Existence and multiplicity of solutions for a class of Hamiltonian systems
UDC 517.9 We investigate a class of Hamiltonian systems \begin{gather*} {-q''}(t)+(L(t)-\xi)q(t)= a(t)|q(t)|^{p-2}q(t)+\eta f(t),\\ q\in H^1(\mathbb{R},\mathbb{R}^N),\end{gather*} where $(t,q)\in \mathbb{R}\times \mathbb{R}^N$, $p>2,$ $a\in C(\mathbb{R},(0,+\infty)),$ $f...
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| Date: | 2024 |
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| Format: | Article |
| Language: | English |
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Institute of Mathematics, NAS of Ukraine
2024
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/7497 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512676040409088 |
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| author | Khachnaoui, Khaled Khachnaoui, Khaled |
| author_facet | Khachnaoui, Khaled Khachnaoui, Khaled |
| author_sort | Khachnaoui, Khaled |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2024-08-10T06:39:09Z |
| description | UDC 517.9
We investigate a class of Hamiltonian systems \begin{gather*} {-q''}(t)+(L(t)-\xi)q(t)= a(t)|q(t)|^{p-2}q(t)+\eta f(t),\\ q\in H^1(\mathbb{R},\mathbb{R}^N),\end{gather*} where $(t,q)\in \mathbb{R}\times \mathbb{R}^N$, $p>2,$ $a\in C(\mathbb{R},(0,+\infty)),$ $f\in C(\mathbb{R},\mathbb{R}^N),$ $\xi, \eta$ are real parameters, and $L\in C(\mathbb{R},\mathbb{R}^{N^2})$ is a positive definite symmetric matrix for all $t\in \mathbb{R}.$ The main technical approach is based on the Nehari manifold method combined with variational and topological methods. The obtained results extend and complement the results available in the literature. |
| doi_str_mv | 10.3842/umzh.v76i5.7497 |
| first_indexed | 2026-03-24T03:32:34Z |
| format | Article |
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| id | umjimathkievua-article-7497 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:32:34Z |
| publishDate | 2024 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
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| spelling | umjimathkievua-article-74972024-08-10T06:39:09Z Existence and multiplicity of solutions for a class of Hamiltonian systems Existence and multiplicity of solutions for a class of Hamiltonian systems Khachnaoui, Khaled Khachnaoui, Khaled Nehari manifold methods; Hamiltonian systems; Variational methods; Critical point UDC 517.9 We investigate a class of Hamiltonian systems \begin{gather*} {-q''}(t)+(L(t)-\xi)q(t)= a(t)|q(t)|^{p-2}q(t)+\eta f(t),\\ q\in H^1(\mathbb{R},\mathbb{R}^N),\end{gather*} where $(t,q)\in \mathbb{R}\times \mathbb{R}^N$, $p>2,$ $a\in C(\mathbb{R},(0,+\infty)),$ $f\in C(\mathbb{R},\mathbb{R}^N),$ $\xi, \eta$ are real parameters, and $L\in C(\mathbb{R},\mathbb{R}^{N^2})$ is a positive definite symmetric matrix for all $t\in \mathbb{R}.$ The main technical approach is based on the Nehari manifold method combined with variational and topological methods. The obtained results extend and complement the results available in the literature. УДК 517.9 Існування та множинність розв’язків для одного класу гамільтонових систем Mи досліджуємо клас гамільтонoвих систем \begin{gather} {-q''}(t)+(L(t)-\xi)q(t)= a(t)|q(t)|^{p-2}q(t)+\eta f(t), \\ q\in H^1(\mathbb{R},\mathbb{R}^N),\end{gather} де $(t,q)\in \mathbb{R}\times \mathbb{R}^N$, $p>2,$ $a\inC(\mathbb{R},(0,+\infty)),$ $f\in C(\mathbb{R},\mathbb{R}^N),$ $\xi$ та $\eta$ --- дійсні параметри і$L\in C(\mathbb{R},\mathbb{R}^{N^2})$ — додатно визначена симетрична матриця для всіх $t\in \mathbb{R}.$ Основний технічний підхід базується на методі многовиду Нехарі спільно з варіаційними і топологічними методами. Отримані результати розширюють і доповнюють результати, доступні в літературі. Institute of Mathematics, NAS of Ukraine 2024-07-03 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/7497 10.3842/umzh.v76i5.7497 Ukrains’kyi Matematychnyi Zhurnal; Vol. 76 No. 6 (2024); 915–930 Український математичний журнал; Том 76 № 6 (2024); 915–930 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7497/10037 Copyright (c) 2024 Khaled Khachnaoui |
| spellingShingle | Khachnaoui, Khaled Khachnaoui, Khaled Existence and multiplicity of solutions for a class of Hamiltonian systems |
| title | Existence and multiplicity of solutions for a class of Hamiltonian systems |
| title_alt | Existence and multiplicity of solutions for a class of Hamiltonian systems |
| title_full | Existence and multiplicity of solutions for a class of Hamiltonian systems |
| title_fullStr | Existence and multiplicity of solutions for a class of Hamiltonian systems |
| title_full_unstemmed | Existence and multiplicity of solutions for a class of Hamiltonian systems |
| title_short | Existence and multiplicity of solutions for a class of Hamiltonian systems |
| title_sort | existence and multiplicity of solutions for a class of hamiltonian systems |
| topic_facet | Nehari manifold methods Hamiltonian systems Variational methods Critical point |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7497 |
| work_keys_str_mv | AT khachnaouikhaled existenceandmultiplicityofsolutionsforaclassofhamiltoniansystems AT khachnaouikhaled existenceandmultiplicityofsolutionsforaclassofhamiltoniansystems |