Existence of ground state solutions to Kirchhoff--Choquard system in R³ with constant potentials

In this paper, we consider the following linearly coupled Kirchhoff--Choquard system in R³: {align*} {cases} -(a_1 + b_1∫R^3 |∇ u|^2\,dx)Δu + V_1 u = μ(I_α * |u|^p) |u|p - 2 u + λv, \ \ x^3\\ -(a_2 + b_2∫R^3 |∇ v|^2\,dx)Δv + V_2 v = ν(I_α * |v|^q) |v|q - 2 v + λu,\ \ x^3 \\ u, v ∈ H^1(R^3), {cases} {align*} where a₁, a₂, b₁, b₂, V₁, V₂, $λ$, $μ$ and $ν$ are positive constants. The function I_α : R³ \0\ → R denotes the Riesz potential with α∈ (0, 3). We study the existence of positive ground state solutions under the conditions 3 + α/3 < p ≤ q < 3 + α, or 3 + α/3 < p < q = 3 + α, or 3 + α/3 = p < q < 3 + α. Assuming suitable conditions on V₁, V₂, and $λ$, 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 3 + α/3 < p < q = 3 + α and the lower half critical case 3 + α/3 = p < q < 3 + α, a ground state solution can still be obtained by taking $μ$ or $ν$ 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.