Homoclinic solutions for subquadratic Hamiltonian systems with competition potentials
In this paper, we consider of the following second-order Hamiltonian system \begin{equation*} \ddot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0,\qquad \forall t \in \mathbb{R}, \end{equation*} where $W(t,x)$ is subquadratic at infinity. With a competition condition, we establish the existence of homoclinic s...
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University of Szeged
2024-01-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
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| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=10529 |
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| author | Ruiqi Liu Dong-Lun Wu Jia-Feng Liao |
| author_facet | Ruiqi Liu Dong-Lun Wu Jia-Feng Liao |
| author_sort | Ruiqi Liu |
| collection | DOAJ |
| description | In this paper, we consider of the following second-order Hamiltonian system
\begin{equation*}
\ddot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0,\qquad \forall t \in \mathbb{R},
\end{equation*}
where $W(t,x)$ is subquadratic at infinity. With a competition condition, we establish the existence of homoclinic solutions by using the variational methods. In our theorem, the smallest eigenvalue function $l(t)$ of $L(t)$ is not necessarily coercive or bounded from above and $W(t,x)$ is not necessarily integrable on $\mathbb{R}$ with respect to $t$. Our theorem generalizes many known results in the references. |
| format | Article |
| id | doaj-art-db5b7b8ec31f48f3ad7479c7aa799fd9 |
| institution | Kabale University |
| issn | 1417-3875 |
| language | English |
| publishDate | 2024-01-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj-art-db5b7b8ec31f48f3ad7479c7aa799fd92025-01-15T21:24:58ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752024-01-012024511310.14232/ejqtde.2024.1.510529Homoclinic solutions for subquadratic Hamiltonian systems with competition potentialsRuiqi Liu0Dong-Lun WuJia-Feng Liao1Meishan High School, Meishan, P.R. ChinaCivil Aviation Flight University of China, Guanghan, P.R. ChinaIn this paper, we consider of the following second-order Hamiltonian system \begin{equation*} \ddot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0,\qquad \forall t \in \mathbb{R}, \end{equation*} where $W(t,x)$ is subquadratic at infinity. With a competition condition, we establish the existence of homoclinic solutions by using the variational methods. In our theorem, the smallest eigenvalue function $l(t)$ of $L(t)$ is not necessarily coercive or bounded from above and $W(t,x)$ is not necessarily integrable on $\mathbb{R}$ with respect to $t$. Our theorem generalizes many known results in the references.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=10529hamiltonian systemshomoclinic solutionssubquadratic potentialscompetition conditionvariational methods |
| spellingShingle | Ruiqi Liu Dong-Lun Wu Jia-Feng Liao Homoclinic solutions for subquadratic Hamiltonian systems with competition potentials Electronic Journal of Qualitative Theory of Differential Equations hamiltonian systems homoclinic solutions subquadratic potentials competition condition variational methods |
| title | Homoclinic solutions for subquadratic Hamiltonian systems with competition potentials |
| title_full | Homoclinic solutions for subquadratic Hamiltonian systems with competition potentials |
| title_fullStr | Homoclinic solutions for subquadratic Hamiltonian systems with competition potentials |
| title_full_unstemmed | Homoclinic solutions for subquadratic Hamiltonian systems with competition potentials |
| title_short | Homoclinic solutions for subquadratic Hamiltonian systems with competition potentials |
| title_sort | homoclinic solutions for subquadratic hamiltonian systems with competition potentials |
| topic | hamiltonian systems homoclinic solutions subquadratic potentials competition condition variational methods |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=10529 |
| work_keys_str_mv | AT ruiqiliu homoclinicsolutionsforsubquadratichamiltoniansystemswithcompetitionpotentials AT donglunwu homoclinicsolutionsforsubquadratichamiltoniansystemswithcompetitionpotentials AT jiafengliao homoclinicsolutionsforsubquadratichamiltoniansystemswithcompetitionpotentials |