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The nonlinear Schrödinger equation in the half-space
The present paper is concerned with the half-space Dirichlet problem where R + N : = { x ∈ R N : x N > 0 } for some N ≥ 1 and p > 1 , c > 0 are constants. We analyse the existence, non-existence and multiplicity of bounded positive solutions to ( P c ). We prove that the existence and multi...
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| Published in: | Mathematische annalen 2022-06, Vol.383 (1-2), p.361-397 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
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| cites | cdi_FETCH-LOGICAL-c363t-d0236cec8c42e51b6d931e848517a9bde9bc474ff15438ac953bfa00ed2fea983 |
| container_end_page | 397 |
| container_issue | 1-2 |
| container_start_page | 361 |
| container_title | Mathematische annalen |
| container_volume | 383 |
| creator | Fernández, Antonio J. Weth, Tobias |
| description | The present paper is concerned with the half-space Dirichlet problem
where
R
+
N
:
=
{
x
∈
R
N
:
x
N
>
0
}
for some
N
≥
1
and
p
>
1
,
c
>
0
are constants. We analyse the existence, non-existence and multiplicity of bounded positive solutions to (
P
c
). We prove that the existence and multiplicity of bounded positive solutions to (
P
c
) depend in a striking way on the value of
c
>
0
and also on the dimension
N
. We find an explicit number
c
p
∈
(
1
,
e
)
, depending only on
p
, which determines the threshold between existence and non-existence. In particular, in dimensions
N
≥
2
, we prove that, for
0
<
c
<
c
p
, problem (
P
c
) admits infinitely many bounded positive solutions, whereas, for
c
>
c
p
, there are no bounded positive solutions to (
P
c
). |
| doi_str_mv | 10.1007/s00208-020-02129-8 |
| format | article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2680276752</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2680276752</sourcerecordid><originalsourceid>FETCH-LOGICAL-c363t-d0236cec8c42e51b6d931e848517a9bde9bc474ff15438ac953bfa00ed2fea983</originalsourceid><addsrcrecordid>eNp9kD1OxDAQhS0EEmHhAlSRqA1jO06cEq34k1aiYKktxxmTrIKza2cLLsYFuBhegkRHMW-K-eaN5hFyyeCaAVQ3EYCDoklSMV5TdUQyVghOmYLqmGRpLqlUgp2Ssxg3ACAAZEbkusPcj37oPZqQv9gufH22vX_DkONub6Z-9Hnv8ylhnRkcjVtj8ZycODNEvPjtC_J6f7dePtLV88PT8nZFrSjFRFvgorRolS04StaUbS0YqkJJVpm6abFubFEVzjFZCGVsLUXjDAC23KGplViQq9l3G8bdHuOkN-M--HRS81IBr8pK8kTxmbJhjDGg09vQv5vwoRnoQzx6jkcn0T_x6IO1mJdigg_v_ln_s_UNCfpnsQ</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2680276752</pqid></control><display><type>article</type><title>The nonlinear Schrödinger equation in the half-space</title><source>Springer Nature</source><creator>Fernández, Antonio J. ; Weth, Tobias</creator><creatorcontrib>Fernández, Antonio J. ; Weth, Tobias</creatorcontrib><description>The present paper is concerned with the half-space Dirichlet problem
where
R
+
N
:
=
{
x
∈
R
N
:
x
N
>
0
}
for some
N
≥
1
and
p
>
1
,
c
>
0
are constants. We analyse the existence, non-existence and multiplicity of bounded positive solutions to (
P
c
). We prove that the existence and multiplicity of bounded positive solutions to (
P
c
) depend in a striking way on the value of
c
>
0
and also on the dimension
N
. We find an explicit number
c
p
∈
(
1
,
e
)
, depending only on
p
, which determines the threshold between existence and non-existence. In particular, in dimensions
N
≥
2
, we prove that, for
0
<
c
<
c
p
, problem (
P
c
) admits infinitely many bounded positive solutions, whereas, for
c
>
c
p
, there are no bounded positive solutions to (
P
c
).</description><identifier>ISSN: 0025-5831</identifier><identifier>EISSN: 1432-1807</identifier><identifier>DOI: 10.1007/s00208-020-02129-8</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Dirichlet problem ; Half spaces ; Mathematics ; Mathematics and Statistics ; Schrodinger equation</subject><ispartof>Mathematische annalen, 2022-06, Vol.383 (1-2), p.361-397</ispartof><rights>The Author(s) 2021</rights><rights>The Author(s) 2021. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c363t-d0236cec8c42e51b6d931e848517a9bde9bc474ff15438ac953bfa00ed2fea983</citedby><cites>FETCH-LOGICAL-c363t-d0236cec8c42e51b6d931e848517a9bde9bc474ff15438ac953bfa00ed2fea983</cites><orcidid>0000-0001-5347-8057</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,776,780,27903,27904</link.rule.ids></links><search><creatorcontrib>Fernández, Antonio J.</creatorcontrib><creatorcontrib>Weth, Tobias</creatorcontrib><title>The nonlinear Schrödinger equation in the half-space</title><title>Mathematische annalen</title><addtitle>Math. Ann</addtitle><description>The present paper is concerned with the half-space Dirichlet problem
