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

TL;DR

This paper proves the existence of ground state solutions for a coupled Kirchhoff--Choquard system in three-dimensional space using variational methods and Nehari--Pohozaev manifold techniques, even in critical cases.

Contribution

It extends the Nehari--Pohozaev manifold approach to coupled Kirchhoff--Choquard systems and handles critical exponent cases by adjusting parameters.

Findings

Existence of positive ground state solutions under various exponent conditions.

Ground states exist even in critical cases with large coupling parameters.

Extension of regularity results for coupled systems to validate Pohozaev identity.

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.