Existence of ground state solutions to Kirchhoff--Choquard system in ${\mathbb{R}}^3$ with nonconstant potentials

In this paper, we study the following linearly coupled Kirchhoff-Choquard system in ${\mathbb{R}}^3$: \begin{align*}\left{\begin{array}{l} &-\left(a_1+b_1\int_{$\mathbb{R}$^3}|\nabla u|^2,dx\right)Δu+V_1(x)u=μ(I_α*|u|^p)|u|^{p-2}u+λv,\quad x \in $\mathbb{R}$^3,\cr &-\left(a_2+b_2\int_{$\mathbb{R}$^3}|\nabla v|^2,dx\right)Δv+V_2(x)v=ν(I_α*|v|^q)|v|^{q-2}v+λu,\quad x \in $\mathbb{R}$^3,\cr &u, v \in H^1($\mathbb{R}$^3), \end{array}\right. \end{align*} where $a_1,a_2,b_1,b_2,\lambda ,\mu ,$ and $\nu$ are positive constants. When the potentials are constant functions, the author previously proved the existence of positive ground state solutions in the following cases: the noncritical case $\frac{3+\alpha }{3}<p\le q<3+\alpha$, the upper half critical case $\frac{3+\alpha }{3}<p<q=3+\alpha$, and the lower half critical case $\frac{3+\alpha }{3}=p<q<3+\alpha$, by using the Nehari-Pohozaev manifold method (NoDEA Nonlinear Differential Equations Appl.33(2026)). In the present paper, we extend these results to the case of nonconstant potentials. Under suitable assumptions on $V_1(x)$, $V_2(x)$, and $\lambda$, we prove the existence of nontrivial ground state solutions. In the noncritical and upper half critical cases, the main tools are Jeanjean’s monotonicity trick and a global compactness lemma. For these cases, we establish a refined version of the splitting lemma by relaxing the standard growth assumption at the origin from $o(\mathrm{\mid }t\mathrm{\mid })$ to the optimal $o(\mathrm{\mid }t{\mathrm{\mid }}^{\alpha \mathrm{/}3})$, thereby significantly broadening the applicable class of nonlinearities. In contrast, for the lower half critical case $p=\frac{3+\alpha }{3}$, the splitting lemma is no longer valid. To overcome this essential difficulty, we obtain a ground state solution directly as a minimizer on the Nehari-Pohozaev manifold by imposing a slightly stronger condition on the potentials.