Critical Choquard Equation Overview

• Critical Choquard equations are nonlocal elliptic problems defined via convolution with Riesz or logarithmic kernels, exhibiting threshold criticality dictated by Hardy–Littlewood–Sobolev inequalities.
• Their analysis employs variational methods, explicit bubble profiles, and concentration-compactness techniques to overcome challenges from scaling invariance and loss of compactness.
• Extensions to fractional, polyharmonic, and normalized settings reveal rich structures with applications in semiclassical analysis, ground state classification, and critical phenomenon exploration.

Critical Choquard equations are nonlocal elliptic equations in which the nonlinear term is generated by convolution with a Riesz, logarithmic, or related kernel, and the exponent is critical either in the sense of the Hardy–Littlewood–Sobolev inequality or, in limiting Sobolev settings, in the sense of critical exponential growth. A representative model is

$-\Delta u+V u=(I_\alpha*|u|^p)|u|^{p-2}u,$

with lower critical exponent $p=\frac{N+\alpha}{N}$ and upper critical exponent $p=\frac{N+\alpha}{N-2}$. Higher-order conformally invariant counterparts include

$$(-\Delta)^{\frac n2}u(x)=\left(\int_{\mathbb R^n}\frac{e^{\frac{2n-\mu}{2}u(y)}}{|x-y|^\mu}\,dy\right)e^{\frac{2n-\mu}{2}u(x)}.$$

Across these settings, the theory combines classification of entire solutions, variational existence of ground states, threshold non-existence, multiplicity, and concentration analysis in autonomous, perturbed, semiclassical, fractional, and higher-order regimes (Moroz et al., 2014, Li et al., 2018, Huang et al., 2023).

1. Criticality and canonical formulations

The adjective “critical” refers to threshold exponents dictated by the Hardy–Littlewood–Sobolev inequality. For the Choquard energy

$\int_{\mathbb R^N}(I_\alpha*|u|^p)|u|^p\,dx,$

the lower critical exponent is $p=\frac{N+\alpha}{N}$, while the upper critical exponent is $p=\frac{N+\alpha}{N-2}$. In the lower-critical case, the convolution integral is invariant under translations and certain scalings, and a loss of compactness may occur; in the upper-critical case, the nonlocal term saturates the Sobolev embedding dictated by the Laplacian (Moroz et al., 2014, Liu et al., 2024).

A parallel terminology is used for more elaborate nonlinearities. In the doubly critical case one takes

$$F(u)=|u|^p+|u|^q,\qquad p=\frac{N+\alpha}{N},\quad q=\frac{N+\alpha}{N-2},$$

so that both critical Hardy–Littlewood–Sobolev thresholds are present simultaneously. This produces two distinct scaling mechanisms and a correspondingly delicate compactness problem (Seok, 2017).

In limiting Sobolev regimes, criticality may be exponential rather than polynomial. This occurs in the one-dimensional fractional logarithmic Choquard equation

$$(-\Delta)^{\frac12}u+a u+\lambda (\ln|\cdot|*|u|^2)u=f(u)\quad \text{in }\mathbb R,$$

and in zero-mass weighted $N$-Laplacian problems in $p=\frac{N+\alpha}{N}$0, where the admissible growth is controlled by Moser–Trudinger-type inequalities rather than power-type Sobolev embeddings (Böer et al., 2020, Romani, 2024).

Regime	Critical expression	Representative sources
Lower HLS criticality	$p=\frac{N+\alpha}{N}$1	(Moroz et al., 2014, Li et al., 2022)
Upper HLS criticality	$p=\frac{N+\alpha}{N}$2, $p=\frac{N+\alpha}{N}$3	(Li et al., 2018, Gao et al., 2017)
Doubly criticality	$p=\frac{N+\alpha}{N}$4 with both critical exponents	(Seok, 2017)
Critical exponential growth	Moser–Trudinger-type growth	(Böer et al., 2020, Romani, 2024)

2. Explicit solutions, bubbles, and classification

A recurrent feature of critical Choquard theory is the appearance of explicit bubble profiles. In the Hardy–Littlewood–Sobolev critical minimization problem,

$p=\frac{N+\alpha}{N}$5

the infimum is attained by

$p=\frac{N+\alpha}{N}$6

for constants $p=\frac{N+\alpha}{N}$7, $p=\frac{N+\alpha}{N}$8, and $p=\frac{N+\alpha}{N}$9. These functions serve both as exact optimizers and as the basic concentration profiles in blow-up and semiclassical analysis (Moroz et al., 2014).

