Nonlinear Fractional Choquard Equations

• Nonlinear fractional Choquard equations are nonlocal PDEs coupling a fractional Laplacian with a Hartree–Choquard convolution term, characterized by critical exponents and variational structures.
• The models integrate fractional diffusion, Riesz potential interactions, and critical phenomena such as Sobolev and Hardy exponents to address existence, multiplicity, and concentration of solutions.
• Advanced formulations include fractional p-Laplacian, weighted Kirchhoff effects, and singular terms, with applications extending to hyperbolic spaces and lattice graphs.

Nonlinear fractional Choquard equations form a class of nonlocal equations in which a fractional operator is coupled to a Hartree–Choquard convolution term. A representative model is

$$(-\Delta)^s u + u = (|x|^{-\mu} * F(u))\,f(u)\quad \text{in }\mathbb{R}^N,$$

while power-type forms include

$$(-\Delta)^s u + w\,u = (K_\alpha * |u|^p)\,|u|^{p-2}u,$$

with $K_\alpha(x)=|x|^{\alpha-N}$ or $I_\alpha(x)=c_{N,\alpha}|x|^{-(N-\alpha)}$ (Shen et al., 2014, d'Avenia et al., 2014). In contemporary work, the class includes fractional $p$-Laplacian and $p$-Kirchhoff operators, singular weights, critical Sobolev and Hardy terms, critical exponential nonlinearities, normalized constraints, magnetic perturbations, and analogues on hyperbolic space, groups of polynomial growth, and lattice graphs (Assunção et al., 2024, Song et al., 31 May 2025, Gupta et al., 2024, Li et al., 2022, Wang, 30 Jul 2025).

1. Canonical models and parameter regimes

The Euclidean whole-space formulation is usually built from a fractional diffusion term and a Riesz-type interaction. In the autonomous setting treated by Shen, Gao, and Yang, the equation is

$$(-\Delta)^s u + u = (|x|^{-\mu} * F(u))\,f(u)\quad \text{in }\mathbb{R}^N,$$

with $N\ge 3$, $s\in(0,1)$, and $\mu\in(0,N)$ (Shen et al., 2014). In the power case studied by D’Avenia, Siciliano, and Squassina, the stationary equation takes the form

$$(-\Delta)^s u + w\,u = (K_\alpha * |u|^p)\,|u|^{p-2}u,$$0

and the natural admissible range is

$$(-\Delta)^s u + w\,u = (K_\alpha * |u|^p)\,|u|^{p-2}u,$$1

This interval is dictated by the interaction between Hardy–Littlewood–Sobolev estimates and the fractional Sobolev embedding (d'Avenia et al., 2014).

A lower HLS-critical threshold occurs at

$$(-\Delta)^s u + w\,u = (K_\alpha * |u|^p)\,|u|^{p-2}u,$$2

which is the scale-invariant exponent for the nonlocal energy under the $$(-\Delta)^s u + w\,u = (K_\alpha * |u|^p)\,|u|^{p-2}u,$$3-preserving dilation. At this exponent, loss of compactness may occur, and the associated minimization problem requires concentration–compactness and nonlocal Brezis–Lieb splitting (Moroz et al., 2014). At the upper end, the Sobolev critical exponent changes with the order of the diffusion; in the fractional framework it is tied to $$(-\Delta)^s u + w\,u = (K_\alpha * |u|^p)\,|u|^{p-2}u,$$4 or to its weighted or quasilinear analogues (Mukherjee et al., 2016, d'Avenia et al., 2014).

Weighted and quasilinear formulations introduce further critical exponents. In the weighted fractional $$(-\Delta)^s u + w\,u = (K_\alpha * |u|^p)\,|u|^{p-2}u,$$5-Kirchhoff setting, the local critical term is governed by

$$(-\Delta)^s u + w\,u = (K_\alpha * |u|^p)\,|u|^{p-2}u,$$6

and the Choquard part is controlled by the lower and upper HLS exponents

$$(-\Delta)^s u + w\,u = (K_\alpha * |u|^p)\,|u|^{p-2}u,$$7

The nonlinearity is assumed to grow between these two thresholds (Assunção et al., 2024).

