Rabinowitz Alternative for Non-cooperative Elliptic Systems on Geodesic Balls
The purpose of this paper is to study properties of continua (closed connected sets) of nontrivial solutions of non-cooperative elliptic systems considered on geodesic balls in Sn{S^{n}}. In particular, we show that if the geodesic ball is a hemisphere, then all these continua are unbounded. It is a...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2018-11-01
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| Series: | Advanced Nonlinear Studies |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/ans-2018-0012 |
| _version_ | 1811185906595397632 |
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| author | Rybicki Sławomir Shioji Naoki Stefaniak Piotr |
| author_facet | Rybicki Sławomir Shioji Naoki Stefaniak Piotr |
| author_sort | Rybicki Sławomir |
| collection | DOAJ |
| description | The purpose of this paper is to study properties of continua (closed connected sets) of nontrivial solutions of non-cooperative elliptic systems considered on geodesic balls in Sn{S^{n}}. In particular, we show that if the geodesic ball is a hemisphere, then all these continua are unbounded. It is also shown that the phenomenon of global symmetry-breaking bifurcation of such solutions occurs. Since the problem is variational and SO(n){\operatorname{SO}(n)}-symmetric, we apply the techniques of equivariant bifurcation theory to prove the main results of this article. As the topological tool, we use the degree theory for SO(n){\operatorname{SO}(n)}-invariant strongly indefinite functionals defined in [A. Gołȩbiewska and S. A. Rybicki, Global bifurcations of critical orbits of G-invariant strongly indefinite functionals, Nonlinear Anal. 74 2011, 5, 1823–1834]. |
| first_indexed | 2024-04-11T13:37:19Z |
| format | Article |
| id | doaj.art-259fa2f27aa94de0889361868a136807 |
| institution | Directory Open Access Journal |
| issn | 1536-1365 2169-0375 |
| language | English |
| last_indexed | 2024-04-11T13:37:19Z |
| publishDate | 2018-11-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Advanced Nonlinear Studies |
| spelling | doaj.art-259fa2f27aa94de0889361868a1368072022-12-22T04:21:24ZengDe GruyterAdvanced Nonlinear Studies1536-13652169-03752018-11-0118484586210.1515/ans-2018-0012Rabinowitz Alternative for Non-cooperative Elliptic Systems on Geodesic BallsRybicki Sławomir0Shioji Naoki1Stefaniak Piotr2Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, 87-100Toruń, ul. Chopina 12/18, PolandDepartment of Mathematics, Faculty of Engineering, Yokohama National University, Tokiwadai, Hodogaya-ku, Yokohama240-8501, JapanSchool of Mathematics, West Pomeranian University of Technology, 70-310Szczecin, al. Piastów 48/49, PolandThe purpose of this paper is to study properties of continua (closed connected sets) of nontrivial solutions of non-cooperative elliptic systems considered on geodesic balls in Sn{S^{n}}. In particular, we show that if the geodesic ball is a hemisphere, then all these continua are unbounded. It is also shown that the phenomenon of global symmetry-breaking bifurcation of such solutions occurs. Since the problem is variational and SO(n){\operatorname{SO}(n)}-symmetric, we apply the techniques of equivariant bifurcation theory to prove the main results of this article. As the topological tool, we use the degree theory for SO(n){\operatorname{SO}(n)}-invariant strongly indefinite functionals defined in [A. Gołȩbiewska and S. A. Rybicki, Global bifurcations of critical orbits of G-invariant strongly indefinite functionals, Nonlinear Anal. 74 2011, 5, 1823–1834].https://doi.org/10.1515/ans-2018-0012symmetric rabinowitz alternativeglobal symmetry-breaking bifurcationsnon-cooperative elliptic systemsequivariant degree35b32 35j20 |
| spellingShingle | Rybicki Sławomir Shioji Naoki Stefaniak Piotr Rabinowitz Alternative for Non-cooperative Elliptic Systems on Geodesic Balls Advanced Nonlinear Studies symmetric rabinowitz alternative global symmetry-breaking bifurcations non-cooperative elliptic systems equivariant degree 35b32 35j20 |
| title | Rabinowitz Alternative for Non-cooperative Elliptic Systems on Geodesic Balls |
| title_full | Rabinowitz Alternative for Non-cooperative Elliptic Systems on Geodesic Balls |
| title_fullStr | Rabinowitz Alternative for Non-cooperative Elliptic Systems on Geodesic Balls |
| title_full_unstemmed | Rabinowitz Alternative for Non-cooperative Elliptic Systems on Geodesic Balls |
| title_short | Rabinowitz Alternative for Non-cooperative Elliptic Systems on Geodesic Balls |
| title_sort | rabinowitz alternative for non cooperative elliptic systems on geodesic balls |
| topic | symmetric rabinowitz alternative global symmetry-breaking bifurcations non-cooperative elliptic systems equivariant degree 35b32 35j20 |
| url | https://doi.org/10.1515/ans-2018-0012 |
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