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 - (a₁ + b₁ₑ℃ | u|²\, dx) Δu + V₁ u = μ (I_α * |u|ᵖ) |u|^p - 2 u + λv, \ \ x³\\ - (a₂ + b₂ₑ℃ | v|²\, dx) Δv + V₂ v = ν (I_α * |v|q) |v|^q - 2 v + λu, \ \ x³ \\ u, v H¹ (R³), 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 + α3 < p q < 3 + α, or 3 + α3 < p < q = 3 + α, or 3 + α3 = 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 + α3 < p < q = 3 + α and the lower half critical case 3 + α3 = 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.