For the higher-order critical Choquard equation in $p=\frac{N+\alpha}{N-2}$0,

$p=\frac{N+\alpha}{N-2}$1

Huang and Niu proved a full classification under the assumptions $p=\frac{N+\alpha}{N-2}$2, $p=\frac{N+\alpha}{N-2}$3 at infinity for $p=\frac{N+\alpha}{N-2}$4, and the finiteness conditions

$p=\frac{N+\alpha}{N-2}$5

$p=\frac{N+\alpha}{N-2}$6

Every such solution has the explicit form

$p=\frac{N+\alpha}{N-2}$7

The proof uses the method of moving spheres, an equivalent integral representation, Pohozaev-type identities, and the asymptotic estimate

$p=\frac{N+\alpha}{N-2}$8

The result is presented as a Liouville-type classification of bubble solutions (Huang et al., 2023).

The same bubble paradigm governs asymptotic analysis in perturbed critical problems. For

$p=\frac{N+\alpha}{N-2}$9

positive ground states, after a suitable rescaling, converge as $$(-\Delta)^{\frac n2}u(x)=\left(\int_{\mathbb R^n}\frac{e^{\frac{2n-\mu}{2}u(y)}}{|x-y|^\mu}\,dy\right)e^{\frac{2n-\mu}{2}u(x)}.$$0 to a particular solution of the critical equation

$$(-\Delta)^{\frac n2}u(x)=\left(\int_{\mathbb R^n}\frac{e^{\frac{2n-\mu}{2}u(y)}}{|x-y|^\mu}\,dy\right)e^{\frac{2n-\mu}{2}u(x)}.$$1

with dimension-dependent sharp rescaling laws in the cases $$(-\Delta)^{\frac n2}u(x)=\left(\int_{\mathbb R^n}\frac{e^{\frac{2n-\mu}{2}u(y)}}{|x-y|^\mu}\,dy\right)e^{\frac{2n-\mu}{2}u(x)}.$$2, $$(-\Delta)^{\frac n2}u(x)=\left(\int_{\mathbb R^n}\frac{e^{\frac{2n-\mu}{2}u(y)}}{|x-y|^\mu}\,dy\right)e^{\frac{2n-\mu}{2}u(x)}.$$3, and $$(-\Delta)^{\frac n2}u(x)=\left(\int_{\mathbb R^n}\frac{e^{\frac{2n-\mu}{2}u(y)}}{|x-y|^\mu}\,dy\right)e^{\frac{2n-\mu}{2}u(x)}.$$4 (Liu et al., 2024).

3. Variational structure and compactness mechanisms

The standard variational framework associates the equation with an energy functional on a natural Sobolev-type space. Depending on the model, this space may be $$(-\Delta)^{\frac n2}u(x)=\left(\int_{\mathbb R^n}\frac{e^{\frac{2n-\mu}{2}u(y)}}{|x-y|^\mu}\,dy\right)e^{\frac{2n-\mu}{2}u(x)}.$$5, $$(-\Delta)^{\frac n2}u(x)=\left(\int_{\mathbb R^n}\frac{e^{\frac{2n-\mu}{2}u(y)}}{|x-y|^\mu}\,dy\right)e^{\frac{2n-\mu}{2}u(x)}.$$6, $$(-\Delta)^{\frac n2}u(x)=\left(\int_{\mathbb R^n}\frac{e^{\frac{2n-\mu}{2}u(y)}}{|x-y|^\mu}\,dy\right)e^{\frac{2n-\mu}{2}u(x)}.$$7, $$(-\Delta)^{\frac n2}u(x)=\left(\int_{\mathbb R^n}\frac{e^{\frac{2n-\mu}{2}u(y)}}{|x-y|^\mu}\,dy\right)e^{\frac{2n-\mu}{2}u(x)}.$$8, a weighted homogeneous Sobolev space, or $$(-\Delta)^{\frac n2}u(x)=\left(\int_{\mathbb R^n}\frac{e^{\frac{2n-\mu}{2}u(y)}}{|x-y|^\mu}\,dy\right)e^{\frac{2n-\mu}{2}u(x)}.$$9. Ground states are then sought by mountain-pass arguments, constrained minimization, or minimization on Nehari or Pohožaev manifolds (Ao, 2016, Li et al., 2018, Böer et al., 2020).

