Ground state solutions to a coupled nonlinear logarithmic Hartree system

In this paper, we study the following coupled nonlinear logarithmic Hartree system \begin{align*} \left{ \displaystyle \begin{array}{ll} \displaystyle -Δu+ λ_1 u =μ_1\left( -\frac{1}{2π}\ln(|x|) \ast u^2 \right)u+β\left( -\frac{1}{2π}\ln(|x|) \ast v^2 \right)u, & x \in ~ \mathbb R^2, \vspace{.4cm}\ -Δv+ λ_2 v =μ_2\left( -\frac{1}{2π}\ln(|x|) \ast v^2 \right)v +β\left( -\frac{1}{2π}\ln(|x|) \ast u^2 \right)v, & x \in ~ \mathbb R^2, \end{array} \right.\hspace{1cm} \end{align*} where $\beta ,{\mu }_i,{\lambda }_i(i=1,2)$ are positive constants, $\ast$ denotes the convolution in ${\mathbb{R}}^2$. By considering the constraint minimum problem on the Nehari manifold, we prove the existence of ground state solutions for $\beta >0$ large enough. Moreover, we also show that every positive solution is radially symmetric and decays exponentially.