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

RESUMO

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)\Delta u+V_1(x)u=\mu(I_{\alpha}*|u|^p)|u|^{p-2}u+\lambda v,\quad x \in \mathbb{R}^3,\cr &-\left(a_2+b_2\int_{\mathbb{R}^3}|\nabla v|^2\,dx\right)\Delta v+V_2(x)v=\nu(I_{\alpha}*|v|^q)|v|^{q-2}v+\lambda u,\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 $o(|t|)$ to the optimal $o(|t|^{\alpha/3})$, 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 In contrast, for the lower half critical case $p=\frac{3+\alpha}{3}$, the splitting lemma is no longer valid.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.