Critical Choquard Equations

• Critical Choquard equations are nonlinear nonlocal elliptic problems featuring Hartree-type convolution terms at Hardy–Littlewood–Sobolev and Sobolev critical thresholds.
• They encompass various regimes including lower, upper, doubly critical, and triply critical settings through combinations of local and nonlocal nonlinearities.
• The variational framework relies on natural constraints like the Nehari manifold and Pohožaev identities to overcome loss of compactness and secure existence and multiplicity of solutions.

Critical Choquard equations are nonlinear nonlocal elliptic problems in which the Choquard, or Hartree, convolution term reaches a limiting growth regime dictated by the Hardy–Littlewood–Sobolev inequality, the Sobolev embedding, or both. In the literature represented here, the topic includes the lower Hardy–Littlewood–Sobolev critical exponent $p=\frac{N+\alpha}{N}$, the upper Hardy–Littlewood–Sobolev critical exponent $p=\frac{N+\alpha}{N-2}$, combinations with the Sobolev critical local exponent $2^*=\frac{2N}{N-2}$, genuinely doubly critical nonlocal nonlinearities containing both Hardy–Littlewood–Sobolev endpoints at once, and several related extensions such as fractional $p$-Choquard equations, normalized critical systems, bounded-domain upper critical problems, and zero-mass equations with critical exponential growth (Moroz et al., 2014, Li et al., 2018, Seok, 2017, Sakuma, 2023).

1. Criticality regimes and canonical models

The basic Choquard equation takes the form

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

or, more generally,

$-\Delta u+u=(I_\alpha*F(u))F'(u),$

with $I_\alpha(x)=A_\alpha(N)|x|^{-(N-\alpha)}$. The relevant admissible range for the power nonlinearity is

$\frac{N+\alpha}{N}\le p\le \frac{N+\alpha}{N-2},$

and the two endpoint values are the lower and upper Hardy–Littlewood–Sobolev critical exponents (Moroz et al., 2014, Li et al., 2018).

A central distinction in the subject is that “critical” does not have a single meaning. In the whole-space power-type theory, the lower endpoint $p=\frac{N+\alpha}{N}$ is critical in the $L^2$-based Hardy–Littlewood–Sobolev sense, whereas the upper endpoint $p=\frac{N+\alpha}{N-2}$0 is critical with respect to the homogeneous Sobolev scaling of $p=\frac{N+\alpha}{N-2}$1 (Moroz et al., 2014, Li et al., 2018). In mixed local/nonlocal problems, criticality may also refer to the local Sobolev exponent $p=\frac{N+\alpha}{N-2}$2 (Li et al., 2018, Ma, 2024). In the doubly critical Choquard literature, the nonlocal nonlinearity itself may simultaneously contain both Hardy–Littlewood–Sobolev critical powers,

$p=\frac{N+\alpha}{N-2}$3

so that the functional exhibits two distinct noncompact critical scalings at once (Seok, 2017). In the fractional $p=\frac{N+\alpha}{N-2}$4-Choquard setting, the nonlocal term can be “doubly critical” in the Hardy–Littlewood–Sobolev sense while the local perturbation is allowed to be Sobolev-critical as well, producing the “triply critical” configuration emphasized in the paper (Sakuma, 2023).

Criticality notion	Defining regime	Representative source
Lower HLS criticality	$p=\frac{N+\alpha}{N-2}$5	(Moroz et al., 2014)
Upper HLS criticality	$p=\frac{N+\alpha}{N-2}$6	(Li et al., 2018)
Doubly critical nonlocality	$p=\frac{N+\alpha}{N-2}$7	(Seok, 2017)
Mixed local/nonlocal criticality	Choquard critical term plus $p=\frac{N+\alpha}{N-2}$8 or $p=\frac{N+\alpha}{N-2}$9	(Li et al., 2018)
Fractional “triply critical” setting	doubly HLS-critical $2^*=\frac{2N}{N-2}$0 plus local critical $2^*=\frac{2N}{N-2}$1	(Sakuma, 2023)
Critical exponential growth	limiting $2^*=\frac{2N}{N-2}$2-Laplacian/Trudinger–Moser regime	(Romani, 2024)

A common misconception is to identify critical Choquard equations exclusively with the upper exponent $2^*=\frac{2N}{N-2}$3. The current literature is broader. It includes lower-critical problems (Moroz et al., 2014), upper-critical problems (Li et al., 2018), combined local/nonlocal critical models (Li et al., 2018, Ma, 2024), and genuinely doubly critical nonlocal equations (Seok, 2017, Sakuma, 2023). It also includes settings in which “critical” refers not to the exponent but to the frequency, as in semiclassical problems with $2^*=\frac{2N}{N-2}$4 (Schaftingen et al., 2016).

