---
title: Linearly Coupled Kirchhoff–Choquard System
url: https://www.emergentmind.com/topics/linearly-coupled-kirchhoff-choquard-system
type: topic
---

# Linearly Coupled Kirchhoff–Choquard System

Searching arXiv for the specified paper and closely related Kirchhoff–Choquard work.
arXiv.search_query({"query":"ti:\"Existence of ground state solutions to Kirchhoff--Choquard system in R^3 with constant potentials\" OR id:2507.09163","max_results":5})
arXiv.search_query({"query":"all:\"Kirchhoff Choquard system\"","max_results":10})
arXiv.search_query({"query":"all:\"Kirchhoff Choquard\" AND (all:\"Nehari Pohozaev\" OR all:\"Chen Liu\")","max_results":10})
The linearly coupled Kirchhoff--Choquard system is a nonlinear, nonlocal elliptic system in which Kirchhoff-type global coefficients and Choquard-type convolution nonlinearities are combined with a linear intercomponent coupling. In the constant-potential setting studied in \(\mathbb{R}^3\), the model takes the form [2507.09163]
\[
\begin{cases}
-\left(a_1 + b_1\displaystyle\int_{\mathbb{R}^3} |\nabla u|^2\,dx\right)\Delta u + V_1 u
= \mu (I_{\alpha} * |u|^p) |u|^{p - 2} u + \lambda v,\\[0.5em]
-\left(a_2 + b_2\displaystyle\int_{\mathbb{R}^3} |\nabla v|^2\,dx\right)\Delta v + V_2 v
= \nu (I_{\alpha} * |v|^q) |v|^{q - 2} v + \lambda u,\\[0.3em]
u, v \in H^1(\mathbb{R}^3),
\end{cases}
\]
where \(a_i,b_i,V_i,\lambda,\mu,\nu>0\) and \(\alpha\in(0,3)\). The analysis in [2507.09163] is centered on the existence of positive ground state solutions, the role of the Nehari--Pohozaev manifold as a natural variational constraint, the recovery of compactness in half-critical regimes through large-parameter conditions, and nonexistence at the simultaneous critical endpoints.

## 1. Structural form of the system

The system is simultaneously nonlocal in two distinct senses. The Kirchhoff contribution enters through the coefficients
\[
a_i+b_i\int_{\mathbb{R}^3}|\nabla w|^2\,dx,
\]
which depend on the global Dirichlet energy of the unknown; this is the source of the Kirchhoff character. The Choquard contribution enters through the Riesz-potential convolution
\[
(I_\alpha*|w|^r)(x)=\int_{\mathbb{R}^3}I_\alpha(x-y)|w(y)|^r\,dy,
\]
which generates the Hartree-type nonlocal interaction. The linear terms \(\lambda v\) and \(\lambda u\) provide the intercomponent coupling [2507.09163].

In \(\mathbb{R}^3\), the Riesz kernel is
\[
I_\alpha(x)=A_{3,\alpha}|x|^{\alpha-3},
\qquad
A_{3,\alpha}=
\frac{\Gamma((3-\alpha)/2)}{\Gamma(\alpha/2)\pi^{3/2}2^\alpha}.
\]
Accordingly, the Choquard energy can also be written in symmetric double-integral form; for example,
\[
\frac{1}{2p}\int_{\mathbb{R}^3}(I_\alpha*|u|^p)|u|^p\,dx
=
\frac{A_{3,\alpha}}{2p}
\iint_{\mathbb{R}^3\times\mathbb{R}^3}
\frac{|u(x)|^p|u(y)|^p}{|x-y|^{3-\alpha}}\,dx\,dy.
\]

The exponent structure is decisive. The treatment in [2507.09163] covers three regimes:
\[
\frac{3+\alpha}{3}<p\le q<3+\alpha,
\qquad
\frac{3+\alpha}{3}<p<q=3+\alpha,
\qquad
p=\frac{3+\alpha}{3}<q<3+\alpha.
\]
The first is the strictly subcritical regime, the second the upper half critical regime, and the third the lower half critical regime. By contrast, the simultaneous endpoints \(p=q=3+\alpha\) and \(p=q=(3+\alpha)/3\) are shown to admit no nontrivial solution, while the mixed endpoint \((p,q)=(3+\alpha,(3+\alpha)/3)\) remains open [2507.09163].

