Fractional Kirchhoff–Choquard System

• Fractional Kirchhoff–Choquard systems are nonlocal elliptic problems coupling fractional diffusion, energy-modulated Kirchhoff coefficients, and Hartree-type convolution interactions.
• They employ variational methods, fractional Sobolev embeddings, and concentration–compactness to tackle challenges of critical growth and lack of compactness.
• The framework extends to Euclidean and Heisenberg-group settings, including weighted, singular, variable-exponent, and magnetic regimes for broader applications.

Searching arXiv for recent and foundational papers on fractional Kirchhoff–Choquard systems. A fractional Kirchhoff–Choquard system is a class of doubly nonlocal elliptic problems in which the principal operator is modulated by a Kirchhoff coefficient depending on a global energy, while the reaction term contains a Choquard or Hartree convolution with a Riesz-type kernel. In the fractional setting, the diffusion is governed by a fractional Laplacian, a fractional $p$-Laplacian, a weighted fractional $p$-Laplacian, or variable-order fractional operators; in several works the model is studied either as a single equation or as a genuinely coupled system for $(u,v)$. Across these variants, the common structure is the coexistence of nonlocal diffusion, nonlocal global coupling through the Kirchhoff term, and nonlocal long-range interaction through the Choquard term, often at Hardy–Littlewood–Sobolev critical growth (Chen, 2018, Bai et al., 2024, Goel et al., 9 Sep 2025).

1. Defining structure and canonical forms

The defining Kirchhoff feature is that the coefficient multiplying the principal operator depends on a global norm or seminorm of the solution. In the whole-space fractional $p$-Kirchhoff–Choquard equation, the model takes the form

$$M\!\left(\|u\|_{W}^{p}\right)(-\Delta)^s_p u + V(x)|u|^{p-2}u = \lambda \,(I_\mu * F(u))\, f(u) \quad \text{in } \mathbb{R}^N,$$

with

$$\|u\|_{W}^{p} = [u]_{s,p}^{p} + \int_{\mathbb{R}^N} V(x)|u|^p\,dx, \qquad (I_\mu * F(u))(x)=\int_{\mathbb{R}^N} \frac{F(u(y))}{|x-y|^\mu}\,dy$$

(Chen, 2018). This formulation exhibits the standard threefold nonlocality: the fractional operator, the energy-dependent Kirchhoff coefficient, and the Choquard convolution.

A critical Heisenberg-group variant replaces Euclidean geometry by $\mathbb H^N$ and the Euclidean Riesz kernel by the group kernel $|\eta^{-1}\xi|^{-\lambda}$. The equation studied there is

$$M\!\left(\|u\|_\mu^{p}\right) \Big(\mu\,(-\Delta)^s_p u + V(\xi)|u|^{p-2}u\Big) = f(\xi,u) + \left(\int_{\mathbb H^N}\frac{|u(\eta)|^{Q_\lambda^*}}{|\eta^{-1}\xi|^\lambda}\,d\eta\right) |u(\xi)|^{Q_\lambda^*-2}u(\xi),$$

with

$$\|u\|_\mu^p = \mu [u]_{s,p}^p+\int_{\mathbb H^N}V(\xi)|u|^p\,d\xi$$

(Bai et al., 2024).

A normalized coupled system on $p$0 introduces two unknowns and $p$1-mass constraints: $p$2 with $p$3, $p$4, and $p$5 as Lagrange multipliers (Goel et al., 9 Sep 2025).

These models admit several extensions. Weighted singular versions combine Choquard terms with Hardy-type and Sobolev-critical singular weights (Assunção et al., 2024). Variable-exponent and variable-order formulations replace constant $p$6, $p$7, and $p$8 by spatially dependent exponents and kernels (Bahrouni et al., 2020, Biswas et al., 2020). Exponential-growth versions replace polynomial reaction terms by Trudinger–Moser type nonlinearities (Goyal et al., 2019). This suggests that the term “fractional Kirchhoff–Choquard system” is best understood as a structural category rather than a single canonical equation.

2. Fractional operators, Kirchhoff coefficients, and Choquard interactions

The fractional diffusion component is represented in several inequivalent but related ways. In the constant-exponent fractional $p$9-Kirchhoff theory, the operator is the fractional $(u,v)$0-Laplacian

$(u,v)$1

with Gagliardo seminorm

$(u,v)$2

(Chen, 2018). On the Heisenberg group, the analogous nonlocal seminorm uses $(u,v)$3 and the homogeneous dimension $(u,v)$4 (Bai et al., 2024). In weighted singular problems, the operator becomes

$(u,v)$5

(Assunção et al., 2024). In variable-order settings, the principal term is

$(u,v)$6

(Biswas et al., 2020).