2. Operators, kernels, and functional settings

The class is unified by the coexistence of two nonlocal mechanisms: the diffusion operator and the Choquard interaction. The diffusion may be the linear fractional Laplacian, the fractional $$(-\Delta)^s u + w\,u = (K_\alpha * |u|^p)\,|u|^{p-2}u,$$8-Laplacian,

$$(-\Delta)^s u + w\,u = (K_\alpha * |u|^p)\,|u|^{p-2}u,$$9

or a weighted variant

$K_\alpha(x)=|x|^{\alpha-N}$0

with $K_\alpha(x)=|x|^{\alpha-N}$1 (Assunção et al., 2024, Sakuma, 2023).

The Choquard interaction is typically generated by a Riesz potential,

$K_\alpha(x)=|x|^{\alpha-N}$2

and enters through terms such as

$K_\alpha(x)=|x|^{\alpha-N}$3

or weighted variants involving $K_\alpha(x)=|x|^{\alpha-N}$4 (Assunção et al., 2024, Shen et al., 2014). In non-Euclidean settings, the same role is played by a Green kernel. On hyperbolic space, the convolution uses the kernel $K_\alpha(x)=|x|^{\alpha-N}$5 of $K_\alpha(x)=|x|^{\alpha-N}$6 (Gupta et al., 2024). On groups of polynomial growth and on lattice graphs, the kernel is the Green’s function $K_\alpha(x)=|x|^{\alpha-N}$7 of a discrete fractional Laplacian, with asymptotics comparable to a Riesz potential (Li et al., 2022, Wang, 30 Jul 2025).

The natural function spaces reflect the operator. Whole-space linear problems use $K_\alpha(x)=|x|^{\alpha-N}$8 (Shen et al., 2014, d'Avenia et al., 2014), quasilinear weighted problems use

$K_\alpha(x)=|x|^{\alpha-N}$9

with

$I_\alpha(x)=c_{N,\alpha}|x|^{-(N-\alpha)}$0

(Assunção et al., 2024). Hyperbolic problems are set in $I_\alpha(x)=c_{N,\alpha}|x|^{-(N-\alpha)}$1 with norm

$I_\alpha(x)=c_{N,\alpha}|x|^{-(N-\alpha)}$2

while lattice and graph problems use discrete analogues of fractional Sobolev or energy spaces adapted to the counting measure and graph kernels (Gupta et al., 2024, Li et al., 2022, Wang, 30 Jul 2025).

A further layer is added by Kirchhoff dependence. In that case the diffusion is multiplied by a norm-dependent coefficient $I_\alpha(x)=c_{N,\alpha}|x|^{-(N-\alpha)}$3 or $I_\alpha(x)=c_{N,\alpha}|x|^{-(N-\alpha)}$4, so the equation becomes nonlocal both through the operator and through the coefficient itself (Assunção et al., 2024, Chen, 2018).

3. Criticality, singularity, and nonlinear structures

The subject is organized by several distinct notions of criticality. One is HLS criticality for the convolution. In the bounded-domain Brezis–Nirenberg-type problem, the critical Choquard exponent is

$I_\alpha(x)=c_{N,\alpha}|x|^{-(N-\alpha)}$5

and the equation

$I_\alpha(x)=c_{N,\alpha}|x|^{-(N-\alpha)}$6

exhibits the same compactness breakdown as local critical problems, but now in a doubly nonlocal form (Mukherjee et al., 2016).

