In this paper, we study the following coupled nonlinear logarithmic Hartree system \begin{ align*} \left\{ \displaystyle \begin{ array}{ ll} \displaystyle -\Delta u+ \lambda_1 u =\mu_1\left( -\frac{ 1}{ 2\pi}\ln(|x|) \ast u^2 \right)u+\beta \left( -\frac{ 1}{ 2\pi}\ln(|x|) \ast v^2 \right)u, & x \in ~ \mathbb R^2, \vspace{ .4cm}\\ -\Delta v+ \lambda_2 v =\mu_2\left( -\frac{ 1}{ 2\pi}\ln(|x|) \ast v^2 \right)v +\beta\left( -\frac{ 1}{ 2\pi}\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.
Categories: math.AP