Normalized solutions to the Schrödinger systems with double critical growth and weakly attractive potentials
In this paper, we look for solutions to the following critical Schrödinger system { − Δ u + ( V 1 + λ 1 ) u = | u | 2 ∗ − 2 u + | u | p 1 − 2 u + β r 1 | u | r 1 − 2 u | v | r 2 i n R N , − Δ v + ( V 2 + λ 2 ) v = | v | 2 ∗ − 2 v + | v | p 2 − 2 v + β r 2 | u | r 1 | v | r 2 − 2 v i n R N , havi...
Saved in:
Published in | Electronic journal of qualitative theory of differential equations Vol. 2023; no. 42; pp. 1 - 22 |
---|---|
Main Authors | , |
Format | Journal Article |
Language | English |
Published |
University of Szeged
01.01.2023
|
Subjects | |
Online Access | Get full text |
Cover
Loading…
Abstract | In this paper, we look for solutions to the following critical Schrödinger system
{
−
Δ
u
+
(
V
1
+
λ
1
)
u
=
|
u
|
2
∗
−
2
u
+
|
u
|
p
1
−
2
u
+
β
r
1
|
u
|
r
1
−
2
u
|
v
|
r
2
i
n
R
N
,
−
Δ
v
+
(
V
2
+
λ
2
)
v
=
|
v
|
2
∗
−
2
v
+
|
v
|
p
2
−
2
v
+
β
r
2
|
u
|
r
1
|
v
|
r
2
−
2
v
i
n
R
N
,
having prescribed mass
∫
R
N
u
2
=
a
1
>
0
and
∫
R
N
v
2
=
a
2
>
0
, where
λ
1
,
λ
2
∈
R
will arise as Lagrange multipliers,
N
⩾
3
,
2
∗
=
2
N
/
(
N
−
2
)
is the Sobolev critical exponent,
r
1
,
r
2
>
1
,
p
1
,
p
2
,
r
1
+
r
2
∈
(
2
+
4
/
N
,
2
∗
)
and
β
>
0
is a coupling constant. Under suitable conditions on the potentials
V
1
and
V
2
,
β
∗
>
0
exists such that the above Schrödinger system admits a positive radial normalized solution when
β
⩾
β
∗
. The proof is based on comparison argument and minmax method. |
---|---|
AbstractList | In this paper, we look for solutions to the following critical Schrödinger system
{
−
Δ
u
+
(
V
1
+
λ
1
)
u
=
|
u
|
2
∗
−
2
u
+
|
u
|
p
1
−
2
u
+
β
r
1
|
u
|
r
1
−
2
u
|
v
|
r
2
i
n
R
N
,
−
Δ
v
+
(
V
2
+
λ
2
)
v
=
|
v
|
2
∗
−
2
v
+
|
v
|
p
2
−
2
v
+
β
r
2
|
u
|
r
1
|
v
|
r
2
−
2
v
i
n
R
N
,
having prescribed mass
∫
R
N
u
2
=
a
1
>
0
and
∫
R
N
v
2
=
a
2
>
0
, where
λ
1
,
λ
2
∈
R
will arise as Lagrange multipliers,
N
⩾
3
,
2
∗
=
2
N
/
(
N
−
2
)
is the Sobolev critical exponent,
r
1
,
r
2
>
1
,
p
1
,
p
2
,
r
1
+
r
2
∈
(
2
+
4
/
N
,
2
∗
)
and
β
>
0
is a coupling constant. Under suitable conditions on the potentials
V
1
and
V
2
,
β
∗
>
0
exists such that the above Schrödinger system admits a positive radial normalized solution when
β
⩾
β
∗
