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

Abstract

In this paper, we consider the following linearly coupled Kirchhoff--Choquard system in $\mathbb{R}^3$: \begin{align*} \begin{cases} -\left(a_1 + b_1\int_{\mathbb{R}^3} |\nabla u|^2\,dx\right)\Delta u + V_1 u = \mu (I_{\alpha} * |u|^p) |u|^{p - 2} u + \lambda v, \ \ x\in\mathbb{R}^3\\ -\left(a_2 + b_2\int_{\mathbb{R}^3} |\nabla v|^2\,dx\right)\Delta v + V_2 v = \nu (I_{\alpha} * |v|^q) |v|^{q - 2} v + \lambda u,\ \ x\in\mathbb{R}^3 \\ u, v \in H^1(\mathbb{R}^3), \end{cases} \end{align*} where $a_1, a_2, b_1, b_2, V_1, V_2$, $\lambda$, $\mu$ and $\nu$ are positive constants. The function $I_{\alpha} : \mathbb{R}^3 \setminus \{0\} \to \mathbb{R}$ 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.