Criticality obstructs compactness through translation, dilation, and, in some problems, conformal invariance. A central technical response is concentration-compactness together with nonlocal splitting results. For the Hardy–Littlewood–Sobolev critical exponent, Moroz and Van Schaftingen used concentration-compactness and a nonlocal version of the Brezis–Lieb lemma to recover compactness under the strict inequality $\int_{\mathbb R^N}(I_\alpha*|u|^p)|u|^p\,dx,$0 (Moroz et al., 2014). In semiclassical critical-frequency problems, Ding, Gao, and Yang established a nonlocal global compactness lemma analogous to the Lions–Struwe result, together with energy splitting criteria for Palais–Smale sequences (Ding et al., 2017).

Another recurrent device is the Pohožaev constraint. In the autonomous upper-critical equation with general nonlinearity,

$\int_{\mathbb R^N}(I_\alpha*|u|^p)|u|^p\,dx,$1

the ground state is obtained as a minimizer of the energy on the manifold defined by the Pohožaev identity, combined with subcritical approximation and Strauss’s compactness lemma for radially symmetric decreasing sequences (Li et al., 2018). In polyharmonic critical Choquard problems, explicit minimizers for the best Hardy–Littlewood–Sobolev constant are used to place the mountain-pass level below the threshold

$\int_{\mathbb R^N}(I_\alpha*|u|^p)|u|^p\,dx,$2

thereby recovering Palais–Smale compactness (Abhishek et al., 7 Jul 2026).

4. Existence and ground-state theory

Existence theory splits according to the form of the potential and the type of criticality. For the nonlinear Choquard equation

$\int_{\mathbb R^N}(I_\alpha*|u|^p)|u|^p\,dx,$3

with $\int_{\mathbb R^N}(I_\alpha*|u|^p)|u|^p\,dx,$4 and $\int_{\mathbb R^N}(I_\alpha*|u|^p)|u|^p\,dx,$5, Moroz and Van Schaftingen proved that if $\int_{\mathbb R^N}(I_\alpha*|u|^p)|u|^p\,dx,$6 and $\int_{\mathbb R^N}(I_\alpha*|u|^p)|u|^p\,dx,$7, then the minimization problem is attained and the equation has a ground state. A sufficient condition for $\int_{\mathbb R^N}(I_\alpha*|u|^p)|u|^p\,dx,$8 is

$\int_{\mathbb R^N}(I_\alpha*|u|^p)|u|^p\,dx,$9

When $p=\frac{N+\alpha}{N}$0, one has $p=\frac{N+\alpha}{N}$1, and no ground state exists (Moroz et al., 2014).

For autonomous upper-critical equations, existence is known both for pure and for combined nonlinearities. The equation

$p=\frac{N+\alpha}{N}$2

with $p=\frac{N+\alpha}{N}$3 admits a nontrivial solution for $p=\frac{N+\alpha}{N}$4, $p=\frac{N+\alpha}{N}$5, and for $p=\frac{N+\alpha}{N}$6, $p=\frac{N+\alpha}{N}$7; the corresponding minimizing problem also has a minimizer (Ao, 2016). More generally, “Choquard equations with critical nonlinearities” establishes positive, radially symmetric, radially nonincreasing ground states for both the lower and upper critical exponents by combining subcritical approximation with the Pohožaev constraint method (Li et al., 2018). For the doubly critical nonlinearity $p=\frac{N+\alpha}{N}$8, a nontrivial solution exists when $p=\frac{N+\alpha}{N}$9 and $p=\frac{N+\alpha}{N-2}$0 (Seok, 2017).

Bounded-domain existence at the upper critical exponent exhibits a Brezis–Nirenberg-type structure. In dimension three, for

$p=\frac{N+\alpha}{N-2}$1

the following are equivalent: the existence of a least energy solution, the strict inequality $p=\frac{N+\alpha}{N-2}$2, and positivity of the Robin function $p=\frac{N+\alpha}{N-2}$3 somewhere in $p=\frac{N+\alpha}{N-2}$4. This equivalence leads to a notion of critical potential $p=\frac{N+\alpha}{N-2}$5, and under perturbation $p=\frac{N+\alpha}{N-2}$6 with $p=\frac{N+\alpha}{N-2}$7 critical, least energy solutions exist if the set

$p=\frac{N+\alpha}{N-2}$8

is nonempty (Gao, 25 Mar 2026).

