
/
(
N
-
1
)
, and
ω
N
-
1
denotes the surface area of the unit sphere in
R
N
. More precisely, we study the effect of the parameter
a
to the attainability of
M
T
N
(
a
,
β
)
. We will prove that for each
β
∈
[
0
,
N
)
there exist the thresholds
a
∗
(
β
)
and
a
∗
(
β
)
such that
M
T
N
(
a
,
β
)
is attained for any
a
∈
(
a
∗
(
β
)
,
a
∗
(
β
)
)
and is not attained for
a
<
a
∗
(
β
)
or
a
>
a
∗
(
β
)
. We also give some qualitative estimates for
a
∗
(
β
)
and
a
∗
(
β
)
. Our results complete the recent studies on the sharp Moser–Trudinger type inequality under constraints due to do Ó, Sani and Tarsi (Commun Contemp Math 19:27, 2016), Lam (Proc Am Math Soc 145:4885–4892, 2017; Math Nachr 291(14–15):2272–2287, 2018) and Ikoma, Ishiwata and Wadade (Math Ann 373(1–2):831–851, 2019).</description><identifier>ISSN: 0025-5831</identifier><identifier>EISSN: 1432-1807</identifier><identifier>DOI: 10.1007/s00208-020-02010-8</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Mathematics ; Mathematics and Statistics ; Thresholds</subject><ispartof>Mathematische annalen, 2021-08, Vol.380 (3-4), p.1933-1958</ispartof><rights>Springer-Verlag GmbH Germany, part of Springer Nature 2020</rights><rights>Springer-Verlag GmbH Germany, part of Springer Nature 2020.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c319t-afd66011c0240b7a4c59fa84ef9bd5a99bced3b8405593fffce4b2fdeb465c113</citedby><cites>FETCH-LOGICAL-c319t-afd66011c0240b7a4c59fa84ef9bd5a99bced3b8405593fffce4b2fdeb465c113</cites><orcidid>0000-0002-0030-5811</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00208-020-02010-8$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00208-020-02010-8$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Nguyen, Van Hoang</creatorcontrib><title>The thresholds of the existence of maximizers for the critical sharp singular Moser–Trudinger inequality under constraints</title><title>Mathematische annalen</title><addtitle>Math. Ann</addtitle><description>This paper is addressed to study the existence of maximizers for the singular Moser–Trudinger supremum under constraints in the critical case
M
T
N
(
a
,
β
)
=
sup
u
∈
W
1
,
N
(
R
N
)
,
‖
∇
u
‖
N
a
+
‖
u
‖
N
N
=
1
∫
R
N
Φ
N
(
1
-
β
/
N
)
α
N
|
u
|
N
N
-
1
|
x
|
-
β
d
x
,
where
a
>
0
,
β
∈
[
0
,
N
)
,
Φ
N
(
t
)
=
e
t
-
∑
k
=
0
N
-
2
t
k
k
!
,
α
N
=
N
ω
N
-
1
1
/
(
N
-
1
)
, and
ω
N
-
1
denotes the surface area of the unit sphere in
R
N
. More precisely, we study the effect of the parameter
a
to the attainability of
M
T
N
(
a
,
β
)
. We will prove that for each
β
∈
[
0
,
N
)
there exist the thresholds
a
∗
(
β
)
and
a
∗
(
β
)
such that
M
T
N
(
a
,
β
)
is attained for any
a
∈
(
a
∗
(
β
)
,
a
∗
(
β
)
)
and is not attained for
a
<
a
∗
(
β
)
or
a
>
a
∗
(
β
)
. We also give some qualitative estimates for
a
∗
(
β
)
and
a
∗
(
β
)
. Our results complete the recent studies on the sharp Moser–Trudinger type inequality under constraints due to do Ó, Sani and Tarsi (Commun Contemp Math 19:27, 2016), Lam (Proc Am Math Soc 145:4885–4892, 2017; Math Nachr 291(14–15):2272–2287, 2018) and Ikoma, Ishiwata and Wadade (Math Ann 373(1–2):831–851, 2019).