Odd symmetry of ground state solutions for the Choquard system
This paper is dedicated to the following Choquard system: $ \left\{\begin{aligned}&-\Delta u+u = \frac{2p}{p+q}\bigl(I_\alpha\ast|v|^q\bigr)|u|^{p-2}u, \\ &-\Delta v+v = \frac{2q}{p+q}\bigl(I_\alpha\ast|u|^p\bigr)|v|^{q-2}v, \\ &u(x)\to 0, \ \ v(x)\to 0\ \ \hbox{as}\ |x|\...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-05-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2023898?viewType=HTML |
| _version_ | 1797810646068232192 |
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| author | Jianqing Chen Qihua Ruan Qian Zhang |
| author_facet | Jianqing Chen Qihua Ruan Qian Zhang |
| author_sort | Jianqing Chen |
| collection | DOAJ |
| description | This paper is dedicated to the following Choquard system:
$ \left\{\begin{aligned}&-\Delta u+u = \frac{2p}{p+q}\bigl(I_\alpha\ast|v|^q\bigr)|u|^{p-2}u, \\ &-\Delta v+v = \frac{2q}{p+q}\bigl(I_\alpha\ast|u|^p\bigr)|v|^{q-2}v, \\ &u(x)\to 0, \ \ v(x)\to 0\ \ \hbox{as}\ |x|\to\infty, \end{aligned}\right. $
where $ N\geq 1 $, $ \alpha\in(0, N) $ and $ \frac{N+\alpha}{N} < p, \ q < 2_*^\alpha $, in which $ 2_*^\alpha $ denotes $ \frac{N+\alpha}{N-2} $ if $ N\geq3 $ and $ 2_*^\alpha: = \infty $ if $ N = 1, \ 2 $. $ I_\alpha $ is a Riesz potential. We obtain the odd symmetry of ground state solutions via a variant of Nehari constraint. Our results can be looked on as a partial generalization to results by Ghimenti and Schaftingen (Nodal solutions for the Choquard equation, J. Funct. Anal. 271 (2016), 107). |
| first_indexed | 2024-03-13T07:11:55Z |
| format | Article |
| id | doaj.art-79bc3fc77ce7479b95589b16d6774530 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-03-13T07:11:55Z |
| publishDate | 2023-05-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-79bc3fc77ce7479b95589b16d67745302023-06-06T01:15:28ZengAIMS PressAIMS Mathematics2473-69882023-05-0188176031761910.3934/math.2023898Odd symmetry of ground state solutions for the Choquard systemJianqing Chen0Qihua Ruan1Qian Zhang 21. School of Mathematics and Statistics, Fujian Normal University, Fuzhou 350117, China2. Provincial Key Laboratory of Applied Mathematics, Putian University, Putian 351100, China3. Department of Mathematical Sciences, Tsinghua University, Beijing 100084, ChinaThis paper is dedicated to the following Choquard system: $ \left\{\begin{aligned}&-\Delta u+u = \frac{2p}{p+q}\bigl(I_\alpha\ast|v|^q\bigr)|u|^{p-2}u, \\ &-\Delta v+v = \frac{2q}{p+q}\bigl(I_\alpha\ast|u|^p\bigr)|v|^{q-2}v, \\ &u(x)\to 0, \ \ v(x)\to 0\ \ \hbox{as}\ |x|\to\infty, \end{aligned}\right. $ where $ N\geq 1 $, $ \alpha\in(0, N) $ and $ \frac{N+\alpha}{N} < p, \ q < 2_*^\alpha $, in which $ 2_*^\alpha $ denotes $ \frac{N+\alpha}{N-2} $ if $ N\geq3 $ and $ 2_*^\alpha: = \infty $ if $ N = 1, \ 2 $. $ I_\alpha $ is a Riesz potential. We obtain the odd symmetry of ground state solutions via a variant of Nehari constraint. Our results can be looked on as a partial generalization to results by Ghimenti and Schaftingen (Nodal solutions for the Choquard equation, J. Funct. Anal. 271 (2016), 107). https://www.aimspress.com/article/doi/10.3934/math.2023898?viewType=HTMLchoquard systemodd symmetryground state solutionnonlocal brézis-lieb lemma |
| spellingShingle | Jianqing Chen Qihua Ruan Qian Zhang Odd symmetry of ground state solutions for the Choquard system AIMS Mathematics choquard system odd symmetry ground state solution nonlocal brézis-lieb lemma |
| title | Odd symmetry of ground state solutions for the Choquard system |
| title_full | Odd symmetry of ground state solutions for the Choquard system |
| title_fullStr | Odd symmetry of ground state solutions for the Choquard system |
| title_full_unstemmed | Odd symmetry of ground state solutions for the Choquard system |
| title_short | Odd symmetry of ground state solutions for the Choquard system |
| title_sort | odd symmetry of ground state solutions for the choquard system |
| topic | choquard system odd symmetry ground state solution nonlocal brézis-lieb lemma |
| url | https://www.aimspress.com/article/doi/10.3934/math.2023898?viewType=HTML |
| work_keys_str_mv | AT jianqingchen oddsymmetryofgroundstatesolutionsforthechoquardsystem AT qihuaruan oddsymmetryofgroundstatesolutionsforthechoquardsystem AT qianzhang oddsymmetryofgroundstatesolutionsforthechoquardsystem |