## 2. Variational framework and notions of solution

The natural energy space is
\[
H=H^1(\mathbb{R}^3)\times H^1(\mathbb{R}^3),
\]
equipped with the norm
\[
\|(u,v)\|^2=\|u\|_{a_1,V_1}^2+\|v\|_{a_2,V_2}^2,
\]
where
\[
\|w\|_{a_i,V_i}
=
\left(
a_i\int |\nabla w|^2\,dx+V_i\int |w|^2\,dx
\right)^{1/2}.
\]
This is an equivalent norm on \(H^1(\mathbb{R}^3)\) for each component [2507.09163].

A weak solution is a pair \((u,v)\in H\) that is a critical point of the associated energy functional \(I\), meaning that
\[
\langle I'(u,v),(\phi,\psi)\rangle=0
\quad
\text{for all }
(\phi,\psi)\in C_0^\infty(\mathbb{R}^3)\times C_0^\infty(\mathbb{R}^3).
\]
A nontrivial weak solution \((u^*,v^*)\neq(0,0)\) is a ground state if
\[
I(u^*,v^*)\le I(u,v)
\]
for every other nontrivial weak solution \((u,v)\in H\setminus\{(0,0)\}\) [2507.09163].

The energy functional is
\[
\begin{aligned}
I(u,v)
={}&
\frac12\left(a_1\int |\nabla u|^2+a_2\int |\nabla v|^2\right)
+\frac12\left(V_1\int u^2+V_2\int v^2\right)\\
&+\frac14\left(
b_1\left(\int |\nabla u|^2\right)^2
+b_2\left(\int |\nabla v|^2\right)^2
\right)
-\frac{\mu}{2p}\int (I_\alpha*|u|^p)|u|^p\\
&-\frac{\nu}{2q}\int (I_\alpha*|v|^q)|v|^q
-\lambda\int uv.
\end{aligned}
\]
Using the Hardy--Littlewood--Sobolev inequality together with the Sobolev embeddings \(H^1(\mathbb{R}^3)\hookrightarrow L^s(\mathbb{R}^3)\) for \(s\in[2,6]\), [2507.09163] shows that \(I\) is finite and \(C^1\) on \(H\) when
\[
\frac{3+\alpha}{3}\le p,q\le 3+\alpha.
\]

A structural assumption on the linear coupling is imposed throughout:
\[
\lambda\le \delta\sqrt{V_1V_2}
\quad\text{for some }\delta\in(0,1).
\]
Under this condition,
\[
\int (V_1u^2+V_2v^2)-2\lambda\int uv
\ge
(1-\delta)\int (V_1u^2+V_2v^2),
\]
so the coupling is controlled by the positive potential terms. This coercivity mechanism is fundamental for the lower bounds on the variational constraint set and for positivity of the minimization level [2507.09163].

## 3. Nehari--Pohozaev manifold and scaling geometry

The variational analysis is organized around a scaling tailored to the Kirchhoff structure:
\[
w^t(x)=t\,w(t^{-2}x),
\qquad t>0.
\]
For a fixed nonzero pair \((u,v)\), one defines the fiber map
\[
\zeta(t)=I(u^t,v^t).
\]
Direct computation yields
\[
\begin{aligned}
\zeta(t)
={}&
\frac12 t^4(a_1\|\nabla u\|_2^2+a_2\|\nabla v\|_2^2)
+\frac12 t^8(V_1\|u\|_2^2+V_2\|v\|_2^2)\\
&+\frac14 t^8(b_1\|\nabla u\|_2^4+b_2\|\nabla v\|_2^4)
-\frac{\mu}{2p}t^{2(p+\alpha+3)}\int (I_\alpha*|u|^p)|u|^p\\
&-\frac{\nu}{2q}t^{2(q+\alpha+3)}\int (I_\alpha*|v|^q)|v|^q
-\lambda t^8\int uv.
\end{aligned}
\]
For \((3+\alpha)/3\le p\le q\le 3+\alpha\), the fiber map has a unique critical point \(t(u,v)>0\), and that point is the global maximum on \((0,\infty)\) [2507.09163].