. The proof is based on comparison argument and minmax method. In this paper, we look for solutions to the following critical Schrödinger system $$\begin{cases} -\Delta u+(V_1+\lambda_1)u=|u|^{2^*-2}u+|u|^{p_1-2}u+\beta r_1|u|^{r_1-2}u|v|^{r_2}&{\rm in}\ \mathbb{R}^N,\\ -\Delta v+(V_2+\lambda_2)v=|v|^{2^*-2}v+|v|^{p_2-2}v+\beta r_2|u|^{r_1}|v|^{r_2-2}v&{\rm in}\ \mathbb{R}^N, \end{cases}$$ having prescribed mass $\int_{\mathbb{R}^N}u^2=a_1>0$ and $\int_{\mathbb{R}^N}v^2=a_2>0$, where $\lambda_1,\lambda_2\in\mathbb{R}$ will arise as Lagrange multipliers, $N\geqslant3$, $2^*=2N/(N-2)$ is the Sobolev critical exponent, $r_1,r_2>1$, $p_1,p_2,r_1+r_2\in(2+4/N,2^*)$ and $\beta>0$ is a coupling constant. Under suitable conditions on the potentials $V_1$ and $V_2$, $\beta_*>0$ exists such that the above Schrödinger system admits a positive radial normalized solution when $\beta\geqslant\beta_*$. The proof is based on comparison argument and minmax method. |
Author | Feng, Xiaojing Long, Lei |
Author_xml | – sequence: 1 givenname: Lei surname: Long fullname: Long, Lei – sequence: 2 givenname: Xiaojing surname: Feng fullname: Feng, Xiaojing |
BookMark | eNpNkE1u2zAQRonCBWInPUB2vIBd_oiUtAyCNg0QtIs2a2LEGdl0ZTEh6RjuwXKBXKxqHARdfYO3eBi8BZuNcSTGLqVYyUpp9Zm2jwVppYTSK7mq1Ac2l5Wsl7qpzey_-4wtct4KoZQ1ds5232PawRD-EPIch30Jccy8RF42xH_6TXp5xjCuKfF8zIV2mR9C2XCM-24g7lMowcPA1ykeJgwj8gPB7-HIoZQEvoQn4g-x0FgCDPmCfeynoU9ve87uv375df1teffj5vb66m7ptbBlSVTXbYVdpzvZmqohwr5vewStrWwaQqts672G1iO0tULE3uq6BWNBqJb0Obs9eTHC1j2ksIN0dBGCewUxrR2k6fOBHKEyIADJCFtp2XTQ1CiM9mC86ZSYXPLk8inmnKh_90nhXtu7U3v3r72TbkJ_AeF5f4U |
Cites_doi | 10.1090/S0002-9947-2014-06237-5 10.1088/1361-6544/aab0bf 10.1007/s00526-021-02020-7 10.1016/j.jde.2022.06.012 10.1080/03605302.2021.1893747 10.1016/S0362-546X(96)00021-1 10.1007/s00229-013-0627-9 10.1007/s00013-012-0468-x 10.1016/j.matpur.2016.03.004 10.1016/j.jmaa.2021.125013 10.1016/j.jde.2022.06.013 10.1007/s00526-020-1703-0 10.57262/die036-0102-133 10.48550/arXiv.2107.08708 10.1007/s11784-021-00878-w 10.1007/s00033-022-01757-1 10.1007/BF00166815 10.1016/j.jfa.2017.01.025 10.1017/CBO9780511551703 10.1088/1751-8113/43/21/213001 10.1016/j.jde.2020.05.016 10.1017/S0308210517000087 10.1515/ans-2007-0306 10.1007/978-3-642-61798-0 10.1016/s0294-1449(16)30422-x 10.1016/j.jfa.2020.108610 10.1007/978-1-4612-4146-1 10.1016/j.na.2016.05.016 10.1063/5.0077931 10.1007/s00526-018-1476-x 10.1103/PhysRevLett.79.2217 10.1007/BF01403504 10.1515/ans-2014-0104 |
ContentType | Journal Article |
DBID | AAYXX CITATION DOA |
DOI | 10.14232/ejqtde.2023.1.42 |
DatabaseName | CrossRef DOAJ Directory of Open Access Journals |
DatabaseTitle | CrossRef |
DatabaseTitleList | CrossRef |
Database_xml | – sequence: 1 dbid: DOA name: DOAJ Directory of Open Access Journals url: https://www.doaj.org/ sourceTypes: Open Website |
DeliveryMethod | fulltext_linktorsrc |
Discipline | Mathematics |
EISSN | 1417-3875 |
EndPage | 22 |
ExternalDocumentID | oai_doaj_org_article_ed25a0ade5064318ba87d053ca5c5b20 10_14232_ejqtde_2023_1_42 |
GroupedDBID | -~9 29G 2WC 5GY 5VS AAYXX ABDBF ACGFO ACIPV ADBBV AEGXH AENEX AIAGR ALMA_UNASSIGNED_HOLDINGS B0M BCNDV C1A CITATION E3Z EAP EBS EJD EMK EN8 EOJEC EPL EST ESX FRJ FRP GROUPED_DOAJ J9A KQ8 LO0 OBODZ OK1 P2P REM RNS TR2 TUS XSB ~8M |
ID | FETCH-LOGICAL-c306t-ee7794dbb3b19548eedff9fda336188ed6269cc3a9cda972dddf6379a56a029e3 |
IEDL.DBID | DOA |
ISSN | 1417-3875 |
IngestDate | Thu Jul 04 21:03:10 EDT 2024 Fri Aug 23 01:22:44 EDT 2024 |
IsDoiOpenAccess | true |
IsOpenAccess | true |
IsPeerReviewed | true |
IsScholarly | true |
Issue | 42 |
Language | English |
LinkModel | DirectLink |
MergedId | FETCHMERGED-LOGICAL-c306t-ee7794dbb3b19548eedff9fda336188ed6269cc3a9cda972dddf6379a56a029e3 |
OpenAccessLink | https://doaj.org/article/ed25a0ade5064318ba87d053ca5c5b20 |
PageCount | 22 |
ParticipantIDs | doaj_primary_oai_doaj_org_article_ed25a0ade5064318ba87d053ca5c5b20 crossref_primary_10_14232_ejqtde_2023_1_42 |
PublicationCentury | 2000 |
PublicationDate | 2023-01-01 |
PublicationDateYYYYMMDD | 2023-01-01 |
PublicationDate_xml | – month: 01 year: 2023 text: 2023-01-01 day: 01 |
PublicationDecade | 2020 |
PublicationTitle | Electronic journal of qualitative theory of differential equations |
PublicationYear | 2023 |
Publisher | University of Szeged |
Publisher_xml | – name: University of Szeged |
References | ref13 ref12 ref15 ref14 ref31 ref30 ref11 ref33 ref10 ref32 ref2 ref1 ref17 ref16 ref19 ref18 ref24 ref23 ref26 ref25 ref20 ref22 ref21 ref28 ref27 ref29 ref8 ref7 ref9 ref4 ref3 ref6 ref5 |
References_xml | – ident: ref11 doi: 10.1090/S0002-9947-2014-06237-5 – ident: ref18 doi: 10.1088/1361-6544/aab0bf – ident: ref23 doi: 10.1007/s00526-021-02020-7 – ident: ref28 doi: 10.1016/j.jde.2022.06.012 – ident: ref5 doi: 10.1080/03605302.2021.1893747 – ident: ref21 doi: 10.1016/S0362-546X(96)00021-1 – ident: ref29 doi: 10.1007/s00229-013-0627-9 – ident: ref2 doi: 10.1007/s00013-012-0468-x – ident: ref4 doi: 10.1016/j.matpur.2016.03.004 – ident: ref10 doi: 10.1016/j.jmaa.2021.125013 – ident: ref12 doi: 10.1016/j.jde.2022.06.013 – ident: ref20 doi: 10.1007/s00526-020-1703-0 – ident: ref33 doi: 10.57262/die036-0102-133 – ident: ref27 doi: 10.48550/arXiv.2107.08708 – ident: ref22 doi: 10.1007/s11784-021-00878-w – ident: ref26 doi: 10.1007/s00033-022-01757-1 – ident: ref1 doi: 10.1007/BF00166815 – ident: ref6 doi: 10.1016/j.jfa.2017.01.025 – ident: ref15 doi: 10.1017/CBO9780511551703 – ident: ref13 doi: 10.1088/1751-8113/43/21/213001 – ident: ref30 doi: 10.1016/j.jde.2020.05.016 – ident: ref3 doi: 10.1017/S0308210517000087 – ident: ref8 doi: 10.1515/ans-2007-0306 – ident: ref16 doi: 10.1007/978-3-642-61798-0 – ident: ref24 doi: 10.1016/s0294-1449(16)30422-x – ident: ref31 doi: 10.1016/j.jfa.2020.108610 – ident: ref32 doi: 10.1007/978-1-4612-4146-1 – ident: ref17 doi: 10.1016/j.na.2016.05.016 – ident: ref25 doi: 10.1063/5.0077931 – ident: ref7 doi: 10.1007/s00526-018-1476-x – ident: ref14 doi: 10.1103/PhysRevLett.79.2217 – ident: ref9 doi: 10.1007/BF01403504 – ident: ref19 doi: 10.1515/ans-2014-0104 |
