*preprint*

**Inserted:** 3 feb 2024

**Last Updated:** 3 feb 2024

**Year:** 2023

**Abstract:**

This article focuses on the existence and non-existence of solutions for the
following system of local and nonlocal type
\begin{equation**}
\left\{
\begin{aligned}
-\partial _{{xx}u} + (-\Delta)_{{y}}^{{s}_{{1}}} u + u - u^{{2}_{{s}_{{1}}}^{{}}-1} = \kappa
\alpha h(x,y) u^{{\alpha}-1}v^{{\beta}} & \quad \mbox{in} ~ \mathbb{R}^{{2},
}
-\partial_{{xx}v} + (-\Delta)_{{y}}^{{s}_{{2}}} v + v- v^{{2}_{{s}_{{2}}}^{{}}-1} = \kappa
\beta h(x,y) u^{{\alpha}v}^{{\beta}-1} & \quad \mbox{in} ~ \mathbb{R}^{{2},
}
u,v ~ \geq ~0 \quad \mbox{in} ~ \mathbb{R}^{{2},
}
\end{aligned}
\right. \end{equation**}
where $s_{1},s_{2} \in (0,1),~\alpha,\beta>1,~\alpha+\beta \leq \min \{
2_{s_{1}}^{},2_{s_{2}}^{}\}$, and $2_{s_i}^{} = \frac{2(1+s_i)}{1-s_i}, i=1,2$.
The existence of a ground state solution entirely depends on the behaviour of
the parameter $\kappa>0$ and on the function $h$. In this article, we prove
that a ground state solution exists in the subcritical case if $\kappa$ is
large enough and $h$ satisfies (1.3). Further, if $\kappa$ becomes very small
in this case then there does not exist any solution to our system. The study in
the critical case, i.e. $s_1=s_2=s, \alpha+\beta=2_s$, is more complex and the
solution exists only for large $\kappa$ and radial $h$ satisfying (H1).
Finally, we establish a Pohozaev identity which enables us to prove the
non-existence results under some smooth assumptions on $h$.