2. Variational formulation and natural constraints

The dominant framework is variational. For the standard autonomous Choquard equation with local mass term, the natural space is $2^*=\frac{2N}{N-2}$5, and a typical functional is

$2^*=\frac{2N}{N-2}$6

or its mixed version

$2^*=\frac{2N}{N-2}$7

(Seok, 2017, Li et al., 2018). On bounded domains with Dirichlet boundary conditions, the natural space becomes $2^*=\frac{2N}{N-2}$8, and the critical nonlocal term is built from the upper Hardy–Littlewood–Sobolev exponent $2^*=\frac{2N}{N-2}$9 (Gao et al., 2016).

At criticality, the Nehari manifold and the Pohožaev constraint become structurally important. In the upper critical autonomous case, the ground-state level is compared with

$p$0

where $p$1 is the Pohožaev functional, and every nonzero configuration can be uniquely scaled onto the Pohožaev manifold (Li et al., 2018). In mixed Brezis–Nirenberg-type critical problems, the Pohožaev identity supplies the natural codimension-one constraint on which the least-energy level is defined and compared with critical thresholds (Li et al., 2018).

Several extensions require different ambient spaces. The fractional $p$2-Choquard theory works in $p$3 and $p$4, with Gagliardo seminorm $p$5 and fractional $p$6-Laplacian $p$7 (Sakuma, 2023). Higher-order critical Choquard–Kirchhoff equations are formulated in $p$8 (Goel et al., 30 Aug 2025). The zero-mass weighted $p$9-Laplacian problem uses a weighted homogeneous space

$-\Delta u + Vu = (I_\alpha*|u|^p)|u|^{p-2}u,$0

because the left-hand side contains no lower-order mass term (Romani, 2024). Normalized solutions are sought on mass spheres

$-\Delta u + Vu = (I_\alpha*|u|^p)|u|^{p-2}u,$1

or on product mass spheres for systems (Li et al., 2022, Pei et al., 25 Oct 2025).

This variational diversity reflects a stable theme of the subject: the critical Choquard term is rarely treated in isolation. It is analyzed through a natural constraint adapted to the relevant scaling—Nehari, Pohožaev, mountain-pass, constrained minimization, or normalized mass-preserving fibers—depending on whether the problem is autonomous, bounded-domain, fractional, normalized, or quasilinear (Li et al., 2018, Sakuma, 2023, Pei et al., 25 Oct 2025).

3. Loss of compactness, threshold levels, and concentration mechanisms

The analytical core of critical Choquard theory is the failure of compactness. On $-\Delta u + Vu = (I_\alpha*|u|^p)|u|^{p-2}u,$2, translation invariance already prevents compact Sobolev embedding. At the critical exponents, this is compounded by scaling invariance. In the lower HLS-critical problem,

$-\Delta u + Vu = (I_\alpha*|u|^p)|u|^{p-2}u,$3

preserves both the $-\Delta u + Vu = (I_\alpha*|u|^p)|u|^{p-2}u,$4-norm and the critical Choquard interaction, which is the source of the threshold level $-\Delta u + Vu = (I_\alpha*|u|^p)|u|^{p-2}u,$5 in the lower-critical theory (Moroz et al., 2014). In the doubly critical problem,

$-\Delta u + Vu = (I_\alpha*|u|^p)|u|^{p-2}u,$6

the functional simultaneously carries the lower-critical scaling $-\Delta u + Vu = (I_\alpha*|u|^p)|u|^{p-2}u,$7 and the upper-critical Sobolev scaling $-\Delta u + Vu = (I_\alpha*|u|^p)|u|^{p-2}u,$8 (Seok, 2017).