SSID | ssj0022656 |
Score | 2.3052216 |
Snippet | In this paper, we look for solutions to the following critical Schrödinger system
{
−
Δ
u
+
(
V
1
+
λ
1
)
u
=
|
u
|
2
∗
−
2
u
+
|
u
|
p
1
−
2
u
+
β
r
1
|
u
|
r... In this paper, we look for solutions to the following critical Schrödinger system $$\begin{cases} -\Delta u+(V_1+\lambda_1)u=|u|^{2^*-2}u+|u|^{p_1-2}u+\beta... |
SourceID | doaj crossref |
SourceType | Open Website Aggregation Database |
StartPage | 1 |
SubjectTerms | normalized solutions positive solutions schrödinger systems weakly attractive potentials |
Title | Normalized solutions to the Schrödinger systems with double critical growth and weakly attractive potentials |
URI | https://doaj.org/article/ed25a0ade5064318ba87d053ca5c5b20 |
Volume | 2023 |
hasFullText | 1 |
inHoldings | 1 |
isFullTextHit | |
isPrint | |
link | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwrV1LTwIxEG4MXvRgfEZ8kB48mSyw7T6PoBJighcl4bZpO7O-EBAWjf4w_4B_zOnuQvDkxWvTbJpvpjvzTdtvGDvTViVNEzvRIrSlG6GdSEboSBGlLnqUMRj7wLl3E3T73vXAH6y0-rJ3wgp54AK4BoLwVVMBWmU1ckCtohDIc4zyja9FwdZdf0GmSqolKE0pzzDtSWQDn14zsKqYQtbduid-RaEVsf48qnS22VaZDvJWsYwdtoajXbbZW2qpzvbYS973fvj4icCXjsKzMac5_NY8TL-_IK_N8UKVecZtbZXDeK6HyE3ZyoDfE9-mYTUC_o7qefjBVZblT6TekE_Gmb01RK64z_qdq7uLrlM2SXAMZfuZgxjSlgKtpc7F2yjmpWmcgpIycKMIgRhLbIxUsQEVhwIA0kCGsfID1RQxygNWGY1HeMh4oCXEQlG0Cn0PldCuEE2NSgrwI_oTVNn5ArRkUmhhJJZDWISTAuHEIpy4iSeqrG1hXU60Mtb5ABk3KY2b_GXco__4yDHbsKsq6iYnrJJN53hKmUSma2y91b5sd2q58_wA0MDMtA |
link.rule.ids | 315,786,790,870,2115,27957,27958 |
linkProvider | Directory of Open Access Journals |
openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Normalized+solutions+to+the+Schr%C3%B6dinger+systems+with+double+critical+growth+and+weakly+attractive+potentials&rft.jtitle=Electronic+journal+of+qualitative+theory+of+differential+equations&rft.au=Long%2C+Lei&rft.au=Feng%2C+Xiaojing&rft.date=2023-01-01&rft.issn=1417-3875&rft.eissn=1417-3875&rft.issue=42&rft.spage=1&rft.epage=22&rft_id=info:doi/10.14232%2Fejqtde.2023.1.42&rft.externalDBID=n%2Fa&rft.externalDocID=10_14232_ejqtde_2023_1_42 |
thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=1417-3875&client=summon |
thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=1417-3875&client=summon |
thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=1417-3875&client=summon |