The Kirchhoff coefficient is the second defining ingredient. Several prototypes recur: nondegenerate lower-bounded coefficients such as $(u,v)$7 or $(u,v)$8 (Chen, 2018, Goel et al., 9 Sep 2025); power-type degenerate coefficients such as $(u,v)$9, which satisfy $p$0 and create additional analytical difficulties (Rawat et al., 2021); and generalized coefficients controlled by inequalities of the form

$p$1

or

$p$2

(Chen, 2018, Assunção et al., 2024). In all of these cases, the diffusion strength depends on the total energy rather than pointwise data, which is the classical Kirchhoff effect in a fractional nonlocal setting.

The Choquard term introduces long-range interaction. In Euclidean models it is usually written with the Riesz potential $p$3, $p$4, so that

$p$5

(Goel et al., 9 Sep 2025). The resulting nonlinearities include generalized Hartree terms $p$6 (Chen, 2018), critical terms of the form

$p$7

(Goel et al., 2022), and coupled cross-interaction terms involving both $p$8 and $p$9 (Goel et al., 9 Sep 2025). The Heisenberg-group analogue replaces $$M\!\left(\|u\|_{W}^{p}\right)(-\Delta)^s_p u + V(x)|u|^{p-2}u = \lambda \,(I_\mu * F(u))\, f(u) \quad \text{in } \mathbb{R}^N,$$0 by $$M\!\left(\|u\|_{W}^{p}\right)(-\Delta)^s_p u + V(x)|u|^{p-2}u = \lambda \,(I_\mu * F(u))\, f(u) \quad \text{in } \mathbb{R}^N,$$1 and the Euclidean critical exponent by the corresponding HLS critical exponent on $$M\!\left(\|u\|_{W}^{p}\right)(-\Delta)^s_p u + V(x)|u|^{p-2}u = \lambda \,(I_\mu * F(u))\, f(u) \quad \text{in } \mathbb{R}^N,$$2 (Bai et al., 2024).

3. Critical exponents, geometry, and noncompactness

A central organizing principle is the distinction between subcritical and critical Choquard growth. In the fractional Euclidean setting, the upper Hardy–Littlewood–Sobolev critical exponent is

$$M\!\left(\|u\|_{W}^{p}\right)(-\Delta)^s_p u + V(x)|u|^{p-2}u = \lambda \,(I_\mu * F(u))\, f(u) \quad \text{in } \mathbb{R}^N,$$3

while the lower exponent is

$$M\!\left(\|u\|_{W}^{p}\right)(-\Delta)^s_p u + V(x)|u|^{p-2}u = \lambda \,(I_\mu * F(u))\, f(u) \quad \text{in } \mathbb{R}^N,$$4

(Goel et al., 9 Sep 2025). These exponents determine whether the convolution term lies in a compact regime or at the threshold where concentration phenomena occur.

For bounded-domain problems with scalar unknown, the critical Choquard exponent is again

$$M\!\left(\|u\|_{W}^{p}\right)(-\Delta)^s_p u + V(x)|u|^{p-2}u = \lambda \,(I_\mu * F(u))\, f(u) \quad \text{in } \mathbb{R}^N,$$5

and the associated term is critical in the sense of the Hardy–Littlewood–Sobolev inequality (Goel et al., 2022, Rawat et al., 2021). In the Heisenberg-group framework, the critical exponent is written

$$M\!\left(\|u\|_{W}^{p}\right)(-\Delta)^s_p u + V(x)|u|^{p-2}u = \lambda \,(I_\mu * F(u))\, f(u) \quad \text{in } \mathbb{R}^N,$$6

and marks the critical threshold for the fractional Sobolev embedding on $$M\!\left(\|u\|_{W}^{p}\right)(-\Delta)^s_p u + V(x)|u|^{p-2}u = \lambda \,(I_\mu * F(u))\, f(u) \quad \text{in } \mathbb{R}^N,$$7 (Bai et al., 2024).

