In this paper, we consider the following linearly coupled Kirchhoff–Choquard system in \(\mathbb {R}^3\) : where \(a_1, a_2, b_1, b_2, V_1, V_2\) , \(\lambda \) , \(\mu \) and \(\nu \) are positive constants. The function \(I_{\alpha }: \mathbb {R}^3 \setminus \{0\} \rightarrow \mathbb {R}\) denotes the Riesz potential with \(\alpha \in (0, 3)\) . We study the existence of positive ground state solutions under the conditions \(\frac{3 + \alpha }{3}< p \le q < 3 + \alpha \) , or \(\frac{3 + \alpha }{3}< p < q = 3 + \alpha \) , or \(\frac{3 + \alpha }{3} = p< q < 3 + \alpha \) . Assuming suitable conditions on \(V_1\) , \(V_2\) , and \(\lambda \) , we obtain a ground state solution by employing a variational approach based on the Nehari–Pohozaev manifold, inspired by the works of Ueno (Commun. Pure Appl. Anal. 24 (2025)) and Chen–Liu (J. Math. Anal. 473 (2019)). In particular, we emphasize that in the upper half critical case \(\frac{3 + \alpha }{3}< p < q = 3 + \alpha \) and the lower half critical case \(\frac{3 + \alpha }{3} = p< q < 3 + \alpha \) , a ground state solution can still be obtained by taking \(\mu \) or \(\nu \) sufficiently large to control the energy level of the minimization problem. To employ the Nehari–Pohozaev manifold we extend a regularity result to the linearly coupled system, which is essential for the validity of the Pohozaev identity.