The Pohozaev identity plays a central role. Every weak solution satisfies
\[
\begin{aligned}
P(u,v)\coloneqq{}&
\frac12(a_1\|\nabla u\|_2^2+a_2\|\nabla v\|_2^2)
+\frac32(V_1\|u\|_2^2+V_2\|v\|_2^2)\\
&+\frac12(b_1\|\nabla u\|_2^4+b_2\|\nabla v\|_2^4)
-\frac{3+\alpha}{2p}\mu\int (I_\alpha*|u|^p)|u|^p\\
&-\frac{3+\alpha}{2q}\nu\int (I_\alpha*|v|^q)|v|^q
-3\lambda\int uv
=0.
\end{aligned}
\]
Differentiating \(\zeta\) at \(t=1\) produces the Nehari--Pohozaev functional
\[
\begin{aligned}
J(u,v)\coloneqq{}&
2(a_1\|\nabla u\|_2^2+a_2\|\nabla v\|_2^2)
+4(V_1\|u\|_2^2+V_2\|v\|_2^2)\\
&+2(b_1\|\nabla u\|_2^4+b_2\|\nabla v\|_2^4)
-\frac{p+\alpha+3}{p}\mu\int (I_\alpha*|u|^p)|u|^p\\
&-\frac{q+\alpha+3}{q}\nu\int (I_\alpha*|v|^q)|v|^q
-8\lambda\int uv.
\end{aligned}
\]
It satisfies
\[
J(u,v)=\langle I'(u,v),(u,v)\rangle+2P(u,v).
\]

The associated constraint is the Nehari--Pohozaev manifold
\[
M=\{(u,v)\in H\setminus\{(0,0)\}: J(u,v)=0\}.
\]
Every nontrivial critical point of \(I\) belongs to \(M\), and for every \((u,v)\neq(0,0)\) there exists a unique \(t=t(u,v)>0\) such that \((u^t,v^t)\in M\). On \(M\),
\[
I(u,v)=\max_{t>0}I(u^t,v^t).
\]
This converts the search for ground states into a constrained minimization problem on a set adapted to both the Euler--Lagrange equation and the Pohozaev identity [2507.09163].

Under the coupling condition \((V)\), the manifold is separated from the origin: there exists \(\underline C>0\) such that
\[
\|(u,v)\|\ge \underline C
\quad\text{for all }(u,v)\in M.
\]
Moreover, for \(p\le q\),
\[
I(u,v)=I(u,v)-\frac{1}{2(p+\alpha+3)}J(u,v)
\ge
\frac{p+\alpha-1}{2(p+\alpha+3)}(1-\delta)\|(u,v)\|^2,
\]
and hence
\[
m\coloneqq \inf_{(u,v)\in M}I(u,v)>0.
\]

## 4. Existence, half-critical thresholds, and endpoint nonexistence

The ground-state problem is formulated as
\[
m=\inf_{(u,v)\in M}I(u,v)
=
\inf_{(u,v)\neq(0,0)}\max_{t>0}I(u^t,v^t).
\]
Within this framework, [2507.09163] establishes the following existence and nonexistence picture.

| Exponent regime | Assumptions | Conclusion |
|---|---|---|
| \(\frac{3+\alpha}{3}<p,q<3+\alpha\) | \((V)\), any \(\mu,\nu>0\) | positive ground state solution |
| \(\frac{3+\alpha}{3}<p<3+\alpha,\ q=3+\alpha\) | \((V)\), fixed \(\nu>0\), \(\mu\ge\mu_0\) | ground state solution |
| \(p=\frac{3+\alpha}{3},\ \frac{3+\alpha}{3}<q<3+\alpha\) | \((V)\), fixed \(\mu>0\), \(\nu\ge\nu_0\) | ground state solution |
| \(p=q=3+\alpha\) or \(p=q=\frac{3+\alpha}{3}\) | \((V)\) | no nontrivial solution |
| \(p=3+\alpha,\ q=\frac{3+\alpha}{3}\) | — | open |

