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| In this paper, we study the following coupled Choquard type system with Hardy--Littlewood--Sobolev lower critical exponents and a local nonlinear perturbation:\begin{equation*}\left\{ \arraycolsep=1.5pt \begin{array}{ll}-\Delta u+V(x)u=\big(I_\alpha*|u|^{\frac{\alpha}{N}+1}\big)|u|^{\frac{\alpha}{N}-1}u+ \lambda(I_\alpha*|v|^{p})|u|^{p-2}u, &\ \text{ in } \mathbb{R}^N,\\-\Delta v+V(x)v=\big(I_\alpha*|v|^{\frac{\alpha}{N}+1}\big)|v|^{\frac{\alpha}{N}-1}v+ \lambda(I_\alpha*|u|^{p})|v|^{p-2}v, &\ \text{ in } \mathbb{R}^N,\\ \end{array} \right.\end{equation*}where $N\geq 3$, $ \alpha \in (0,N)$, $I_{\alpha}:\mathbb{R}^N\backslash{\{0\}}\to\mathbb{R}$ is a Riesz potential,$V\in C(\mathbb{R}^N,[0,\infty))$ and satisfies some suitable conditions. In the case when $\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N+1}$, $p=\frac{N+\alpha+2}{N+1}$, and $\frac{N+\alpha+2}{N+1}<p<\frac{N+\alpha}{N-2}$, respectively, we investigate the existence of positive ground states of this system if $\lambda>\lambda_{*}$ by variational approaches. |
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| Keywords:Fundamental Mathematics, Choquard system, groundstate, lower critical exponent |
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