</description><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Thresholds</subject><issn>0025-5831</issn><issn>1432-1807</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><recordid>eNp9UE1PwzAMjRBIjMEf4BSJc8Fp2q09ookvaYjLOEdp6tBMXbvFrbQhDvwH_iG_hHRD4sbBtvz8ni0_xi4FXAuA6Q0BxJBFIQ0hIMqO2EgkMo5EBtNjNgrzNEozKU7ZGdESACRAOmIfiwp5V3mkqq1L4q0NHXLcOuqwMTgAK711K_eOnrht_X5uvOuc0TWnSvs1J9e89bX2_Lkl9N-fXwvflwFDz12Dm17XrtvxvikDYNqGOq9d09E5O7G6Jrz4rWP2en-3mD1G85eHp9ntPDJS5F2kbTmZgBAG4gSKqU5MmludJWjzokx1nhcGS1lkCaRpLq21BpMitiUWySQ1QsgxuzrsXft20yN1atn2vgknVTxIQEgBgRUfWMa3RB6tWnu30n6nBKjBZXVwWYWk9i6rLIjkQUSBPDz8t_of1Q-EWoQu</recordid><startdate>20210801</startdate><enddate>20210801</enddate><creator>Nguyen, Van Hoang</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-0030-5811</orcidid></search><sort><creationdate>20210801</creationdate><title>The thresholds of the existence of maximizers for the critical sharp singular Moser–Trudinger inequality under constraints</title><author>Nguyen, Van Hoang</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-afd66011c0240b7a4c59fa84ef9bd5a99bced3b8405593fffce4b2fdeb465c113</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Thresholds</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Nguyen, Van Hoang</creatorcontrib><collection>CrossRef</collection><jtitle>Mathematische annalen</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Nguyen, Van Hoang</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The thresholds of the existence of maximizers for the critical sharp singular Moser–Trudinger inequality under constraints</atitle><jtitle>Mathematische annalen</jtitle><stitle>Math. Ann</stitle><date>2021-08-01</date><risdate>2021</risdate><volume>380</volume><issue>3-4</issue><spage>1933</spage><epage>1958</epage><pages>1933-1958</pages><issn>0025-5831</issn><eissn>1432-1807</eissn><abstract>This paper is addressed to study the existence of maximizers for the singular Moser–Trudinger supremum under constraints in the critical case
M
T
N
(
a
,
β
)
=
sup
u
∈
W
1
,
N
(
R
N
)
,
‖
∇
u
‖
N
a
+
‖
u
‖
N
N
=
1
∫
R
N
Φ
N
(
1
-
β
/
N
)
α
N
|
u
|
N
N
-
1
|
x
|
-
β
d
x
,
where
a
>
0
,
β
∈
[
0
,
N
)
,
Φ
N
(
t
)
=
e
t
-
∑
k
=
0
N
-
2
t
k
k
!
,
α
N
=
N
ω
N
-
1
1
/
(
N
-
1
)
, and
ω
N
-
1
denotes the surface area of the unit sphere in
R
N
. More precisely, we study the effect of the parameter
a
to the attainability of
M
T
N
(
a
,
β
)
. We will prove that for each
β
∈
[
0
,
N
)
there exist the thresholds
a
∗
(
β
)
and
a
∗
(
β
)
such that
M
T
N
(
a
,
β
)
is attained for any
a
∈
(
a
∗
(
β
)
,
a
∗
(
β
)
)
and is not attained for
a
<
a
∗
(
β
)
or
a
>
a
∗
(
β
)
. We also give some qualitative estimates for
a
∗
(
β
)
and
a
∗
(
β
)
. Our results complete the recent studies on the sharp Moser–Trudinger type inequality under constraints due to do Ó, Sani and Tarsi (Commun Contemp Math 19:27, 2016), Lam (Proc Am Math Soc 145:4885–4892, 2017; Math Nachr 291(14–15):2272–2287, 2018) and Ikoma, Ishiwata and Wadade (Math Ann 373(1–2):831–851, 2019).</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00208-020-02010-8</doi><tpages>26</tpages><orcidid>https://orcid.org/0000-0002-0030-5811</orcidid></addata></record> |
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issn | 0025-5831 1432-1807 |
language | eng |
recordid | cdi_proquest_journals_2559301310 |
source | SpringerNature Journals |
subjects | Mathematics Mathematics and Statistics Thresholds |
title | The thresholds of the existence of maximizers for the critical sharp singular Moser–Trudinger inequality under constraints |
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