Positive ground state solutions for a class of fractional coupled Choquard systems

Kexin Ouyang; Yu Wei; Huiqin Lu; Kexin Ouyang; Yu Wei; Huiqin Lu

doi:10.3934/math.2023806

AIMS Mathematics

2023, Volume 8, Issue 7: 15789-15804. doi: 10.3934/math.2023806

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Research article

Positive ground state solutions for a class of fractional coupled Choquard systems

1.
School of Mathematics and Statistics, Shandong Normal University, Jinan, 250358, China
2.
Mathematics and Science College, Shanghai Normal University, Shanghai 200234, China

Received: 01 March 2023 Revised: 15 April 2023 Accepted: 18 April 2023 Published: 28 April 2023
MSC : 35J65, 47J05, 47J30

In this paper, we combine the critical point theory and variational method to investigate the following a class of coupled fractional systems of Choquard type

$ \begin{equation*} \left\{ \begin{array}{l} (-\Delta)^{s}u+\lambda_{1}u& = (I_{\alpha}*|u|^{p})|u|^{p-2}u+\beta v \quad &&\text{in}\\ \mathbb{R}^{N}, \ (-\Delta)^{s}v+\lambda_{2}v& = (I_{\alpha}*|v|^{p})|v|^{p-2}v+\beta u \quad &&\text{in}\ \mathbb{R}^{N}, \end{array} \right. \end{equation*} $

with $ s\in(0, 1), \ N\geq 3, \ \alpha\in(0, N), \ p > 1 $, $ \lambda_{i} > 0 $ are constants for $ i = 1, \ 2 $, $ \beta > 0 $ is a parameter, and $ I_{\alpha}(x) $ is the Riesz Potential. We prove the existence and asymptotic behaviour of positive ground state solutions of the systems by using constrained minimization method and Hardy-Littlewood-Sobolev inequality. Moreover, nonexistence of nontrivial solutions is also obtained.
- fractional Choquard systems,
- ground state solution,
- asymptotic behaviour
Citation: Kexin Ouyang, Yu Wei, Huiqin Lu. Positive ground state solutions for a class of fractional coupled Choquard systems[J]. AIMS Mathematics, 2023, 8(7): 15789-15804. doi: 10.3934/math.2023806

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Abstract

In this paper, we combine the critical point theory and variational method to investigate the following a class of coupled fractional systems of Choquard type

$ \begin{equation*} \left\{ \begin{array}{l} (-\Delta)^{s}u+\lambda_{1}u& = (I_{\alpha}*|u|^{p})|u|^{p-2}u+\beta v \quad &&\text{in}\\ \mathbb{R}^{N}, \ (-\Delta)^{s}v+\lambda_{2}v& = (I_{\alpha}*|v|^{p})|v|^{p-2}v+\beta u \quad &&\text{in}\ \mathbb{R}^{N}, \end{array} \right. \end{equation*} $

with $ s\in(0, 1), \ N\geq 3, \ \alpha\in(0, N), \ p > 1 $, $ \lambda_{i} > 0 $ are constants for $ i = 1, \ 2 $, $ \beta > 0 $ is a parameter, and $ I_{\alpha}(x) $ is the Riesz Potential. We prove the existence and asymptotic behaviour of positive ground state solutions of the systems by using constrained minimization method and Hardy-Littlewood-Sobolev inequality. Moreover, nonexistence of nontrivial solutions is also obtained.

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