A second notion is combined local and nonlocal criticality. Sakuma studies a $I_\alpha(x)=c_{N,\alpha}|x|^{-(N-\alpha)}$7-fractional Choquard-type equation with a critical local term and “doubly critical” nonlocality, where

$I_\alpha(x)=c_{N,\alpha}|x|^{-(N-\alpha)}$8

with

$I_\alpha(x)=c_{N,\alpha}|x|^{-(N-\alpha)}$9

Here the nonlocal term simultaneously touches the lower and upper HLS critical powers, while the local term may also be critical through $p$0 (Sakuma, 2023).

A third regime replaces power criticality by exponential criticality. In the fractional $p$1-Choquard equation with $p$2, the nonlinearity is governed by a fractional Moser–Trudinger inequality,

$p$3

and the Choquard term is coupled to exponential growth rather than polynomial growth (Song et al., 31 May 2025). In one dimension, a logarithmic kernel produces the equation

$p$4

with critical exponential growth in $p$5 and a weighted space

$p$6

to control the logarithmic convolution (Böer et al., 2020).

Singular structures form another branch of the theory. The weighted fractional $p$7-Kirchhoff equation in $p$8 couples a local Hardy-type term, a local critical Sobolev term, and a generalized nonlocal Choquard term, all with singular weights (Assunção et al., 2024). On bounded domains, singular data may include a Hardy potential, a negative-power term $p$9, and a Radon measure together with a critical Choquard contribution,

$p$0

where

$p$1

The solution concept there is a positive SOLA, that is, a solution obtained as limit of approximations (Panda et al., 2020).

4. Variational formulation and compactness mechanisms

Most existence theories are variational. In the whole-space Berestycki–Lions framework, the functional is

$p$2

and critical points are weak solutions of the fractional Choquard equation (Shen et al., 2014). In the weighted fractional $p$3-Kirchhoff problem, the corresponding energy is

$p$4

again with Euler–Lagrange identity equivalent to the weak formulation (Assunção et al., 2024).

Two compactness obstructions recur. The first is critical growth, either Sobolev, HLS, or Moser–Trudinger. The second is lack of compactness in $p$5 because of translation invariance or concentration. Standard responses include concentration–compactness, nonlocal Brezis–Lieb splitting, and sub-threshold energy estimates (Mukherjee et al., 2016, Moroz et al., 2014, Bhattarai, 2016). In the non-autonomous fractional Choquard equation

$p$6

the asymptotic autonomous level $p$7 acts as a compactness threshold, and strict inequality $p$8 rules out bubbling at infinity (Chen et al., 2015).

The choice of compactness condition depends on the nonlinear structure. For fractional $p$9-Kirchhoff–Choquard equations without Ambrosetti–Rabinowitz, Cerami sequences are used instead of Palais–Smale sequences (Chen, 2018). The weighted singular Kirchhoff problem also replaces Palais–Smale by the Cerami condition and proves boundedness and strong convergence of Cerami sequences through uniform control of the convolution term and Simon-type inequalities (Assunção et al., 2024). By contrast, the critical exponential double-phase problem uses a penalized functional, a homeomorphism from an incomplete sphere to a nonsmooth Nehari-type set, and Ljusternik–Schnirelmann category theory (Song et al., 31 May 2025).

Several specialized devices recur in concrete models. The doubly weighted Stein–Weiss inequality controls singular Choquard integrals (Assunção et al., 2024). Penalization outside a potential well prevents mass leakage in semiclassical or concentration problems (Song et al., 31 May 2025, Ambrosio, 2018). Pohozaev identities serve both as compactness tools and as nonexistence criteria in critical settings (Mukherjee et al., 2016, Moroz et al., 2014). For normalized problems on the $$(-\Delta)^s u + u = (|x|^{-\mu} * F(u))\,f(u)\quad \text{in }\mathbb{R}^N,$$0-sphere, constrained minimization and strict subhomogeneity of the energy level exclude dichotomy (Chen et al., 31 Jul 2025, Bhattarai, 2016).

