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.
Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms
