In this paper, we investigate the existence and nonexistence of results for a class of Hamiltonian-Choquard-type elliptic systems. We show the nonexistence of classical nontrivial solutions for the problem
\[ \begin{cases} -\Delta u + u= ( I_{\alpha} \ast |v|^{p} )v^{p-1} \text{ in } \mathbb{R}^{N},\\ -\Delta v + v= ( I_{\beta} \ast |u|^{q} )u^{q-1} \text{ in } \mathbb{R}^{N}, \\ u(x),v(x) \rightarrow 0 \text{ when } |x|\rightarrow \infty, \end{cases} \]
when
$(N+\alpha )/p + (N+\beta )/q \leq 2(N-2)$
(if
$N\geq 3$
) and
$(N+\alpha )/p + (N+\beta )/q \geq 2N$
(if
$N=2$
), where
$I_{\alpha }$
and
$I_{\beta }$
denote the Riesz potential. Second, via variational methods and the generalized Nehari manifold, we show the existence of a nontrivial non-negative solution or a Nehari-type ground state solution for the problem
\[ \begin{cases} -\Delta u + u= (I_{\alpha} \ast |v|^{\frac{\alpha}{2}+1})|v|^{\frac{\alpha}{2}-1}v + g(v) \hbox{ in } \mathbb{R}^{2},\\ - \Delta v + v= (I_{\beta} \ast |u|^{\frac{\beta}{2}+1})|u|^{\frac{\beta}{2}-1}u + f(u), \hbox{ in } \mathbb{R}^{2},\\ u,v \in H^{1}(\mathbb{R}^{2}), \end{cases} \]
where
$\alpha ,\,\beta \in (0,\,2)$
and
$f,\,g$
have exponential critical growth in the Trudinger–Moser sense.
Ground State Solutions for a Nonlocal Equation in $$\mathbb {R}^2$$ Involving Vanishing Potentials and Exponential Critical Growth
