Existence of ground state solutions to Kirchhoff--Choquard system in ℝ3 with nonconstant potentials
Abstract
In this paper, we study the following linearly coupled Kirchhoff-Choquard system in ℝ3: {l &−(a1+b1∫ℝ3|∇ u|2 dx)Δu+V1(x)u=μ(Iα*|u|p)|u|p−2u+λv, x ∈ ℝ3,cr &−(a2+b2∫ℝ3|∇ v|2 dx)Δv+V2(x)v=ν(Iα*|v|q)|v|q−2v+λu, x ∈ ℝ3,cr &u, v ∈ H1(ℝ3), . where a1, a2, b1, b2, λ, μ, and ν 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 (3+α)/3<p≤ q<3+α, the upper half critical case (3+α)/3<p<q=3+α, and the lower half critical case (3+α)/3=p<q<3+α, 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 V1(x), V2(x), and λ, 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|α/3), thereby significantly broadening the applicable class of nonlinearities. In contrast, for the lower half critical case p=(3+α)/3, the splitting lemma is no longer valid.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.
Related
- Existence of ground state solutions for fractional Choquard systems
- Existence and multiplicity of solutions for mixed local and nonlocal Kirchhoff type problems
- Systems of Kirchhoff quasilinear Schrodinger equations with Hardy potential and critical Choquard-type nonlinearity
- The exact number of nonconstant positive solutions to the planar isotropic Lp dual Minkowski problem
- Symmetry, uniqueness and nondegeneracy of positive solutions of the Choquard equation
BibTeX
@misc{matsuzawa2026existence,
title = {{Existence of ground state solutions to Kirchhoff--Choquard system in $\mathbbR^3$ with nonconstant potentials}},
author = {Hiroshi Matsuzawa},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18405},
eprint = {2609.18405},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18405},
}
RIS
TY - UNPB
TI - Existence of ground state solutions to Kirchhoff--Choquard system in ℝ^3 with nonconstant potentials
AU - Hiroshi Matsuzawa
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18405
UR - https://arxiv.org/abs/2609.18405
AB - In this paper, we study the following linearly coupled Kirchhoff-Choquard system in ℝ^3: {l &−(a_1+b_1∫_ℝ^3|∇ u|^2 dx)Δu+V_1(x)u=μ(I_α*|u|^p)|u|^p−2u+λv, x ∈ ℝ^3,cr &−(a_2+b_2∫_ℝ^3|∇ v|^2 dx)Δv+V_2(x)v=ν(I_α*|v|^q)|v|^q−2v+λu, x ∈ ℝ^3,cr &u, v ∈ H^1(ℝ^3), . where a_1, a_2, b_1, b_2, λ, μ, and ν 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 (3+α)/3<p≤ q<3+α, the upper half critical case (3+α)/3<p<q=3+α, and the lower half critical case (3+α)/3=p<q<3+α, 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 λ, 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|^α/3), thereby significantly broadening the applicable class of nonlinearities. In contrast, for the lower half critical case p=(3+α)/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.
ER -