5. Non-existence, multiplicity, and concentration

Critical Choquard equations exhibit sharp threshold non-existence phenomena as well as multiplicity above threshold. For the combined equation

$p=\frac{N+\alpha}{N-2}$9

when one of $$F(u)=|u|^p+|u|^q,\qquad p=\frac{N+\alpha}{N},\quad q=\frac{N+\alpha}{N-2},$$0 or $$F(u)=|u|^p+|u|^q,\qquad p=\frac{N+\alpha}{N},\quad q=\frac{N+\alpha}{N-2},$$1 is critical, the parameter space $$F(u)=|u|^p+|u|^q,\qquad p=\frac{N+\alpha}{N},\quad q=\frac{N+\alpha}{N-2},$$2 splits into regions of non-existence and multiplicity. In particular, for $$F(u)=|u|^p+|u|^q,\qquad p=\frac{N+\alpha}{N},\quad q=\frac{N+\alpha}{N-2},$$3 and $$F(u)=|u|^p+|u|^q,\qquad p=\frac{N+\alpha}{N},\quad q=\frac{N+\alpha}{N-2},$$4, there exists $$F(u)=|u|^p+|u|^q,\qquad p=\frac{N+\alpha}{N},\quad q=\frac{N+\alpha}{N-2},$$5 such that no positive ground state exists for $$F(u)=|u|^p+|u|^q,\qquad p=\frac{N+\alpha}{N},\quad q=\frac{N+\alpha}{N-2},$$6, while ground states exist for $$F(u)=|u|^p+|u|^q,\qquad p=\frac{N+\alpha}{N},\quad q=\frac{N+\alpha}{N-2},$$7; for $$F(u)=|u|^p+|u|^q,\qquad p=\frac{N+\alpha}{N},\quad q=\frac{N+\alpha}{N-2},$$8, there is a larger threshold $$F(u)=|u|^p+|u|^q,\qquad p=\frac{N+\alpha}{N},\quad q=\frac{N+\alpha}{N-2},$$9 such that at least two positive solutions exist for $$(-\Delta)^{\frac12}u+a u+\lambda (\ln|\cdot|*|u|^2)u=f(u)\quad \text{in }\mathbb R,$$0. The same work notes that uniqueness is largely open except for special cases (Ma, 2024).

Potential-well and semiclassical problems add concentration phenomena. In

$$(-\Delta)^{\frac12}u+a u+\lambda (\ln|\cdot|*|u|^2)u=f(u)\quad \text{in }\mathbb R,$$1

with $$(-\Delta)^{\frac12}u+a u+\lambda (\ln|\cdot|*|u|^2)u=f(u)\quad \text{in }\mathbb R,$$2 nonempty and bounded, ground states exist for $$(-\Delta)^{\frac12}u+a u+\lambda (\ln|\cdot|*|u|^2)u=f(u)\quad \text{in }\mathbb R,$$3 large enough and localize near $$(-\Delta)^{\frac12}u+a u+\lambda (\ln|\cdot|*|u|^2)u=f(u)\quad \text{in }\mathbb R,$$4 as $$(-\Delta)^{\frac12}u+a u+\lambda (\ln|\cdot|*|u|^2)u=f(u)\quad \text{in }\mathbb R,$$5; for $$(-\Delta)^{\frac12}u+a u+\lambda (\ln|\cdot|*|u|^2)u=f(u)\quad \text{in }\mathbb R,$$6, at least $$(-\Delta)^{\frac12}u+a u+\lambda (\ln|\cdot|*|u|^2)u=f(u)\quad \text{in }\mathbb R,$$7 distinct solutions are obtained by Lusternik–Schnirelmann category theory (Gao et al., 2017). In the critical-frequency case $$(-\Delta)^{\frac12}u+a u+\lambda (\ln|\cdot|*|u|^2)u=f(u)\quad \text{in }\mathbb R,$$8 with $$(-\Delta)^{\frac12}u+a u+\lambda (\ln|\cdot|*|u|^2)u=f(u)\quad \text{in }\mathbb R,$$9, the pure critical problem has at least $N$0 semiclassical solutions for small $N$1, and the double-critical problem has arbitrarily many pairs of semiclassical solutions with small energy (Ding et al., 2017). In singularly perturbed critical problems in $N$2, positive ground states concentrate near maxima of $N$3 or minima of $N$4, and the number of positive solutions is bounded below by the corresponding Lusternik–Schnirelmann category (Alves et al., 2016).