Weighted singular models introduce a weighted critical Sobolev exponent

$$M\!\left(\|u\|_{W}^{p}\right)(-\Delta)^s_p u + V(x)|u|^{p-2}u = \lambda \,(I_\mu * F(u))\, f(u) \quad \text{in } \mathbb{R}^N,$$8

and also weighted Choquard windows

$$M\!\left(\|u\|_{W}^{p}\right)(-\Delta)^s_p u + V(x)|u|^{p-2}u = \lambda \,(I_\mu * F(u))\, f(u) \quad \text{in } \mathbb{R}^N,$$9

(Assunção et al., 2024). Variable-exponent settings replace these fixed exponents by pointwise critical bounds $$\|u\|_{W}^{p} = [u]_{s,p}^{p} + \int_{\mathbb{R}^N} V(x)|u|^p\,dx, \qquad (I_\mu * F(u))(x)=\int_{\mathbb{R}^N} \frac{F(u(y))}{|x-y|^\mu}\,dy$$0 and variable HLS balance conditions (Bahrouni et al., 2020, Biswas et al., 2020).

The analytic consequence of criticality is loss of compactness. Several papers state this explicitly: standard embeddings fail to be compact at the critical Sobolev or HLS level (Bai et al., 2024, Assunção et al., 2024, Goel et al., 2022). On unbounded domains, translation invariance and escape of mass add a second source of noncompactness (Chen, 2018, Assunção et al., 2024). A plausible implication is that the Kirchhoff coefficient intensifies the difficulty, because weak convergence of $$\|u\|_{W}^{p} = [u]_{s,p}^{p} + \int_{\mathbb{R}^N} V(x)|u|^p\,dx, \qquad (I_\mu * F(u))(x)=\int_{\mathbb{R}^N} \frac{F(u(y))}{|x-y|^\mu}\,dy$$1 alone does not directly control the nonlinear factor $$\|u\|_{W}^{p} = [u]_{s,p}^{p} + \int_{\mathbb{R}^N} V(x)|u|^p\,dx, \qquad (I_\mu * F(u))(x)=\int_{\mathbb{R}^N} \frac{F(u(y))}{|x-y|^\mu}\,dy$$2; this point is stated directly in the bounded-domain critical Choquard analysis (Goel et al., 2022).

4. Variational formulation and principal analytical tools

The dominant framework is variational. The energy functional usually has a Kirchhoff part minus a Choquard part, with additional local terms when present. For the whole-space generalized model,

$$\|u\|_{W}^{p} = [u]_{s,p}^{p} + \int_{\mathbb{R}^N} V(x)|u|^p\,dx, \qquad (I_\mu * F(u))(x)=\int_{\mathbb{R}^N} \frac{F(u(y))}{|x-y|^\mu}\,dy$$3

(Chen, 2018). For the Heisenberg-group critical problem,

$$\|u\|_{W}^{p} = [u]_{s,p}^{p} + \int_{\mathbb{R}^N} V(x)|u|^p\,dx, \qquad (I_\mu * F(u))(x)=\int_{\mathbb{R}^N} \frac{F(u(y))}{|x-y|^\mu}\,dy$$4

(Bai et al., 2024). For the normalized two-component system,

$$\|u\|_{W}^{p} = [u]_{s,p}^{p} + \int_{\mathbb{R}^N} V(x)|u|^p\,dx, \qquad (I_\mu * F(u))(x)=\int_{\mathbb{R}^N} \frac{F(u(y))}{|x-y|^\mu}\,dy$$5

on the constraint manifold

$$\|u\|_{W}^{p} = [u]_{s,p}^{p} + \int_{\mathbb{R}^N} V(x)|u|^p\,dx, \qquad (I_\mu * F(u))(x)=\int_{\mathbb{R}^N} \frac{F(u(y))}{|x-y|^\mu}\,dy$$6

(Goel et al., 9 Sep 2025).

The principal tools are recurrent across the literature. The Hardy–Littlewood–Sobolev inequality controls the convolution term in essentially every formulation (Chen, 2018, Goel et al., 2022, Rawat et al., 2021). When singular weights are present, a doubly weighted Stein–Weiss inequality plays the same role (Assunção et al., 2024). Fractional Sobolev embeddings provide the local compactness input in subcritical regimes (Chen, 2018), while concentration–compactness handles critical loss of mass and concentration at points or at infinity (Bai et al., 2024, Goel et al., 2022, Liang et al., 2020). Mountain-pass geometry is the standard existence mechanism for nontrivial critical points (Chen, 2018, Assunção et al., 2024). Symmetry-based multiplicity uses Krasnosel’skii genus, Benci pseudo-index, Fountain theorem, or Dual Fountain theorem, depending on the parity structure and the functional setting (Bai et al., 2024, Biswas et al., 2020, Liang et al., 2020).