where
R
+
N
:
=
{
x
∈
R
N
:
x
N
>
0
}
for some
N
≥
1
and
p
>
1
,
c
>
0
are constants. We analyse the existence, non-existence and multiplicity of bounded positive solutions to (
P
c
). We prove that the existence and multiplicity of bounded positive solutions to (
P
c
) depend in a striking way on the value of
c
>
0
and also on the dimension
N
. We find an explicit number
c
p
∈
(
1
,
e
)
, depending only on
p
, which determines the threshold between existence and non-existence. In particular, in dimensions
N
≥
2
, we prove that, for
0
<
c
<
c
p
, problem (
P
c
) admits infinitely many bounded positive solutions, whereas, for
c
>
c
p
, there are no bounded positive solutions to (
P
c
).</description><subject>Dirichlet problem</subject><subject>Half spaces</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Schrodinger equation</subject><issn>0025-5831</issn><issn>1432-1807</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><recordid>eNp9kD1OxDAQhS0EEmHhAlSRqA1jO06cEq34k1aiYKktxxmTrIKza2cLLsYFuBhegkRHMW-K-eaN5hFyyeCaAVQ3EYCDoklSMV5TdUQyVghOmYLqmGRpLqlUgp2Ssxg3ACAAZEbkusPcj37oPZqQv9gufH22vX_DkONub6Z-9Hnv8ylhnRkcjVtj8ZycODNEvPjtC_J6f7dePtLV88PT8nZFrSjFRFvgorRolS04StaUbS0YqkJJVpm6abFubFEVzjFZCGVsLUXjDAC23KGplViQq9l3G8bdHuOkN-M--HRS81IBr8pK8kTxmbJhjDGg09vQv5vwoRnoQzx6jkcn0T_x6IO1mJdigg_v_ln_s_UNCfpnsQ</recordid><startdate>20220601</startdate><enddate>20220601</enddate><creator>Fernández, Antonio J.</creator><creator>Weth, Tobias</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>C6C</scope><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0001-5347-8057</orcidid></search><sort><creationdate>20220601</creationdate><title>The nonlinear Schrödinger equation in the half-space</title><author>Fernández, Antonio J. ; Weth, Tobias</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c363t-d0236cec8c42e51b6d931e848517a9bde9bc474ff15438ac953bfa00ed2fea983</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2022</creationdate><topic>Dirichlet problem</topic><topic>Half spaces</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Schrodinger equation</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Fernández, Antonio J.</creatorcontrib><creatorcontrib>Weth, Tobias</creatorcontrib><collection>Springer Open Access</collection><collection>CrossRef</collection><jtitle>Mathematische annalen</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Fernández, Antonio J.</au><au>Weth, Tobias</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The nonlinear Schrödinger equation in the half-space</atitle><jtitle>Mathematische annalen</jtitle><stitle>Math. Ann</stitle><date>2022-06-01</date><risdate>2022</risdate><volume>383</volume><issue>1-2</issue><spage>361</spage><epage>397</epage><pages>361-397</pages><issn>0025-5831</issn><eissn>1432-1807</eissn><abstract>The present paper is concerned with the half-space Dirichlet problem
where
R
+
N
:
=
{
x
∈
R
N
:
x
N
>
0
}
for some
N
≥
1
and
p
>
1
,
c
>
0
are constants. We analyse the existence, non-existence and multiplicity of bounded positive solutions to (
P
c
). We prove that the existence and multiplicity of bounded positive solutions to (
P
c
) depend in a striking way on the value of
c
>
0
and also on the dimension
N
. We find an explicit number
c
p
∈
(
1
,
e
)
, depending only on
p
, which determines the threshold between existence and non-existence. In particular, in dimensions
N
≥
2
, we prove that, for
0
<
c
<
c
p
, problem (
P
c
) admits infinitely many bounded positive solutions, whereas, for
c
>
c
p
, there are no bounded positive solutions to (
P
c
).</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00208-020-02129-8</doi><tpages>37</tpages><orcidid>https://orcid.org/0000-0001-5347-8057</orcidid><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0025-5831 |
| ispartof | Mathematische annalen, 2022-06, Vol.383 (1-2), p.361-397 |
| issn | 0025-5831 1432-1807 |
| language | eng |
| recordid | cdi_proquest_journals_2680276752 |
| source | Springer Nature |
| subjects | Dirichlet problem Half spaces Mathematics Mathematics and Statistics Schrodinger equation |
| title | The nonlinear Schrödinger equation in the half-space |
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