Threshold energies organize the compactness theory. For the doubly critical equation, the compactness proposition is stated below the minimum of the two pure critical bubble energies associated with the lower and upper critical inequalities (Seok, 2017). For the bounded-domain upper critical Choquard equation, Palais–Smale compactness holds below

$-\Delta u + Vu = (I_\alpha*|u|^p)|u|^{p-2}u,$9

the nonlocal analogue of the Brezis–Nirenberg threshold (Gao et al., 2016). In the fractional $-\Delta u+u=(I_\alpha*F(u))F'(u),$0-Choquard problem, the mountain-pass level must be placed below an explicit critical threshold $-\Delta u+u=(I_\alpha*F(u))F'(u),$1 built from the best constants $-\Delta u+u=(I_\alpha*F(u))F'(u),$2, $-\Delta u+u=(I_\alpha*F(u))F'(u),$3, and, in the local critical case, a quantity $-\Delta u+u=(I_\alpha*F(u))F'(u),$4 (Sakuma, 2023).

Several compactness-recovery devices recur across the literature. One is the nonlocal Brezis–Lieb splitting for Riesz potentials, used in the lower-critical, upper-critical, bounded-domain, and system settings to separate the energy of a weak limit from that of the remainder (Moroz et al., 2014, Gao et al., 2016, Alves et al., 2018, Pei et al., 25 Oct 2025). Another is concentration-compactness, which in the critical Choquard–Kirchhoff setting is upgraded to a full defect-measure principle tracking the Choquard energy itself and the mass at infinity (Goel et al., 30 Aug 2025). A third is the use of radial subspaces: the doubly critical whole-space equation is treated in $-\Delta u+u=(I_\alpha*F(u))F'(u),$5 precisely because the compact radial Sobolev embedding replaces translation invariance (Seok, 2017).

The bounded-domain three-dimensional upper critical problem displays a different compactness mechanism. There, least-energy existence is equivalent, for $-\Delta u+u=(I_\alpha*F(u))F'(u),$6 sufficiently small, to a strict inequality $-\Delta u+u=(I_\alpha*F(u))F'(u),$7, and that strict gap is in turn equivalent to positivity of the Robin function $-\Delta u+u=(I_\alpha*F(u))F'(u),$8 somewhere in the domain (Gao, 25 Mar 2026). This is a low-dimensional global effect rather than a purely local test-bubble condition.

4. Existence, ground states, and multiplicity

The lower HLS-critical equation with external potential,

$-\Delta u+u=(I_\alpha*F(u))F'(u),$9

admits a nontrivial solution if

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

because this strict asymptotic condition yields $I_\alpha(x)=A_\alpha(N)|x|^{-(N-\alpha)}$1 and therefore compactness of minimizing sequences (Moroz et al., 2014). By contrast, if $I_\alpha(x)=A_\alpha(N)|x|^{-(N-\alpha)}$2, then $I_\alpha(x)=A_\alpha(N)|x|^{-(N-\alpha)}$3, and the critical level is not attained (Moroz et al., 2014).

For the upper Hardy–Littlewood–Sobolev critical exponent, the autonomous equation

$I_\alpha(x)=A_\alpha(N)|x|^{-(N-\alpha)}$4

admits a positive radially symmetric ground state under assumptions (f1)–(f4), via the Pohožaev constraint method, subcritical approximation, and Strauss compactness (Li et al., 2018). This complements Brezis–Nirenberg-type whole-space results for

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

where positive radially nonincreasing ground states are obtained at both critical HLS endpoints $I_\alpha(x)=A_\alpha(N)|x|^{-(N-\alpha)}$6 and $I_\alpha(x)=A_\alpha(N)|x|^{-(N-\alpha)}$7 in precise parameter regimes (Li et al., 2018).

The genuinely doubly critical equation

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

admits a nontrivial radial solution when $I_\alpha(x)=A_\alpha(N)|x|^{-(N-\alpha)}$9 and $\frac{N+\alpha}{N}\le p\le \frac{N+\alpha}{N-2},$0, by a mountain-pass argument combined with a compactness threshold excluding both critical concentration mechanisms (Seok, 2017). The fractional $\frac{N+\alpha}{N}\le p\le \frac{N+\alpha}{N-2},$1-Choquard analogue

$\frac{N+\alpha}{N}\le p\le \frac{N+\alpha}{N-2},$2

has a ground state under

$\frac{N+\alpha}{N}\le p\le \frac{N+\alpha}{N-2},$3

either when $\frac{N+\alpha}{N}\le p\le \frac{N+\alpha}{N-2},$4, $\frac{N+\alpha}{N}\le p\le \frac{N+\alpha}{N-2},$5, or when $\frac{N+\alpha}{N}\le p\le \frac{N+\alpha}{N-2},$6, $\frac{N+\alpha}{N}\le p\le \frac{N+\alpha}{N-2},$7, and $\frac{N+\alpha}{N}\le p\le \frac{N+\alpha}{N-2},$8 is sufficiently small (Sakuma, 2023).

Bounded-domain upper critical problems display both existence and multiplicity. For

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

variational methods yield existence and, depending on the perturbation, multiplicity results including an Ambrosetti–Brezis–Cerami-type theorem, infinitely many small-energy solutions, and infinitely many unbounded solutions (Gao et al., 2016). In three dimensions, least-energy existence for the upper critical Choquard equation on a bounded $p=\frac{N+\alpha}{N}$0 domain is characterized by the equivalence

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

for $p=\frac{N+\alpha}{N}$2 sufficiently small (Gao, 25 Mar 2026).

Multiplicity can also occur in whole-space critical potential problems. For

$p=\frac{N+\alpha}{N}$3

with $p=\frac{N+\alpha}{N}$4, $p=\frac{N+\alpha}{N}$5, and a smallness assumption on $p=\frac{N+\alpha}{N}$6, there are two distinct positive bound states for small $p=\frac{N+\alpha}{N}$7, although the constrained infimum $p=\frac{N+\alpha}{N}$8 is not attained, so no ground state exists (Alves et al., 2018).

A later synthesis of combined critical local and nonlocal nonlinearities identifies threshold parameters separating nonexistence of positive ground states, existence of a positive ground state, and existence of two distinct positive solutions in several regimes, including the upper HLS-critical regime, the lower HLS-threshold regime in $p=\frac{N+\alpha}{N}$9, and the Sobolev-critical local regime (Ma, 2024).

5. Normalized, semiclassical, zero-mass, and higher-order variants

Normalized critical Choquard theory studies constrained solutions with prescribed $L^2$0-mass. For the lower critical equation with local perturbation,

$L^2$1

the existence theory is sharp in the ranges reported. If $L^2$2, then the infimum $L^2$3 on the mass sphere is achieved by a positive radial decreasing ground state. If $L^2$4 and

$L^2$5

then no normalized solution exists (Li et al., 2022). The non-autonomous version with coefficient $L^2$6 has at least as many normalized solutions as the number of global maxima of $L^2$7 when $L^2$8 is small (Li et al., 2022).

The normalized critical system with linear and nonlinear couplings,

$L^2$9

with $p=\frac{N+\alpha}{N-2}$00 or $p=\frac{N+\alpha}{N-2}$01, illustrates another critical dichotomy. If

$p=\frac{N+\alpha}{N-2}$02

a positive normalized ground state exists for $p=\frac{N+\alpha}{N-2}$03 and $p=\frac{N+\alpha}{N-2}$04; if

$p=\frac{N+\alpha}{N-2}$05

it exists for $p=\frac{N+\alpha}{N-2}$06 and $p=\frac{N+\alpha}{N-2}$07 (Pei et al., 25 Oct 2025).

Semiclassical critical Choquard equations exhibit concentration phenomena. In the singularly perturbed critical equation in $p=\frac{N+\alpha}{N-2}$08,

$p=\frac{N+\alpha}{N-2}$09

ground states of the autonomous critical problem are first established, and then semiclassical solutions are shown to concentrate near maxima of $p=\frac{N+\alpha}{N-2}$10 or minima of $p=\frac{N+\alpha}{N-2}$11, with multiplicity controlled by $p=\frac{N+\alpha}{N-2}$12 (Alves et al., 2016). By contrast, “critical frequency” in the problem

$p=\frac{N+\alpha}{N-2}$13

means $p=\frac{N+\alpha}{N-2}$14 with $p=\frac{N+\alpha}{N-2}$15, not HLS-criticality of the exponent; the paper derives concentration on the nonstandard scale $p=\frac{N+\alpha}{N-2}$16 or convergence to a Dirichlet Choquard problem on the zero set of $p=\frac{N+\alpha}{N-2}$17 (Schaftingen et al., 2016).

