2025/07/12 by Matsuzawa, Hiroshi
#35J47 #35J50(Primary) 35J20 #35Q55 #45K05(Secondary) #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2507.09163
In this paper, we consider the following linearly coupled Kirchhoff--Choquard system in ℝ3: \begincases -(a1 + b1∫ℝ3 |∇ u|2 dx)Δu + V1 u = μ(Iα * |u|p) |u|p - 2 u + λv, x∈ℝ3
-(a2 + b2∫ℝ3 |∇ v|2 dx)Δv + V2 v = ν(Iα * |v|q) |v|q - 2 v + λu, x∈ℝ3
u, v ∈ H1(ℝ3),
\endcases where a1, a2, b1, b2, V1, V2, λ, μ and ν are positive constants. The function Iα : ℝ3 ∖ \0\ → ℝ denotes the Riesz potential with α∈ (0, 3).
We study the existence of positive ground state solutions under the conditions (3 + α)/(3) < p ≤ q < 3 + α, or (3 + α)/(3) < p < q = 3 + α, or (3 + α)/(3) = p < q < 3 + α. Assuming suitable conditions on V1, V2, and λ, 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 (3 + α)/(3) < p < q = 3 + α and the lower half critical case (3 + α)/(3) = p < q < 3 + α, a ground state solution can still be obtained by taking μ or ν 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.