5. Existence, multiplicity, concentration, and normalization

Across the literature, the basic conclusions range from nontrivial weak solutions to positive ground states, multiplicity, concentration, normalized states, and convergence to limiting problems.

Setting	Representative conclusion	Paper
Weighted fractional $$(-\Delta)^s u + u = (|x|^{-\mu} * F(u))\,f(u)\quad \text{in }\mathbb{R}^N,$$1-Kirchhoff equation in $$(-\Delta)^s u + u = (|x|^{-\mu} * F(u))\,f(u)\quad \text{in }\mathbb{R}^N,$$2 with Hardy, critical Sobolev, and weighted Choquard terms	existence of a nontrivial weak solution for any $$(-\Delta)^s u + u = (|x|^{-\mu} * F(u))\,f(u)\quad \text{in }\mathbb{R}^N,$$3	(Assunção et al., 2024)
Fractional $$(-\Delta)^s u + u = (|x|^{-\mu} * F(u))\,f(u)\quad \text{in }\mathbb{R}^N,$$4-Choquard equation with exponential growth	at least $$(-\Delta)^s u + u = (|x|^{-\mu} * F(u))\,f(u)\quad \text{in }\mathbb{R}^N,$$5 positive weak solutions for $$(-\Delta)^s u + u = (|x|^{-\mu} * F(u))\,f(u)\quad \text{in }\mathbb{R}^N,$$6 small, concentrating near minima of $$(-\Delta)^s u + u = (|x|^{-\mu} * F(u))\,f(u)\quad \text{in }\mathbb{R}^N,$$7	(Song et al., 31 May 2025)
Non-autonomous fractional Choquard equation with $$(-\Delta)^s u + u = (|x|^{-\mu} * F(u))\,f(u)\quad \text{in }\mathbb{R}^N,$$8	existence of a positive ground state solution	(Chen et al., 2015)
Hyperbolic-space Choquard equation with $$(-\Delta)^s u + u = (|x|^{-\mu} * F(u))\,f(u)\quad \text{in }\mathbb{R}^N,$$9	existence of a positive ground state	(Gupta et al., 2024)
Normalized fractional Choquard equation with two Hartree powers and potentials	at least a pair of weak normalized solutions under small $N\ge 3$0, with an $N\ge 3$1-critical mass threshold in the critical case	(Chen et al., 31 Jul 2025)
Fractional magnetic Choquard equation	a nontrivial weak solution for $N\ge 3$2 small, with concentration near $N\ge 3$3	(Ambrosio, 2018)

Other settings extend the same pattern. On bounded domains at the HLS critical exponent, existence holds for all $N\ge 3$4 when $N\ge 3$5, while for $N\ge 3$6 the existence theory requires $N\ge 3$7 sufficiently large and not an eigenvalue; multiplicity is obtained through genus and pseudo-index methods (Mukherjee et al., 2016). On groups of polynomial growth, the discrete Choquard equation

$N\ge 3$8

has a ground state for all sufficiently large $N\ge 3$9, and the ground states converge, as $s\in(0,1)$0, to a ground state of a limiting Dirichlet problem on the potential well (Li et al., 2022). On lattice graphs, a fractional $s\in(0,1)$1-Laplacian Choquard equation admits a strictly positive mountain-pass solution under growth assumptions, and a positive ground state under an additional monotonicity condition (Wang, 30 Jul 2025).

Normalized and dynamical perspectives produce further conclusions. For the fractional Schrödinger–Choquard equation,

$s\in(0,1)$2

ground states arise by constrained minimization at fixed mass, every minimizing sequence is relatively compact up to translation, and the set of minimizers is orbitally stable under the flow, assuming well-posedness and conservation of mass and energy (Bhattarai, 2016).

