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

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.

Keywords

• fractional choquard systems
• ground state solution
• asymptotic behaviour