Brake orbit solutions for semilinear elliptic systems with asymmetric double well potential
We consider a class of semilinear elliptic system of the form: 0.1 - Δ u ( x , y ) + ∇ W ( u ( x , y ) ) = 0 , ( x , y ) ∈ R 2 , where W : R 2 → R is a double well potential with minima a ± ∈ R 2 . We show, via variational methods, that if the set of minimal heteroclinic solutions to the one-dimensi...
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| Veröffentlicht in: | Journal of fixed point theory and applications 2017-03, Vol.19 (1), p.691-717 |
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| creator | Alessio, Francesca Montecchiari, Piero |
| description | We consider a class of semilinear elliptic system of the form:
0.1
-
Δ
u
(
x
,
y
)
+
∇
W
(
u
(
x
,
y
)
)
=
0
,
(
x
,
y
)
∈
R
2
,
where
W
:
R
2
→
R
is a double well potential with minima
a
±
∈
R
2
. We show, via variational methods, that if the set of minimal heteroclinic solutions to the one-dimensional system
-
q
¨
(
x
)
+
∇
W
(
q
(
x
)
)
=
0
,
x
∈
R
, up to translations, is finite and constituted by not degenerate functions, then Eq. (
0.1
) has infinitely many solutions
u
∈
C
2
(
R
2
)
2
, parametrized by an energy value, which are periodic in the variable
y
and satisfy
lim
x
→
±
∞
u
(
x
,
y
)
=
a
±
for any
y
∈
R
. |
| doi_str_mv | 10.1007/s11784-016-0370-4 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_1880744815</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>1880744815</sourcerecordid><originalsourceid>FETCH-LOGICAL-c316t-ffc1d3cab6e3070bdb655ac8fc313444f831272fb98d5b9a5402ee566aec58543</originalsourceid><addsrcrecordid>eNp1kE1LxDAQhoMouK7-AG8Bz9Wk-dyjLn7Bghc9eQhpO9GsbVOTFNl_b5eKePE0w8wz78CD0Dkll5QQdZUoVZoXhMqCMEUKfoAWVEpaKMXl4W_P9DE6SWlLiCQlVQv0ehPtB-AQK59xCu2YfegTdiHiBJ1vfQ82YmhbP2Rf47RLGbqEv3x-xzbtug5ynOZNGKsW8NcE4iFk6LO37Sk6crZNcPZTl-jl7vZ5_VBsnu4f19ebomZU5sK5mjastpUERhSpmkoKYWvtpjXjnDvNaKlKV610I6qVFZyUAEJKC7XQgrMluphzhxg-R0jZbMMY--mloVoTxbmmYqLoTNUxpBTBmSH6zsadocTsHZrZoZkcmr1Ds08u55s0sf0bxD_J_x59A2T_dfM</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>1880744815</pqid></control><display><type>article</type><title>Brake orbit solutions for semilinear elliptic systems with asymmetric double well potential</title><source>SpringerLink Journals - AutoHoldings</source><creator>Alessio, Francesca ; Montecchiari, Piero</creator><creatorcontrib>Alessio, Francesca ; Montecchiari, Piero</creatorcontrib><description>We consider a class of semilinear elliptic system of the form:
0.1
-
Δ
u
(
x
,
y
)
+
∇
W
(
u
(
x
,
y
)
)
=
0
,
(
x
,
y
)
∈
R
2
,
where
W
:
R
2
→
R
is a double well potential with minima
a
±
∈
R
2
. We show, via variational methods, that if the set of minimal heteroclinic solutions to the one-dimensional system
-
q
¨
(
x
)
+
∇
W
(
q
(
x
)
)
=
0
,
x
∈
R
, up to translations, is finite and constituted by not degenerate functions, then Eq. (
0.1
) has infinitely many solutions
u
∈
C
2
(
R
2
)
2
, parametrized by an energy value, which are periodic in the variable
y
and satisfy
lim
x
→
±
∞
u
(
x
,
y
)
=
a
±
for any
y
∈
R
.</description><identifier>ISSN: 1661-7738</identifier><identifier>EISSN: 1661-7746</identifier><identifier>DOI: 10.1007/s11784-016-0370-4</identifier><language>eng</language><publisher>Cham: Springer International Publishing</publisher><subject>Analysis ; Elliptical orbits ; Energy value ; Mathematical Methods in Physics ; Mathematics ; Mathematics and Statistics ; Translations ; Variational methods</subject><ispartof>Journal of fixed point theory and applications, 2017-03, Vol.19 (1), p.691-717</ispartof><rights>Springer International Publishing 2016</rights><rights>Copyright Springer Science & Business Media 2017</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c316t-ffc1d3cab6e3070bdb655ac8fc313444f831272fb98d5b9a5402ee566aec58543</citedby><cites>FETCH-LOGICAL-c316t-ffc1d3cab6e3070bdb655ac8fc313444f831272fb98d5b9a5402ee566aec58543</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s11784-016-0370-4$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s11784-016-0370-4$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Alessio, Francesca</creatorcontrib><creatorcontrib>Montecchiari, Piero</creatorcontrib><title>Brake orbit solutions for semilinear elliptic systems with asymmetric double well potential</title><title>Journal of fixed point theory and applications</title><addtitle>J. Fixed Point Theory Appl</addtitle><description>We consider a class of semilinear elliptic system of the form:
