On Existence of Multiple Normalized Solutions to a Class of Elliptic Problems in Whole RN Via Penalization Method
In this paper we study the existence of multiple normalized solutions to the following class of elliptic problems - ϵ 2 Δ u + V ( x ) u = λ u + f ( u ) , in R N , ∫ R N | u | 2 d x = a 2 ϵ N , where a , ϵ > 0 , λ ∈ R is an unknown parameter that appears as a Lagrange multiplier, V : R N → [ 0 , ∞...
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| Veröffentlicht in: | Potential analysis 2024, Vol.61 (3), p.463-483 |
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| creator | Alves, Claudianor O. Van Thin, Nguyen |
| description | In this paper we study the existence of multiple normalized solutions to the following class of elliptic problems
-
ϵ
2
Δ
u
+
V
(
x
)
u
=
λ
u
+
f
(
u
)
,
in
R
N
,
∫
R
N
|
u
|
2
d
x
=
a
2
ϵ
N
,
where
a
,
ϵ
>
0
,
λ
∈
R
is an unknown parameter that appears as a Lagrange multiplier,
V
:
R
N
→
[
0
,
∞
)
is a continuous function, and
f
is a continuous function with
L
2
-subcritical growth. It is proved that the number of normalized solutions is related to the topological richness of the set where the potential
V
attains its minimum value. In the proof of our main result, we apply minimization techniques, Lusternik-Schnirelmann category and the penalization method due to del Pino and Felmer (Calc. Var. Partial Differential Equations
4
, 121–137
1996
). |
| doi_str_mv | 10.1007/s11118-023-10116-2 |
| format | Article |
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-
ϵ
2
Δ
u
+
V
(
x
)
u
=
λ
u
+
f
(
u
)
,
in
R
N
,
∫
R
N
|
u
|
2
d
x
=
a
2
ϵ
N
,
where
a
,
ϵ
>
0
,
λ
∈
R
is an unknown parameter that appears as a Lagrange multiplier,
V
:
R
N
→
[
0
,
∞
)
is a continuous function, and
f
is a continuous function with
L
2
-subcritical growth. It is proved that the number of normalized solutions is related to the topological richness of the set where the potential
V
attains its minimum value. In the proof of our main result, we apply minimization techniques, Lusternik-Schnirelmann category and the penalization method due to del Pino and Felmer (Calc. Var. Partial Differential Equations
4
, 121–137
1996
).</description><identifier>ISSN: 0926-2601</identifier><identifier>EISSN: 1572-929X</identifier><identifier>DOI: 10.1007/s11118-023-10116-2</identifier><language>eng</language><publisher>Dordrecht: Springer Netherlands</publisher><subject>Continuity (mathematics) ; Functional Analysis ; Geometry ; Lagrange multiplier ; Mathematics ; Mathematics and Statistics ; Partial differential equations ; Potential Theory ; Probability Theory and Stochastic Processes</subject><ispartof>Potential analysis, 2024, Vol.61 (3), p.463-483</ispartof><rights>The Author(s), under exclusive licence to Springer Nature B.V. 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s11118-023-10116-2$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s11118-023-10116-2$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Alves, Claudianor O.</creatorcontrib><creatorcontrib>Van Thin, Nguyen</creatorcontrib><title>On Existence of Multiple Normalized Solutions to a Class of Elliptic Problems in Whole RN Via Penalization Method</title><title>Potential analysis</title><addtitle>Potential Anal</addtitle><description>In this paper we study the existence of multiple normalized solutions to the following class of elliptic problems
-
ϵ
2
Δ
u
+
V
(
x
)
u
=
λ
u
+
f
(
u
)
,
in
R
N
,
∫
R
N
|
u
|
2
d
x
=
a
2
ϵ
N
,
where
a
,
ϵ
>
0
,
λ
∈
R
is an unknown parameter that appears as a Lagrange multiplier,
V
:
R
N
→
[
0
,
∞
)
is a continuous function, and
f
is a continuous function with
L
2
-subcritical growth. It is proved that the number of normalized solutions is related to the topological richness of the set where the potential
V
attains its minimum value. In the proof of our main result, we apply minimization techniques, Lusternik-Schnirelmann category and the penalization method due to del Pino and Felmer (Calc. Var. Partial Differential Equations
4
, 121–137
1996
).</description><subject>Continuity (mathematics)</subject><subject>Functional Analysis</subject><subject>Geometry</subject><subject>Lagrange multiplier</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Partial differential equations</subject><subject>Potential Theory</subject><subject>Probability Theory and Stochastic Processes</subject><issn>0926-2601</issn><issn>1572-929X</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><sourceid/><recordid>eNpFkEtLxDAUhYMoOI7-AVcB19U8JkmzlGF8wLzwvQtpmzoZMk2nSUH89aaO4N1cuJxzOPcD4BKja4yQuAk4TZ4hQjOMMOYZOQIjzATJJJEfx2CEJElHjvApOAthixAiQuQjsF81cPZlQzRNaaCv4aJ30bbOwKXvdtrZb1PBZ-_6aH0TYPRQw6nTIQzamXO2jbaE684XzuwCtA183_jkflrCN6vh2jRDhh7ccGHixlfn4KTWLpiLvz0Gr3ezl-lDNl_dP05v51mbeseszg1NL-W6mPCCCslQVWM0YYQXJcllKRktOGeVkYJLVnLDUCmkrDifMG0qSsfg6pDbdn7fmxDV1vddqhMUxSgXdOCUVPSgCm1nm0_T_aswUgNbdWCrElv1y1YR-gOWrmwG</recordid><startdate>2024</startdate><enddate>2024</enddate><creator>Alves, Claudianor O.</creator><creator>Van Thin, Nguyen</creator><general>Springer Netherlands</general><general>Springer Nature B.V</general><scope/></search><sort><creationdate>2024</creationdate><title>On Existence of Multiple Normalized Solutions to a Class of Elliptic Problems in Whole RN Via Penalization Method</title><author>Alves, Claudianor O. ; Van Thin, Nguyen</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-p157t-f8e31188ab46b37950df104526bc289c953b665de97695c6e50c799d6645aed33</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Continuity (mathematics)</topic><topic>Functional Analysis</topic><topic>Geometry</topic><topic>Lagrange multiplier</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Partial differential equations</topic><topic>Potential Theory</topic><topic>Probability Theory and Stochastic Processes</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Alves, Claudianor O.</creatorcontrib><creatorcontrib>Van Thin, Nguyen</creatorcontrib><jtitle>Potential analysis</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Alves, Claudianor O.</au><au>Van Thin, Nguyen</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On Existence of Multiple Normalized Solutions to a Class of Elliptic Problems in Whole RN Via Penalization Method</atitle><jtitle>Potential analysis</jtitle><stitle>Potential Anal</stitle><date>2024</date><risdate>2024</risdate><volume>61</volume><issue>3</issue><spage>463</spage><epage>483</epage><pages>463-483</pages><issn>0926-2601</issn><eissn>1572-929X</eissn><abstract>In this paper we study the existence of multiple normalized solutions to the following class of elliptic problems
-
ϵ
2
Δ
u
+
V
(
x
)
u
=
λ
u
+
f
(
u
)
,
in
R
N
,
∫
R
N
|
u
|
2
d
x
=
a
2
ϵ
N
,
where
a
,
ϵ
>
0
,
λ
∈
R
is an unknown parameter that appears as a Lagrange multiplier,
V
:
R
N
→
[
0
,
∞
)
is a continuous function, and
f
is a continuous function with
L
2
-subcritical growth. It is proved that the number of normalized solutions is related to the topological richness of the set where the potential
V
attains its minimum value. In the proof of our main result, we apply minimization techniques, Lusternik-Schnirelmann category and the penalization method due to del Pino and Felmer (Calc. Var. Partial Differential Equations
4
, 121–137
1996
).</abstract><cop>Dordrecht</cop><pub>Springer Netherlands</pub><doi>10.1007/s11118-023-10116-2</doi><tpages>21</tpages></addata></record> |
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| ispartof | Potential analysis, 2024, Vol.61 (3), p.463-483 |
| issn | 0926-2601 1572-929X |
| language | eng |
| recordid | cdi_proquest_journals_3108730116 |
| source | SpringerLink Journals - AutoHoldings |
| subjects | Continuity (mathematics) Functional Analysis Geometry Lagrange multiplier Mathematics Mathematics and Statistics Partial differential equations Potential Theory Probability Theory and Stochastic Processes |
| title | On Existence of Multiple Normalized Solutions to a Class of Elliptic Problems in Whole RN Via Penalization Method |
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