On the Existence of Ground State Solutions for Choquard Systems with Nonlinear Couplings

In this paper, we investigate the following coupled nonlinear Choquard system:

$$\begin{cases}-\Delta u + V_1(x)u = (J_{\mu}(x) \ast \vert u\vert^p)\vert u\vert^{p-2}u + \beta \vert u\vert^{q-1}\vert v\vert^q \quad \text{in}\, \mathbb{R}^N,\\-\Delta v + V_2(x)v = (J_{\mu}(x) \ast \vert v\vert^p)\vert v\vert^{p-2}v + \beta \vert v\vert^{q-1}\vert u\vert^q \quad \text{in}\, \mathbb{R}^N,\\u,\, v \in H^1(\mathbb{R}^N),\end{cases}$$

where \(N \geq 3,\ 0 < \mu < N,\ \frac{2N-\mu}{N} < p < \frac{2N-\mu}{N-2},\ 1 < q < p,\ \beta > 0\) is a coupling parameter, and J_μ(x) is convolution kernel, and V_i(x) is nonnegative continuous external potential for i = 1, 2. Using the Pohožaev-Nehari manifold method, we establish the existence of positive ground state solutions for the system. Furthermore, we prove a nonexistence result for nontrivial solutions under certain conditions.