0.1
-
Δ
u
(
x
,
y
)
+
∇
W
(
u
(
x
,
y
)
)
=
0
,
(
x
,
y
)
∈
R
2
,
where
W
:
R
2
→
R
is a double well potential with minima
a
±
∈
R
2
. We show, via variational methods, that if the set of minimal heteroclinic solutions to the one-dimensional system
-
q
¨
(
x
)
+
∇
W
(
q
(
x
)
)
=
0
,
x
∈
R
, up to translations, is finite and constituted by not degenerate functions, then Eq. (
0.1
) has infinitely many solutions
u
∈
C
2
(
R
2
)
2
, parametrized by an energy value, which are periodic in the variable
y
and satisfy
lim
x
→
±
∞
u
(
x
,
y
)
=
a
±
for any
y
∈
R
.</description><subject>Analysis</subject><subject>Elliptical orbits</subject><subject>Energy value</subject><subject>Mathematical Methods in Physics</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Translations</subject><subject>Variational methods</subject><issn>1661-7738</issn><issn>1661-7746</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2017</creationdate><recordtype>article</recordtype><recordid>eNp1kE1LxDAQhoMouK7-AG8Bz9Wk-dyjLn7Bghc9eQhpO9GsbVOTFNl_b5eKePE0w8wz78CD0Dkll5QQdZUoVZoXhMqCMEUKfoAWVEpaKMXl4W_P9DE6SWlLiCQlVQv0ehPtB-AQK59xCu2YfegTdiHiBJ1vfQ82YmhbP2Rf47RLGbqEv3x-xzbtug5ynOZNGKsW8NcE4iFk6LO37Sk6crZNcPZTl-jl7vZ5_VBsnu4f19ebomZU5sK5mjastpUERhSpmkoKYWvtpjXjnDvNaKlKV610I6qVFZyUAEJKC7XQgrMluphzhxg-R0jZbMMY--mloVoTxbmmYqLoTNUxpBTBmSH6zsadocTsHZrZoZkcmr1Ds08u55s0sf0bxD_J_x59A2T_dfM</recordid><startdate>20170301</startdate><enddate>20170301</enddate><creator>Alessio, Francesca</creator><creator>Montecchiari, Piero</creator><general>Springer International Publishing</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20170301</creationdate><title>Brake orbit solutions for semilinear elliptic systems with asymmetric double well potential</title><author>Alessio, Francesca ; Montecchiari, Piero</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c316t-ffc1d3cab6e3070bdb655ac8fc313444f831272fb98d5b9a5402ee566aec58543</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2017</creationdate><topic>Analysis</topic><topic>Elliptical orbits</topic><topic>Energy value</topic><topic>Mathematical Methods in Physics</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Translations</topic><topic>Variational methods</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Alessio, Francesca</creatorcontrib><creatorcontrib>Montecchiari, Piero</creatorcontrib><collection>CrossRef</collection><jtitle>Journal of fixed point theory and applications</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Alessio, Francesca</au><au>Montecchiari, Piero</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Brake orbit solutions for semilinear elliptic systems with asymmetric double well potential</atitle><jtitle>Journal of fixed point theory and applications</jtitle><stitle>J. Fixed Point Theory Appl</stitle><date>2017-03-01</date><risdate>2017</risdate><volume>19</volume><issue>1</issue><spage>691</spage><epage>717</epage><pages>691-717</pages><issn>1661-7738</issn><eissn>1661-7746</eissn><abstract>We consider a class of semilinear elliptic system of the form:
0.1
-
Δ
u
(
x
,
y
)
+
∇
W
(
u
(
x
,
y
)
)
=
0
,
(
x
,
y
)
∈
R
2
,
where
W
:
R
2
→
R
is a double well potential with minima
a
±
∈
R
2
. We show, via variational methods, that if the set of minimal heteroclinic solutions to the one-dimensional system
-
q
¨
(
x
)
+
∇
W
(
q
(
x
)
)
=
0
,
x
∈
R
, up to translations, is finite and constituted by not degenerate functions, then Eq. (
0.1
) has infinitely many solutions
u
∈
C
2
(
R
2
)
2
, parametrized by an energy value, which are periodic in the variable
y
and satisfy
lim
x
→
±
∞
u
(
x
,
y
)
=
a
±
for any
y
∈
R
.</abstract><cop>Cham</cop><pub>Springer International Publishing</pub><doi>10.1007/s11784-016-0370-4</doi><tpages>27</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 1661-7738 |
| ispartof | Journal of fixed point theory and applications, 2017-03, Vol.19 (1), p.691-717 |
| issn | 1661-7738 1661-7746 |
| language | eng |
| recordid | cdi_proquest_journals_1880744815 |
| source | SpringerLink Journals - AutoHoldings |
| subjects | Analysis Elliptical orbits Energy value Mathematical Methods in Physics Mathematics Mathematics and Statistics Translations Variational methods |
| title | Brake orbit solutions for semilinear elliptic systems with asymmetric double well potential |
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