6. Qualitative properties, nonexistence, and broader geometry

Once existence is established, qualitative analysis depends strongly on the setting. In the Euclidean autonomous power case, ground states are positive, radially symmetric, and radially decreasing; they satisfy $s\in(0,1)$3, Hölder or $s\in(0,1)$4 regularity according to $s\in(0,1)$5, and, when $s\in(0,1)$6, the asymptotic law

$s\in(0,1)$7

The same work proves nonexistence for the zero-potential problem unless $s\in(0,1)$8, and classifies fixed-sign solutions in the special case $s\in(0,1)$9 and $\mu\in(0,N)$0 by the bubble profile

$\mu\in(0,N)$1

It also derives infinitely many radial solutions and, in certain dimensions, infinitely many nonradial solutions (d'Avenia et al., 2014).

For general nonlinearities, positivity and symmetry can still be recovered under sign assumptions. If $\mu\in(0,N)$2 is odd and nonnegative on $\mu\in(0,N)$3, then any ground state of

$\mu\in(0,N)$4

is strictly positive, radially symmetric with respect to some point, and radially nonincreasing (Shen et al., 2014). On hyperbolic space, positive ground states are radially symmetric and radially nonincreasing with respect to the hyperbolic distance from some center, and regularity theory yields $\mu\in(0,N)$5 bootstrapping and, under additional assumptions, $\mu\in(0,N)$6 regularity (Gupta et al., 2024).

Nonexistence results are usually tied to Pohozaev identities or virial conditions. For the bounded-domain HLS-critical fractional Choquard problem, a Pohozaev identity implies that if $\mu\in(0,N)$7 and the domain is strictly star-shaped, then there is no nontrivial nonnegative solution (Mukherjee et al., 2016). At the HLS lower critical exponent for the local Choquard equation with variable potential,

$\mu\in(0,N)$8

necessary conditions involve the behavior of $\mu\in(0,N)$9 at infinity and the virial quantity $$(-\Delta)^s u + w\,u = (K_\alpha * |u|^p)\,|u|^{p-2}u,$$00; a sufficient existence condition is

$$(-\Delta)^s u + w\,u = (K_\alpha * |u|^p)\,|u|^{p-2}u,$$01

while Hardy-based bounds yield nonexistence under opposite assumptions (Moroz et al., 2014).

The broader geometry of the subject includes low-regularity and non-Euclidean models. The singular critical Choquard problem with Hardy potential and Radon measure admits a positive SOLA in $$(-\Delta)^s u + w\,u = (K_\alpha * |u|^p)\,|u|^{p-2}u,$$02 for every $$(-\Delta)^s u + w\,u = (K_\alpha * |u|^p)\,|u|^{p-2}u,$$03 and $$(-\Delta)^s u + w\,u = (K_\alpha * |u|^p)\,|u|^{p-2}u,$$04 under small $$(-\Delta)^s u + w\,u = (K_\alpha * |u|^p)\,|u|^{p-2}u,$$05 and coercivity conditions (Panda et al., 2020). The logarithmic one-dimensional equation admits a mountain pass solution, a ground state, and, in the subcritical exponential case, infinitely many solutions via genus theory (Böer et al., 2020). Recent semiclassical and double-phase works identify open directions that include sign-changing solutions, improvements of convergence rates, other ranges of $$(-\Delta)^s u + w\,u = (K_\alpha * |u|^p)\,|u|^{p-2}u,$$06, different potential landscapes, bounded domains, and relaxation of small-parameter assumptions (Song et al., 31 May 2025).

In aggregate, the nonlinear fractional Choquard equation is not a single model but a family of doubly nonlocal problems whose analytic structure is determined by the interaction of fractional diffusion, convolution criticality, and the ambient geometry. The existing literature shows that the decisive ingredients are the precise exponent regime, the compactness mechanism available in the underlying space, and the way additional structures—Kirchhoff coefficients, weights, magnetic phases, normalized constraints, or curvature—alter the variational balance (Assunção et al., 2024, Chen, 2018, Gupta et al., 2024).