Least energy sign-changing solutions for Kirchhoff-Schrödinger-Poisson system on bounded domains
We investigate the following nonlinear system $ \begin{cases} -(a+ b\int _{\Omega}|\nabla u|^{2}dx)\Delta u+\phi u = \lambda u+\mu|u|^{2}u, \; \ x\in\Omega, \\ -\Delta\phi = u^{2}, \; \ x\in\Omega, \\ u = \phi = 0, \; \ x\in \partial\Omega, \end{cases} $ with $ a, b > 0 $, $ \lambd...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-03-01
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| Series: | Electronic Research Archive |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/era.2023149?viewType=HTML |
| _version_ | 1797831783448838144 |
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| author | Xia Su Wen Guan Xia Li |
| author_facet | Xia Su Wen Guan Xia Li |
| author_sort | Xia Su |
| collection | DOAJ |
| description | We investigate the following nonlinear system
$ \begin{cases} -(a+ b\int _{\Omega}|\nabla u|^{2}dx)\Delta u+\phi u = \lambda u+\mu|u|^{2}u, \; \ x\in\Omega, \\ -\Delta\phi = u^{2}, \; \ x\in\Omega, \\ u = \phi = 0, \; \ x\in \partial\Omega, \end{cases} $
with $ a, b > 0 $, $ \lambda, \mu\in\mathbb{R} $, and $ \Omega\subset \mathbb{R}^{3} $ is bounded with smooth boundary. Let $ \lambda_{1} > 0 $ be the first eigenvalue of $ (-\Delta u, H^{1}_{0}(\Omega)) $. We get that for certain $ \widetilde{\mu} > 0 $ there exists at least one least energy sign-changing solution for the above system if $ \lambda < a\lambda_{1} $ and $ \mu > \widetilde{\mu} $. In addition, we remark that the nonlinearity $ \lambda u+\mu|u|^{2}u $ does not satisfy the growth conditions. |
| first_indexed | 2024-04-09T13:57:14Z |
| format | Article |
| id | doaj.art-e1ed3405280e40b8b59387a5b9337cc2 |
| institution | Directory Open Access Journal |
| issn | 2688-1594 |
| language | English |
| last_indexed | 2024-04-09T13:57:14Z |
| publishDate | 2023-03-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Electronic Research Archive |
| spelling | doaj.art-e1ed3405280e40b8b59387a5b9337cc22023-05-08T01:37:15ZengAIMS PressElectronic Research Archive2688-15942023-03-013152959297310.3934/era.2023149Least energy sign-changing solutions for Kirchhoff-Schrödinger-Poisson system on bounded domainsXia Su0Wen Guan1Xia Li21. Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Huai'an 223003, Jiangsu, China2. Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050, Gansu, China2. Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050, Gansu, ChinaWe investigate the following nonlinear system $ \begin{cases} -(a+ b\int _{\Omega}|\nabla u|^{2}dx)\Delta u+\phi u = \lambda u+\mu|u|^{2}u, \; \ x\in\Omega, \\ -\Delta\phi = u^{2}, \; \ x\in\Omega, \\ u = \phi = 0, \; \ x\in \partial\Omega, \end{cases} $ with $ a, b > 0 $, $ \lambda, \mu\in\mathbb{R} $, and $ \Omega\subset \mathbb{R}^{3} $ is bounded with smooth boundary. Let $ \lambda_{1} > 0 $ be the first eigenvalue of $ (-\Delta u, H^{1}_{0}(\Omega)) $. We get that for certain $ \widetilde{\mu} > 0 $ there exists at least one least energy sign-changing solution for the above system if $ \lambda < a\lambda_{1} $ and $ \mu > \widetilde{\mu} $. In addition, we remark that the nonlinearity $ \lambda u+\mu|u|^{2}u $ does not satisfy the growth conditions.https://www.aimspress.com/article/doi/10.3934/era.2023149?viewType=HTMLkirchhoff type equationschrödinger-poisson problemsign-changing solutionsnehari manifoldthe growth conditions |
| spellingShingle | Xia Su Wen Guan Xia Li Least energy sign-changing solutions for Kirchhoff-Schrödinger-Poisson system on bounded domains Electronic Research Archive kirchhoff type equation schrödinger-poisson problem sign-changing solutions nehari manifold the growth conditions |
| title | Least energy sign-changing solutions for Kirchhoff-Schrödinger-Poisson system on bounded domains |
| title_full | Least energy sign-changing solutions for Kirchhoff-Schrödinger-Poisson system on bounded domains |
| title_fullStr | Least energy sign-changing solutions for Kirchhoff-Schrödinger-Poisson system on bounded domains |
| title_full_unstemmed | Least energy sign-changing solutions for Kirchhoff-Schrödinger-Poisson system on bounded domains |
| title_short | Least energy sign-changing solutions for Kirchhoff-Schrödinger-Poisson system on bounded domains |
| title_sort | least energy sign changing solutions for kirchhoff schrodinger poisson system on bounded domains |
| topic | kirchhoff type equation schrödinger-poisson problem sign-changing solutions nehari manifold the growth conditions |
| url | https://www.aimspress.com/article/doi/10.3934/era.2023149?viewType=HTML |
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