In the upper half critical case \(q=3+\alpha\), compactness is recovered by forcing the minimization level below the sharp threshold associated with the best constant \(S^*\), defined by
\[
S^*\left(\int (I_\alpha*|w|^{3+\alpha})|w|^{3+\alpha}\,dx\right)^{1/(3+\alpha)}
\le
\int |\nabla w|^2\,dx.
\]
For each fixed \(\nu\), there exists \(\mu_0>0\) such that, for \(\mu\ge\mu_0\),
\[
m_{\mu,\nu}
<
\frac{\alpha+1}{4(3+\alpha)}
\,\nu\,
\left(\left(\frac{a_2}{\nu}\right)S^*\right)^{(3+\alpha)/(2+\alpha)}.
\]
This strict inequality is the quantitative device ensuring that concentration cannot be lost at the critical level [2507.09163].

In the lower half critical case \(p=(3+\alpha)/3\), the relevant sharp constant is
\[
S_*
\left(\int (I_\alpha*|w|^{(3+\alpha)/3})|w|^{(3+\alpha)/3}\,dx\right)^{3/(3+\alpha)}
\le
\int |w|^2\,dx,
\]
with
\[
S_*=
\inf_{w\in L^2}
\frac{\|w\|_2^2}
{\left(\int (I_\alpha*|w|^{(3+\alpha)/3})|w|^{(3+\alpha)/3}\,dx\right)^{3/(3+\alpha)}}.
\]
For each fixed \(\mu>0\), there exists \(\nu_0>0\) such that, for \(\nu\ge\nu_0\),
\[
m_{\mu,\nu}
<
\frac{\alpha}{2(3+\alpha)}
\,\mu\,
\left(
\frac{4(3+\alpha)}{\mu(12+\alpha)}V_1(1-\delta)S_*
\right)^{(3+\alpha)/3}.
\]
The role of these large-parameter assumptions is not merely technical: they ensure that the ground-state level lies below the critical compactness threshold. This suggests that the subcritical component can be used to dominate the variational level even when one Choquard exponent reaches a critical boundary.

## 5. Compactness, splitting, regularity, and positivity

The existence proof proceeds by minimizing \(I\) on \(M\). Starting from a minimizing sequence \(\{(u_n,v_n)\}\subset M\) with \(I(u_n,v_n)\to m\), the argument combines concentration--compactness in the sense of Lions with a nonlocal Brezis--Lieb splitting for the Choquard terms [2507.09163]. The splitting formula has the form
\[
\begin{aligned}
I(u_n,v_n)
={}&
I(u,v)+I(u_n-u,v_n-v)\\
&+\frac12\Big(
b_1\|\nabla u\|_2^2\|\nabla(u_n-u)\|_2^2
+b_2\|\nabla v\|_2^2\|\nabla(v_n-v)\|_2^2
\Big)+o(1),
\end{aligned}
\]
together with analogous decompositions for \(\langle I'(u_n,v_n),(u_n,v_n)\rangle\) and \(J(u_n,v_n)\). Because of the Kirchhoff terms, these identities contain mixed quadratic--quartic gradient contributions that do not appear in non-Kirchhoff Choquard systems.

Nonvanishing is obtained by contradiction. If local mass vanished, then the Hardy--Littlewood--Sobolev estimate would force the Choquard energies to vanish, contradicting the positive lower bound on \(M\). In the half-critical regimes, the contradiction requires the sharp constants \(S^*\) or \(S_*\) together with the threshold inequalities above. Once a nontrivial weak limit is recovered, a deformation argument and degree theory show that a minimizer on \(M\) is in fact a free critical point of \(I\) in \(H\), so the constrained minimizer is a genuine weak solution [2507.09163].