Several papers replace the Palais–Smale condition by the Cerami condition, especially when growth is slower or compactness is weak. This occurs in the generalized whole-space equation without Ambrosetti–Rabinowitz (Chen, 2018) and in the weighted singular critical problem on $$\|u\|_{W}^{p} = [u]_{s,p}^{p} + \int_{\mathbb{R}^N} V(x)|u|^p\,dx, \qquad (I_\mu * F(u))(x)=\int_{\mathbb{R}^N} \frac{F(u(y))}{|x-y|^\mu}\,dy$$7 (Assunção et al., 2024). In normalized problems, the Pohozaev identity and Pohozaev manifold are central: $$\|u\|_{W}^{p} = [u]_{s,p}^{p} + \int_{\mathbb{R}^N} V(x)|u|^p\,dx, \qquad (I_\mu * F(u))(x)=\int_{\mathbb{R}^N} \frac{F(u(y))}{|x-y|^\mu}\,dy$$8 (Goel et al., 9 Sep 2025). In singular problems, truncation is required because the functional is not $$\|u\|_{W}^{p} = [u]_{s,p}^{p} + \int_{\mathbb{R}^N} V(x)|u|^p\,dx, \qquad (I_\mu * F(u))(x)=\int_{\mathbb{R}^N} \frac{F(u(y))}{|x-y|^\mu}\,dy$$9; the regularized problem

$\mathbb H^N$0

is introduced precisely for that purpose (Rawat et al., 2021).

5. Existence, multiplicity, and normalized solutions

The existence theory spans subcritical, critical, singular, weighted, and constrained regimes. In the generalized whole-space fractional $\mathbb H^N$1-Kirchhoff equation with Choquard nonlinearity, there is a nontrivial weak solution for every $\mathbb H^N$2 under assumptions $\mathbb H^N$3, $\mathbb H^N$4, $\mathbb H^N$5, and $\mathbb H^N$6–$\mathbb H^N$7, without assuming the Ambrosetti–Rabinowitz condition (Chen, 2018). A similar all-$\mathbb H^N$8 existence theorem holds for the weighted critical singular equation in $\mathbb H^N$9: if $|\eta^{-1}\xi|^{-\lambda}$0 and $|\eta^{-1}\xi|^{-\lambda}$1, $|\eta^{-1}\xi|^{-\lambda}$2-$|\eta^{-1}\xi|^{-\lambda}$3, $|\eta^{-1}\xi|^{-\lambda}$4-$|\eta^{-1}\xi|^{-\lambda}$5 hold, then the problem admits at least one nontrivial weak solution for every $|\eta^{-1}\xi|^{-\lambda}$6 (Assunção et al., 2024).

Critical bounded-domain Choquard problems yield richer multiplicity patterns. For

