Odd symmetry of ground state solutions for the Choquard system

Jianqing Chen; Qihua Ruan; Qian Zhang; Jianqing Chen; Qihua Ruan; Qian Zhang

doi:10.3934/math.2023898

AIMS Mathematics

2023, Volume 8, Issue 8: 17603-17619. doi: 10.3934/math.2023898

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Research article

Odd symmetry of ground state solutions for the Choquard system

1.
School of Mathematics and Statistics, Fujian Normal University, Fuzhou 350117, China
2.
Provincial Key Laboratory of Applied Mathematics, Putian University, Putian 351100, China
3.
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Received: 06 February 2023 Revised: 07 May 2023 Accepted: 08 May 2023 Published: 23 May 2023
MSC : 35J20, 35J05, 35J60

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).
- Choquard system,
- odd symmetry,
- ground state solution,
- nonlocal Brézis-Lieb lemma
Citation: Jianqing Chen, Qihua Ruan, Qian Zhang. Odd symmetry of ground state solutions for the Choquard system[J]. AIMS Mathematics, 2023, 8(8): 17603-17619. doi: 10.3934/math.2023898

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Abstract

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).

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