A notable technical contribution is the regularity theorem for coupled Choquard systems. For \(N\ge 3\), \(1+\alpha/N\le p,q\le (N+\alpha)/(N-2)\), if \((u,v)\in H^1\times H^1\) solves
\[
-d_1\Delta u+V_1u=\mu(I_\alpha*|u|^p)|u|^{p-2}u+\lambda v,
\qquad
-d_2\Delta v+V_2v=\nu(I_\alpha*|v|^q)|v|^{q-2}v+\lambda u,
\]
then
\[
u,v\in W^{2,r}_{\mathrm{loc}}(\mathbb{R}^N)\ \text{for all }r\ge 1,
\qquad
u,v\in C^2(\mathbb{R}^N).
\]
This regularity justifies the localized virial-type testing needed for the Pohozaev identity. The proof adapts Moroz--Van Schaftingen bootstrap arguments, a coupled linear estimate, and a Choquard-specific Gagliardo--Nirenberg--HLS interpolation cited in [2507.09163].

Positivity of ground states follows from a variational and PDE argument. From a ground state \((u,v)\), one may take absolute values and rescale along the fiber to obtain \(t_0>0\) such that \((|u|^{t_0},|v|^{t_0})\in M\) and
\[
I(|u|^{t_0},|v|^{t_0})\le I(u,v).
\]
Then the strong maximum principle, applied componentwise in the cooperative setting, yields strict positivity.

## 6. Relation to adjacent theories and unresolved directions

The analysis in [2507.09163] is explicitly positioned at the intersection of two earlier variational developments. Ueno, in *Communications on Pure and Applied Analysis* 24 (2025), established ground states for linearly coupled Kirchhoff--Schrödinger systems with local nonlinearities by means of the Nehari--Pohozaev manifold, including critical exponents \(3<p<q=6\) with large \(\mu\). Chen--Liu, in *Journal of Mathematical Analysis and Applications* 473 (2019), developed the Nehari--Pohozaev method for single Kirchhoff--Choquard equations with
\[
\frac{3+\alpha}{3}<p<3+\alpha.
\]
The linearly coupled Kirchhoff--Choquard system combines these two directions: it incorporates Kirchhoff nonlocality, Choquard nonlocality, and linear coupling in a single framework, while extending the Nehari--Pohozaev machinery to the coupled nonlocal setting [2507.09163].

Relative to standard non-Kirchhoff Choquard systems, the Kirchhoff terms alter both scaling and compactness. The fiber map contains \(t^8\)-terms arising from the quartic gradient contributions, and the energy splitting must accommodate the interaction between weak limits and the global coefficients
\[
\int |\nabla u|^2,\qquad \int |\nabla v|^2.
\]
This changes the variational landscape and the critical threshold analysis. A plausible implication is that even familiar Choquard concentration phenomena must be re-evaluated once Kirchhoff modulation is present, because the natural scaling and the compactness defects are no longer the same as in the purely semilinear Hartree setting.

Several limitations remain explicit. The potentials \(V_1,V_2\) are assumed to be positive constants. Variable potentials are reserved for forthcoming work, where the present concentration--compactness argument would require refinement because additional localization issues arise. Radial symmetry, uniqueness, multiplicity, decay at infinity, and nondegeneracy are not treated. The simultaneous critical endpoints admit only the trivial solution, whereas the mixed endpoint
\[
(p,q)=\left(3+\alpha,\frac{3+\alpha}{3}\right)
\]
remains open [2507.09163].

Within the current constant-potential theory, the main conclusion is precise: under
\[
\lambda\le \delta\sqrt{V_1V_2},\qquad \delta\in(0,1),
\]
the system admits positive ground state solutions throughout the subcritical regime and in the two half-critical regimes under sufficiently large \(\mu\) or \(\nu\), while nontrivial solutions fail at the simultaneous critical endpoints. The combination of the Nehari--Pohozaev manifold, concentration--compactness, sharp HLS-type thresholds, and coupled Choquard regularity constitutes the core framework for this result [2507.09163].

Source: https://www.emergentmind.com/topics/linearly-coupled-kirchhoff-choquard-system