Ground states solutions for some non-autonomous Schrödinger-Bopp-Podolsky system
In this paper we study the existence of ground states solutions for non-autonomous Schrödinger–Bopp–Podolsky system \begin{equation*} \begin{cases} -\Delta u + u +\lambda K(x)\phi u = b(x)|u|^{p-2}u & \text{in} \ \mathbb{R}^{3}, \\ -\Delta \phi + a^2\Delta^2\phi = 4\pi K(x) u^{2}...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2022-10-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=10041 |
| _version_ | 1797830454873686016 |
|---|---|
| author | Chun-Rong Jia Lin Li Shang-Jie Chen |
| author_facet | Chun-Rong Jia Lin Li Shang-Jie Chen |
| author_sort | Chun-Rong Jia |
| collection | DOAJ |
| description | In this paper we study the existence of ground states solutions for non-autonomous Schrödinger–Bopp–Podolsky system
\begin{equation*}
\begin{cases}
-\Delta u + u +\lambda K(x)\phi u = b(x)|u|^{p-2}u & \text{in} \ \mathbb{R}^{3}, \\
-\Delta \phi + a^2\Delta^2\phi = 4\pi K(x) u^{2} & \text{in}\ \mathbb{R}^{3},
\end{cases}
\end{equation*}
where $\lambda>0, 2<p\leq 4$ and both $K(x)$ and $b(x)$ are nonnegative functions in $\mathbb{R}^{3}$. Assuming that ${\lim_{\vert{x}\vert \to +\infty}}K(x)=K_\infty >0$ and ${\lim_{\vert{x}\vert \to +\infty}}b(x)=b_\infty >0$ and satisfying suitable assumptions, but not requiring any symmetry property on them. We show that the existence of a positive solution depends on the parameters $\lambda$ and $p$. We also establish the existence of ground state solutions for the case $3.18\approx\frac{1+\sqrt{73}}{3}<p\le{4}$. |
| first_indexed | 2024-04-09T13:36:20Z |
| format | Article |
| id | doaj.art-a205eb15ab3e4bb1bce3c08815d7d1dc |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:36:20Z |
| publishDate | 2022-10-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-a205eb15ab3e4bb1bce3c08815d7d1dc2023-05-09T07:53:12ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752022-10-0120225112910.14232/ejqtde.2022.1.5110041Ground states solutions for some non-autonomous Schrödinger-Bopp-Podolsky systemChun-Rong Jia0Lin Li1Shang-Jie Chen2School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, ChinaSchool of Mathematics and Statistics & Chongqing Key Laboratory of Economic and Social Application Statistics,, Chongqing Technology and Business University, Chongqing, ChinaSchool of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, ChinaIn this paper we study the existence of ground states solutions for non-autonomous Schrödinger–Bopp–Podolsky system \begin{equation*} \begin{cases} -\Delta u + u +\lambda K(x)\phi u = b(x)|u|^{p-2}u & \text{in} \ \mathbb{R}^{3}, \\ -\Delta \phi + a^2\Delta^2\phi = 4\pi K(x) u^{2} & \text{in}\ \mathbb{R}^{3}, \end{cases} \end{equation*} where $\lambda>0, 2<p\leq 4$ and both $K(x)$ and $b(x)$ are nonnegative functions in $\mathbb{R}^{3}$. Assuming that ${\lim_{\vert{x}\vert \to +\infty}}K(x)=K_\infty >0$ and ${\lim_{\vert{x}\vert \to +\infty}}b(x)=b_\infty >0$ and satisfying suitable assumptions, but not requiring any symmetry property on them. We show that the existence of a positive solution depends on the parameters $\lambda$ and $p$. We also establish the existence of ground state solutions for the case $3.18\approx\frac{1+\sqrt{73}}{3}<p\le{4}$.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=10041non-autonomous schrödinger–bopp–podolsky systemvariational methodspohožaev identitynehari manifold |
| spellingShingle | Chun-Rong Jia Lin Li Shang-Jie Chen Ground states solutions for some non-autonomous Schrödinger-Bopp-Podolsky system Electronic Journal of Qualitative Theory of Differential Equations non-autonomous schrödinger–bopp–podolsky system variational methods pohožaev identity nehari manifold |
| title | Ground states solutions for some non-autonomous Schrödinger-Bopp-Podolsky system |
| title_full | Ground states solutions for some non-autonomous Schrödinger-Bopp-Podolsky system |
| title_fullStr | Ground states solutions for some non-autonomous Schrödinger-Bopp-Podolsky system |
| title_full_unstemmed | Ground states solutions for some non-autonomous Schrödinger-Bopp-Podolsky system |
| title_short | Ground states solutions for some non-autonomous Schrödinger-Bopp-Podolsky system |
| title_sort | ground states solutions for some non autonomous schrodinger bopp podolsky system |
| topic | non-autonomous schrödinger–bopp–podolsky system variational methods pohožaev identity nehari manifold |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=10041 |
| work_keys_str_mv | AT chunrongjia groundstatessolutionsforsomenonautonomousschrodingerbopppodolskysystem AT linli groundstatessolutionsforsomenonautonomousschrodingerbopppodolskysystem AT shangjiechen groundstatessolutionsforsomenonautonomousschrodingerbopppodolskysystem |