Other extensions shift the differential operator rather than the nonlocal term. The $p=\frac{N+\alpha}{N-2}$18-biharmonic Kirchhoff equation with critical Choquard nonlinearity,

$p=\frac{N+\alpha}{N-2}$19

requires a concentration-compactness principle tailored to both the critical Choquard interaction and the Kirchhoff coefficient, and the existence theory depends sharply on the relation between $p=\frac{N+\alpha}{N-2}$20, $p=\frac{N+\alpha}{N-2}$21, and $p=\frac{N+\alpha}{N-2}$22 (Goel et al., 30 Aug 2025). In the zero-mass weighted $p=\frac{N+\alpha}{N-2}$23-Laplacian problem, the critical growth becomes exponential rather than polynomial, and existence of a positive radial weak solution is proved by combining weighted Trudinger–Moser estimates with the Choquard convolution structure (Romani, 2024).

6. Structural themes, distinctions, and unresolved regimes

Three structural themes recur. First, existence is typically obtained only after proving that the variational level lies strictly below a critical threshold associated with a pure bubble or a whole-space limit problem (Moroz et al., 2014, Seok, 2017, Gao et al., 2016). Second, multiplicity usually appears after that threshold has been crossed sufficiently far, often through a large perturbation parameter, barycenter construction, or normalized secondary branch (Ma, 2024, Alves et al., 2018, Pei et al., 25 Oct 2025). Third, critical Choquard theory is highly sensitive to the precise meaning of criticality: lower HLS-critical, upper HLS-critical, doubly HLS-critical, Sobolev-critical local perturbation, critical frequency, and critical exponential growth are analytically distinct categories (Moroz et al., 2014, Seok, 2017, Schaftingen et al., 2016, Romani, 2024).

Several limitations remain explicit in the cited literature. The fractional $p=\frac{N+\alpha}{N-2}$24-Choquard theorem allows the local critical case $p=\frac{N+\alpha}{N-2}$25 only when $p=\frac{N+\alpha}{N-2}$26 and $p=\frac{N+\alpha}{N-2}$27 is sufficiently small (Sakuma, 2023). The bounded-domain three-dimensional Robin-function characterization is stated for $p=\frac{N+\alpha}{N-2}$28 sufficiently small (Gao, 25 Mar 2026). The normalized critical coupled system excludes the exact mixed-coupling threshold

$p=\frac{N+\alpha}{N-2}$29

(Pei et al., 25 Oct 2025). The $p=\frac{N+\alpha}{N-2}$30-biharmonic Kirchhoff paper explicitly leaves open the range

$p=\frac{N+\alpha}{N-2}$31

because its method does not provide minimizers there (Goel et al., 30 Aug 2025). The combined-critical whole-space theory also records an open interval in the $p=\frac{N+\alpha}{N-2}$32 Sobolev-critical regime (Ma, 2024).

A further point of interpretation concerns ground states versus bound states. Some critical problems admit positive solutions but no ground state because the infimum equals the critical whole-space level and is not attained; the positive solutions that exist are then genuine bound states above that unattained floor (Alves et al., 2018). This distinction is fundamental in critical Choquard theory and is one reason why minimization, mountain-pass constructions, Pohožaev constraints, and normalized methods coexist rather than collapsing into a single variational paradigm.

Taken together, these works show that critical Choquard equations form a unified but internally differentiated field. The unifying features are the nonlocal Hartree interaction, the sharp Hardy–Littlewood–Sobolev structure, and the systematic need to control bubbling and translation loss. The differentiation arises from which endpoint is critical, whether a local perturbation is present, whether the setting is whole-space or bounded-domain, whether mass is prescribed, and whether the differential operator is local, fractional, quasilinear, or higher order (Li et al., 2018, Sakuma, 2023, Gao, 25 Mar 2026).