$|\eta^{-1}\xi|^{-\lambda}$7

the bounded-domain fractional Kirchhoff–Choquard problem has at least one positive solution for small $|\eta^{-1}\xi|^{-\lambda}$8 when $|\eta^{-1}\xi|^{-\lambda}$9, at least one positive solution for large $$M\!\left(\|u\|_\mu^{p}\right) \Big(\mu\,(-\Delta)^s_p u + V(\xi)|u|^{p-2}u\Big) = f(\xi,u) + \left(\int_{\mathbb H^N}\frac{|u(\eta)|^{Q_\lambda^*}}{|\eta^{-1}\xi|^\lambda}\,d\eta\right) |u(\xi)|^{Q_\lambda^*-2}u(\xi),$$0 when $$M\!\left(\|u\|_\mu^{p}\right) \Big(\mu\,(-\Delta)^s_p u + V(\xi)|u|^{p-2}u\Big) = f(\xi,u) + \left(\int_{\mathbb H^N}\frac{|u(\eta)|^{Q_\lambda^*}}{|\eta^{-1}\xi|^\lambda}\,d\eta\right) |u(\xi)|^{Q_\lambda^*-2}u(\xi),$$1, at least two positive solutions for sufficiently large $$M\!\left(\|u\|_\mu^{p}\right) \Big(\mu\,(-\Delta)^s_p u + V(\xi)|u|^{p-2}u\Big) = f(\xi,u) + \left(\int_{\mathbb H^N}\frac{|u(\eta)|^{Q_\lambda^*}}{|\eta^{-1}\xi|^\lambda}\,d\eta\right) |u(\xi)|^{Q_\lambda^*-2}u(\xi),$$2 in the superlinear case $$M\!\left(\|u\|_\mu^{p}\right) \Big(\mu\,(-\Delta)^s_p u + V(\xi)|u|^{p-2}u\Big) = f(\xi,u) + \left(\int_{\mathbb H^N}\frac{|u(\eta)|^{Q_\lambda^*}}{|\eta^{-1}\xi|^\lambda}\,d\eta\right) |u(\xi)|^{Q_\lambda^*-2}u(\xi),$$3, and at least two positive solutions in the concave regime $$M\!\left(\|u\|_\mu^{p}\right) \Big(\mu\,(-\Delta)^s_p u + V(\xi)|u|^{p-2}u\Big) = f(\xi,u) + \left(\int_{\mathbb H^N}\frac{|u(\eta)|^{Q_\lambda^*}}{|\eta^{-1}\xi|^\lambda}\,d\eta\right) |u(\xi)|^{Q_\lambda^*-2}u(\xi),$$4 under the stated restrictions on $$M\!\left(\|u\|_\mu^{p}\right) \Big(\mu\,(-\Delta)^s_p u + V(\xi)|u|^{p-2}u\Big) = f(\xi,u) + \left(\int_{\mathbb H^N}\frac{|u(\eta)|^{Q_\lambda^*}}{|\eta^{-1}\xi|^\lambda}\,d\eta\right) |u(\xi)|^{Q_\lambda^*-2}u(\xi),$$5 (Goel et al., 2022). For the degenerate singular critical problem with prototype $$M\!\left(\|u\|_\mu^{p}\right) \Big(\mu\,(-\Delta)^s_p u + V(\xi)|u|^{p-2}u\Big) = f(\xi,u) + \left(\int_{\mathbb H^N}\frac{|u(\eta)|^{Q_\lambda^*}}{|\eta^{-1}\xi|^\lambda}\,d\eta\right) |u(\xi)|^{Q_\lambda^*-2}u(\xi),$$6, there exists $$M\!\left(\|u\|_\mu^{p}\right) \Big(\mu\,(-\Delta)^s_p u + V(\xi)|u|^{p-2}u\Big) = f(\xi,u) + \left(\int_{\mathbb H^N}\frac{|u(\eta)|^{Q_\lambda^*}}{|\eta^{-1}\xi|^\lambda}\,d\eta\right) |u(\xi)|^{Q_\lambda^*-2}u(\xi),$$7 such that for every $$M\!\left(\|u\|_\mu^{p}\right) \Big(\mu\,(-\Delta)^s_p u + V(\xi)|u|^{p-2}u\Big) = f(\xi,u) + \left(\int_{\mathbb H^N}\frac{|u(\eta)|^{Q_\lambda^*}}{|\eta^{-1}\xi|^\lambda}\,d\eta\right) |u(\xi)|^{Q_\lambda^*-2}u(\xi),$$8 the problem has at least two distinct positive weak solutions, and every weak solution is bounded and belongs to $$M\!\left(\|u\|_\mu^{p}\right) \Big(\mu\,(-\Delta)^s_p u + V(\xi)|u|^{p-2}u\Big) = f(\xi,u) + \left(\int_{\mathbb H^N}\frac{|u(\eta)|^{Q_\lambda^*}}{|\eta^{-1}\xi|^\lambda}\,d\eta\right) |u(\xi)|^{Q_\lambda^*-2}u(\xi),$$9 when $$\|u\|_\mu^p = \mu [u]_{s,p}^p+\int_{\mathbb H^N}V(\xi)|u|^p\,d\xi$$0 (Rawat et al., 2021).

The Heisenberg-group critical equation exhibits a parameter-dependent dichotomy. In the critical case $$\|u\|_\mu^p = \mu [u]_{s,p}^p+\int_{\mathbb H^N}V(\xi)|u|^p\,d\xi$$1, the problem has infinitely many solutions for $$\|u\|_\mu^p = \mu [u]_{s,p}^p+\int_{\mathbb H^N}V(\xi)|u|^p\,d\xi$$2 sufficiently large by means of the Krasnosel’skii genus theorem and a Palais–Smale condition above a threshold

$$\|u\|_\mu^p = \mu [u]_{s,p}^p+\int_{\mathbb H^N}V(\xi)|u|^p\,d\xi$$3

(Bai et al., 2024). In the subcritical case $$\|u\|_\mu^p = \mu [u]_{s,p}^p+\int_{\mathbb H^N}V(\xi)|u|^p\,d\xi$$4, for every $$\|u\|_\mu^p = \mu [u]_{s,p}^p+\int_{\mathbb H^N}V(\xi)|u|^p\,d\xi$$5 there exists $$\|u\|_\mu^p = \mu [u]_{s,p}^p+\int_{\mathbb H^N}V(\xi)|u|^p\,d\xi$$6 such that for all $$\|u\|_\mu^p = \mu [u]_{s,p}^p+\int_{\mathbb H^N}V(\xi)|u|^p\,d\xi$$7, the problem has at least $$\|u\|_\mu^p = \mu [u]_{s,p}^p+\int_{\mathbb H^N}V(\xi)|u|^p\,d\xi$$8 pairs of solutions (Bai et al., 2024).

Normalized solutions introduce a different existence paradigm. For the coupled constrained system with $$\|u\|_\mu^p = \mu [u]_{s,p}^p+\int_{\mathbb H^N}V(\xi)|u|^p\,d\xi$$9, the $p$00-subcritical case

$p$01

yields a normalized ground state with negative energy

$p$02

negative multipliers $p$03, and positivity plus radial monotonicity of the solution (Goel et al., 9 Sep 2025). In the $p$04-supercritical case

$p$05

the solution is obtained by a mountain-pass argument at a positive energy level

$p$06

(Goel et al., 9 Sep 2025).

Odd nonlinearities and radial symmetry frequently strengthen the results. In the variable-order problem without Ambrosetti–Rabinowitz, oddness of $p$07 yields a sequence of nontrivial weak solutions with energies tending to $p$08 by the Fountain theorem and another sequence with negative critical values converging to $p$09 by the Dual Fountain theorem (Biswas et al., 2020). In the variable-exponent whole-space problem, radial symmetry is used to obtain a nontrivial radial weak solution (Bahrouni et al., 2020).

6. Extensions, special regimes, and conceptual scope

The literature shows that fractional Kirchhoff–Choquard theory is not limited to the standard scalar polynomial model. One direction is geometric generalization: the Heisenberg-group equation replaces Euclidean scaling by sub-Riemannian homogeneous structure, changing the critical exponent to $p$10 and requiring a fractional concentration–compactness principle on $p$11 (Bai et al., 2024). Another direction is weighted singularity: the whole-space weighted equation combines a Hardy-type potential, a Sobolev-critical weighted local term, and a weighted Choquard convolution term with critical singular weights (Assunção et al., 2024).

A further extension concerns the choice of operator and growth law. Variable-exponent and variable-order models employ spaces $p$12 and $p$13, together with variable HLS inequalities and compact radial embeddings of Strauss or Lions type (Bahrouni et al., 2020, Biswas et al., 2020). Exponential-growth problems replace algebraic superlinearity by nonlinearities behaving like $p$14, and the key compactness threshold is then governed by the fractional Trudinger–Moser inequality rather than a polynomial Sobolev exponent (Goyal et al., 2019). Magnetic and semiclassical variants add a fractional magnetic operator $p$15, a magnetic Gagliardo seminorm $p$16, and small-$p$17 concentration phenomena, while retaining the Kirchhoff–Choquard structure (Liang et al., 2020).

Several recurring misconceptions are corrected by the published results. First, the subject is not confined to nondegenerate Kirchhoff coefficients; degenerate cases with $p$18 are treated explicitly in both scalar singular problems and magnetic semiclassical problems (Rawat et al., 2021, Liang et al., 2020). Second, Ambrosetti–Rabinowitz is not structurally necessary for existence theory: both constant-exponent and variable-order models establish existence and multiplicity without that assumption, replacing it with superlinearity and monotonicity conditions tailored to Cerami compactness or Nehari geometry (Chen, 2018, Biswas et al., 2020). Third, “system” need not mean only two-component coupling. Some papers use a genuinely coupled pair $p$19 (Goel et al., 9 Sep 2025), while others study a single equation with multiple interacting nonlocal mechanisms; a plausible implication is that the term has acquired a broader structural meaning in the recent literature.

Taken together, these works define fractional Kirchhoff–Choquard systems as a research area centered on three analytic themes: nonlocal diffusion, global Kirchhoff coupling, and nonlocal HLS-type interaction. The principal technical questions are compactness at critical growth, the effect of degeneracy in $p$20, and the construction of critical points under symmetry, mass constraints, singularity, or weighted geometry. The available results cover existence, ground states, multiplicity, regularity, normalized solutions, and semiclassical states across Euclidean, weighted, magnetic, variable-exponent, and Heisenberg-group settings (Goel et al., 2022, Rawat et al., 2021, Goel et al., 9 Sep 2025).