In this paper, we study the following Hamiltonian Choquard-type elliptic systems involving singular weights \begin{ eqnarray*} \begin{ aligned}\displaystyle \left\{ \arraycolsep=1.5pt \begin{ array}{ ll} -\Delta u + V(x)u = \Big(I_{ \mu_{ 1}}\ast \frac{ G(v)}{ |x|^{ \alpha}}\Big)\frac{ g(v)}{ |x|^{ \alpha}} \ \ \ & \mbox{ in} \ \mathbb{ R}^{ 2},\\[2mm] -\Delta v + V(x)v = \Big(I_{ \mu_{ 2}}\ast \frac{ F(u)}{ |x|^{ \beta}}\Big)\frac{ f(u)}{ |x|^{ \beta}} \ \ \ & \mbox{ in} \ \mathbb{ R}^{ 2}, \end{ array} \right. \end{ aligned} \end{ eqnarray*} where $\mu_{ 1},\mu_{ 2}\in(0,2)$, $0<\alpha \leq \frac{ \mu_{ 1}}{ 2}$, $0<\beta \leq \frac{ \mu_{ 2}}{ 2}$, $V(x)$ is a continuous positive potential, $I_{ \mu_{ 1}}$ and $I_{ \mu_{ 2}}$ denote the Riesz potential, $\ast$ indicates the convolution operator, $F(s),G(s)$ are the primitive of $f(s),g(s)$ with $f(s),g(s)$ have exponential growth in $\mathbb{ R}^{ 2}$. Using the linking theorem and variational methods, we establish the existence of solutions to the above problem.
Categories: math.AP