On a class of singular Hamiltonian Choquard-type elliptic systems with critical exponential growth

Abstract

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} -Δu + V(x)u = \Big(I_{μ_{1}}\ast \frac{G(v)}{|x|^α}\Big)\frac{g(v)}{|x|^α} \ \ \ & \mbox{in} \ \mathbb{R}^{2},\\[2mm] -Δv + V(x)v = \Big(I_{μ_{2}}\ast \frac{F(u)}{|x|^β}\Big)\frac{f(u)}{|x|^β} \ \ \ & \mbox{in} \ \mathbb{R}^{2}, \end{array} \right. \end{aligned} \end{eqnarray*} where $μ_{1},μ_{2}\in(0,2)$, $0<α\leq \frac{μ_{1}}{2}$, $0<β\leq \frac{μ_{2}}{2}$, $V(x)$ is a continuous positive potential, $I_{μ_{1}}$ and $I_{μ_{2}}$ 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.