Existence and multiplicity of homoclinic solutions for a second-order Hamiltonian system
In this paper, we find new conditions to ensure the existence of one nontrivial homoclinic solution and also infinitely many homoclinic solutions for the second order Hamiltonian system $$ \ddot{u}-a(t)|u|^{p-2}u+\nabla W(t,u)=0,\qquad t\in \mathbb{R}, $$ where $p>2$, $a\in C(\mathbb{R}, \mathbb...
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| Format: | Article |
| Language: | English |
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University of Szeged
2019-02-01
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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=6918 |
| _version_ | 1797830542900592640 |
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| author | Yiwei Ye |
| author_facet | Yiwei Ye |
| author_sort | Yiwei Ye |
| collection | DOAJ |
| description | In this paper, we find new conditions to ensure the existence of one nontrivial homoclinic solution and also infinitely many homoclinic solutions for the second order Hamiltonian system
$$
\ddot{u}-a(t)|u|^{p-2}u+\nabla W(t,u)=0,\qquad t\in
\mathbb{R},
$$
where $p>2$, $a\in C(\mathbb{R}, \mathbb{R})$ with $\inf_{t\in \mathbb{R}}a(t)>0$ and $\int_\mathbb{R}\big(\frac{1}{a(t)}\big)^{2/(p-2)} dt<+\infty$, and $W(t,x)$ is, as $|x|\rightarrow \infty$, superquadratic or subquadratic with certain hypotheses different from those used in previous related studies. Our approach is variational and we use the Cerami condition instead of the Palais–Smale one for deformation arguments. |
| first_indexed | 2024-04-09T13:37:46Z |
| format | Article |
| id | doaj.art-e59971b0db5149528f3c028794d75775 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:37:46Z |
| publishDate | 2019-02-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-e59971b0db5149528f3c028794d757752023-05-09T07:53:09ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752019-02-0120191112610.14232/ejqtde.2019.1.116918Existence and multiplicity of homoclinic solutions for a second-order Hamiltonian systemYiwei Ye0Department of Mathematics, Chongqing Normal University, Chongqing, PR ChinaIn this paper, we find new conditions to ensure the existence of one nontrivial homoclinic solution and also infinitely many homoclinic solutions for the second order Hamiltonian system $$ \ddot{u}-a(t)|u|^{p-2}u+\nabla W(t,u)=0,\qquad t\in \mathbb{R}, $$ where $p>2$, $a\in C(\mathbb{R}, \mathbb{R})$ with $\inf_{t\in \mathbb{R}}a(t)>0$ and $\int_\mathbb{R}\big(\frac{1}{a(t)}\big)^{2/(p-2)} dt<+\infty$, and $W(t,x)$ is, as $|x|\rightarrow \infty$, superquadratic or subquadratic with certain hypotheses different from those used in previous related studies. Our approach is variational and we use the Cerami condition instead of the Palais–Smale one for deformation arguments.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=6918homoclinic solutionshamiltonian systemsvariational methodsweighted $l^p$ space |
| spellingShingle | Yiwei Ye Existence and multiplicity of homoclinic solutions for a second-order Hamiltonian system Electronic Journal of Qualitative Theory of Differential Equations homoclinic solutions hamiltonian systems variational methods weighted $l^p$ space |
| title | Existence and multiplicity of homoclinic solutions for a second-order Hamiltonian system |
| title_full | Existence and multiplicity of homoclinic solutions for a second-order Hamiltonian system |
| title_fullStr | Existence and multiplicity of homoclinic solutions for a second-order Hamiltonian system |
| title_full_unstemmed | Existence and multiplicity of homoclinic solutions for a second-order Hamiltonian system |
| title_short | Existence and multiplicity of homoclinic solutions for a second-order Hamiltonian system |
| title_sort | existence and multiplicity of homoclinic solutions for a second order hamiltonian system |
| topic | homoclinic solutions hamiltonian systems variational methods weighted $l^p$ space |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=6918 |
| work_keys_str_mv | AT yiweiye existenceandmultiplicityofhomoclinicsolutionsforasecondorderhamiltoniansystem |