Multiplicity can also arise without semiclassical scaling. For

$N$5

there are at least two distinct positive solutions for small $N$6 under a smallness assumption on $N$7, but the paper emphasizes that these are not ground states and that ground state solutions do not exist under its hypotheses (Alves et al., 2018). A different phenomenon appears in the non-radial setting: if $N$8 is bounded, nonnegative, radially symmetric, $N$9, $p=\frac{N+\alpha}{N}$00, and $p=\frac{N+\alpha}{N}$01 has an isolated local maximum or isolated local minimum at some $p=\frac{N+\alpha}{N}$02 with $p=\frac{N+\alpha}{N}$03, then

$p=\frac{N+\alpha}{N}$04

has infinitely many non-radial solutions with arbitrarily large energies, constructed by Lyapunov–Schmidt finite dimensional reduction. The same source states that if $p=\frac{N+\alpha}{N}$05 has constant sign and is not identically zero, then the Pohozaev identity yields non-existence (Caputo et al., 21 Jul 2025).

6. Fractional, normalized, logarithmic, exponential, and polyharmonic extensions

The critical Choquard framework extends well beyond the standard Laplacian on $p=\frac{N+\alpha}{N}$06. Fractional and singular models include

$p=\frac{N+\alpha}{N}$07

on bounded domains, where positive and sign-changing solutions are obtained under suitable assumptions on the critical Hardy–Sobolev and Hardy–Littlewood–Sobolev exponents (Yang et al., 2019). An even more singular problem combines a fractional Laplacian, a Hardy potential, a singular term $p=\frac{N+\alpha}{N}$08, a critical Choquard term, and a Radon measure; in that setting, one proves existence of a positive SOLA, meaning a solution obtained as the limit of approximations, for sufficiently small $p=\frac{N+\alpha}{N}$09 (Panda et al., 2020).

Normalized and logarithmic variants show that criticality interacts strongly with constraints and kernel choice. For the lower critical normalized problem

$p=\frac{N+\alpha}{N}$10

there is existence of normalized ground states for $p=\frac{N+\alpha}{N}$11, non-existence at the mass-critical value $p=\frac{N+\alpha}{N}$12 under $p=\frac{N+\alpha}{N}$13, and multiplicity in the non-autonomous case, with the number of normalized solutions bounded below by the number of global maxima of the weight $p=\frac{N+\alpha}{N}$14 when $p=\frac{N+\alpha}{N}$15 is small (Li et al., 2022). For the one-dimensional logarithmic equation with critical exponential growth,

$p=\frac{N+\alpha}{N}$16

one obtains a nontrivial mountain-pass solution, a ground state under critical or subcritical exponential growth, and infinitely many pairs of solutions under subcritical growth and an oddness assumption on $p=\frac{N+\alpha}{N}$17 (Böer et al., 2020).

Zero-mass and higher-order settings further broaden the theory. In the weighted $p=\frac{N+\alpha}{N}$18-Laplacian problem

$p=\frac{N+\alpha}{N}$19

variational methods yield a positive, radially symmetric weak solution for both subcritical and critical exponential nonlinearities (Romani, 2024). For the $p=\frac{N+\alpha}{N}$20-harmonic equation with critical Choquard nonlinearity and subcritical perturbation,

$p=\frac{N+\alpha}{N}$21

the existence of nontrivial solutions is obtained by combining minimizers for the sharp critical Choquard inequality with refined threshold estimates; the work states that it is the first article dealing with the polyharmonic equation and critical Choquard type nonlinearity, and that the results are new for $p=\frac{N+\alpha}{N}$22 (Abhishek et al., 7 Jul 2026).

Taken together, these developments show that the phrase “critical Choquard equation” does not denote a single equation but a family of nonlocal critical problems whose common analytic core is the interaction between convolution nonlinearity, critical embedding, and loss of compactness. The explicit bubble profiles, Pohožaev identities, threshold inequalities, and concentration mechanisms occurring across the literature indicate a unified structure, while the fractional, logarithmic, exponential, zero-mass, normalized, and polyharmonic extensions show that the range of admissible critical phenomena is substantially broader than the classical whole-space Hartree model.