Planar Choquard Equation Insights

• The planar Choquard equation is a 2D nonlinear, nonlocal elliptic equation featuring Hartree-type interactions and unique exponential critical growth due to borderline Sobolev embeddings.
• It involves both Riesz and logarithmic kernels, and its study uses variational methods, including mountain-pass and Pohožaev identities, to establish existence and symmetry of solutions.
• Analytical techniques such as Moser–Trudinger inequalities, novel scaling strategies, and compactness arguments address the challenges posed by criticality and nonlocal interactions in the plane.

The planar Choquard equation is a two-dimensional nonlinear, nonlocal elliptic equation of Hartree type. In its standard Riesz-potential form it is written as

$$-\Delta u+u=(I_\alpha*F(u))\,F'(u)\qquad\text{in }\mathbb{R}^2,$$

with $0<\alpha<2$, $F\in C^1(\mathbb R,\mathbb R)$, and $I_\alpha(x)\sim |x|^{\alpha-2}$. In the literature it is also called a Schrödinger–Newton or Hartree-type equation. In dimension two, the subject splits into two essentially different settings: equations with a genuine Riesz kernel $I_\alpha$, and equations with the planar Newtonian kernel, which is logarithmic rather than power-like. The plane is a distinguished dimension because $H^1(\mathbb R^2)$ is borderline for Sobolev embedding, so critical growth is of exponential Trudinger–Moser type rather than a finite power type (Battaglia et al., 2016, Moroz et al., 2016, Cassani et al., 2023).

1. Core formulations in two dimensions

A general Choquard equation has the form

$$-\Delta u+u=\bigl(I_\alpha*G(u)\bigr)\,H(u)\qquad\text{in }\mathbb R^N,$$

and the planar case studied in the general nonlinearity framework is

$$-\Delta u+u=(I_\alpha*F(u))\,F'(u)\qquad\text{in }\mathbb R^2.$$

Here the convolution couples the value of $u$ at a point with its values at all other points through a singular kernel, and the classical Hartree case corresponds to power-type choices of $F$ (Battaglia et al., 2016).

For the autonomous power model, the survey literature uses

$0<\alpha<2$0

with $0<\alpha<2$1. In that setting, the planar problem is framed on $0<\alpha<2$2, and the Riesz-potential term is controlled by the Hardy–Littlewood–Sobolev inequality (Moroz et al., 2016).

The logarithmic planar model replaces the Riesz kernel by the two-dimensional Newtonian kernel. One representative equation is

$0<\alpha<2$3

while another, strongly indefinite version is

$0<\alpha<2$4

These logarithmic models arise because in $0<\alpha<2$5 the fundamental solution of $0<\alpha<2$6 is logarithmic rather than power-like (Cassani et al., 2023, Cabrera et al., 25 Feb 2025).

The planar literature also includes fractional and mixed-diffusion variants. Examples are the fractional Choquard equation

$0<\alpha<2$7

and the mixed local–nonlocal model

$0<\alpha<2$8

with $0<\alpha<2$9 and $F\in C^1(\mathbb R,\mathbb R)$0 (Ma et al., 2017, Chen et al., 29 Jun 2026).

2. Variational structure and solution concepts

For the standard planar Riesz-kernel problem,

$F\in C^1(\mathbb R,\mathbb R)$1

solutions are sought in

$F\in C^1(\mathbb R,\mathbb R)$2

A weak solution $F\in C^1(\mathbb R,\mathbb R)$3 satisfies

$F\in C^1(\mathbb R,\mathbb R)$4

for all $F\in C^1(\mathbb R,\mathbb R)$5. The equation is the Euler–Lagrange equation of

$F\in C^1(\mathbb R,\mathbb R)$6

Its associated Pohožaev functional is

$F\in C^1(\mathbb R,\mathbb R)$7

and sufficiently regular solutions satisfy $F\in C^1(\mathbb R,\mathbb R)$8 (Battaglia et al., 2016).

A groundstate is a nontrivial solution of least energy,

$F\in C^1(\mathbb R,\mathbb R)$9

For the planar general-nonlinearity problem, this level coincides with the mountain-pass value

$I_\alpha(x)\sim |x|^{\alpha-2}$0

and the mountain-pass solution is shown to be a groundstate (Battaglia et al., 2016).

In normalized problems, the variational constraint is the prescribed mass. For

$I_\alpha(x)\sim |x|^{\alpha-2}$1

the relevant manifold is

$I_\alpha(x)\sim |x|^{\alpha-2}$2

and normalized solutions are critical points of the energy functional restricted to $I_\alpha(x)\sim |x|^{\alpha-2}$3. In that setting one again introduces a Pohožaev functional and a Pohožaev manifold $I_\alpha(x)\sim |x|^{\alpha-2}$4 (Huang et al., 2024).

For strongly indefinite logarithmic equations, the natural energy is not defined on all of $I_\alpha(x)\sim |x|^{\alpha-2}$5. One uses instead

$I_\alpha(x)\sim |x|^{\alpha-2}$6

together with the Choquard energy

$I_\alpha(x)\sim |x|^{\alpha-2}$7

where $I_\alpha(x)\sim |x|^{\alpha-2}$8 and

$I_\alpha(x)\sim |x|^{\alpha-2}$9

The corresponding Nehari set is split into $I_\alpha$0, $I_\alpha$1, and $I_\alpha$2 according to the sign of $I_\alpha$3 (Cabrera et al., 25 Feb 2025).

3. Two-dimensional criticality and admissible nonlinearities

The central structural fact in the plane is that $I_\alpha$4 does not embed into $I_\alpha$5, and the natural critical growth is exponential. For the general planar equation, the basic hypotheses are: nontriviality $I_\alpha$6, exponential-type growth control

$I_\alpha$7

and the subcriticality condition at zero

$I_\alpha$8

The corresponding analytic input is the planar Moser–Trudinger inequality

$I_\alpha$9

which replaces the higher-dimensional Sobolev power control (Battaglia et al., 2016).

For pure powers within that general framework, $H^1(\mathbb R^2)$0 satisfies the planar assumptions if and only if $H^1(\mathbb R^2)$1, and it is known that there are no nontrivial solutions for $H^1(\mathbb R^2)$2. For the autonomous power equation

$H^1(\mathbb R^2)$3

the survey literature states that the action functional is well defined on $H^1(\mathbb R^2)$4 for

$H^1(\mathbb R^2)$5

while groundstate existence holds for

$H^1(\mathbb R^2)$6

These two statements refer to different parameterizations of the nonlinearity, and they are presented separately in the literature (Battaglia et al., 2016, Moroz et al., 2016).

Criticality takes a different form for fractional planar Choquard equations. For

$H^1(\mathbb R^2)$7

the critical exponent is

$H^1(\mathbb R^2)$8

In the subcritical range

$H^1(\mathbb R^2)$9

there are no positive solutions, while in the critical case $$-\Delta u+u=\bigl(I_\alpha*G(u)\bigr)\,H(u)\qquad\text{in }\mathbb R^N,$$0, every positive solution is radially symmetric and radially nonincreasing about some point in $$-\Delta u+u=\bigl(I_\alpha*G(u)\bigr)\,H(u)\qquad\text{in }\mathbb R^N,$$1 (Ma et al., 2017).

For logarithmic models, criticality is again exponential. In the standard $$-\Delta u+u=\bigl(I_\alpha*G(u)\bigr)\,H(u)\qquad\text{in }\mathbb R^N,$$2 approach to the planar logarithmic Choquard equation, the nonlinearity is allowed to have Moser–Trudinger-critical growth, typified by

$$-\Delta u+u=\bigl(I_\alpha*G(u)\bigr)\,H(u)\qquad\text{in }\mathbb R^N,$$3

together with $$-\Delta u+u=\bigl(I_\alpha*G(u)\bigr)\,H(u)\qquad\text{in }\mathbb R^N,$$4 as $$-\Delta u+u=\bigl(I_\alpha*G(u)\bigr)\,H(u)\qquad\text{in }\mathbb R^N,$$5. In normalized problems, criticality is encoded by the asymptotic law

$$-\Delta u+u=\bigl(I_\alpha*G(u)\bigr)\,H(u)\qquad\text{in }\mathbb R^N,$$6

and the mountain-pass threshold must be kept below $$-\Delta u+u=\bigl(I_\alpha*G(u)\bigr)\,H(u)\qquad\text{in }\mathbb R^N,$$7 (Cassani et al., 2023, Huang et al., 2024).

4. Existence, compactness, and symmetry for standard planar equations

The basic existence theorem for the planar Riesz-kernel equation states that if $$-\Delta u+u=\bigl(I_\alpha*G(u)\bigr)\,H(u)\qquad\text{in }\mathbb R^N,$$8 satisfies $$-\Delta u+u=\bigl(I_\alpha*G(u)\bigr)\,H(u)\qquad\text{in }\mathbb R^N,$$9, $$-\Delta u+u=(I_\alpha*F(u))\,F'(u)\qquad\text{in }\mathbb R^2.$$0, and $$-\Delta u+u=(I_\alpha*F(u))\,F'(u)\qquad\text{in }\mathbb R^2.$$1, then

$$-\Delta u+u=(I_\alpha*F(u))\,F'(u)\qquad\text{in }\mathbb R^2.$$2

has a groundstate solution $$-\Delta u+u=(I_\alpha*F(u))\,F'(u)\qquad\text{in }\mathbb R^2.$$3. Under the same assumptions there is at least one nontrivial solution; every solution belongs to $$-\Delta u+u=(I_\alpha*F(u))\,F'(u)\qquad\text{in }\mathbb R^2.$$4 for every $$-\Delta u+u=(I_\alpha*F(u))\,F'(u)\qquad\text{in }\mathbb R^2.$$5; every solution satisfies the Pohožaev identity; and the set of groundstates is compact in $$-\Delta u+u=(I_\alpha*F(u))\,F'(u)\qquad\text{in }\mathbb R^2.$$6 up to translations (Battaglia et al., 2016).

Under the additional assumptions that $$-\Delta u+u=(I_\alpha*F(u))\,F'(u)\qquad\text{in }\mathbb R^2.$$7 is even and nondecreasing on $$-\Delta u+u=(I_\alpha*F(u))\,F'(u)\qquad\text{in }\mathbb R^2.$$8, any groundstate has constant sign and is radially symmetric with respect to some point $$-\Delta u+u=(I_\alpha*F(u))\,F'(u)\qquad\text{in }\mathbb R^2.$$9. The sign conclusion is obtained from the facts that $u$0 is again a groundstate and that a strong maximum principle applies; radial symmetry is derived by polarization arguments (Battaglia et al., 2016).

The variational proof in the plane differs from the higher-dimensional one because pure dilations do not suffice: in $u$1, the kinetic term $u$2 is invariant under dilation. The construction of the optimal mountain-pass path therefore mixes dilations and amplitude scalings. On the Palais–Smale side, the crucial compactness object is a Pohožaev–Palais–Smale sequence,

$u$3

obtained by Jeanjean’s scaling trick on the enlarged space $u$4 (Battaglia et al., 2016).

For the autonomous power equation, the general survey theory adds the standard qualitative conclusions: ground states are nontrivial weak solutions minimizing the action on the Nehari manifold, they can be chosen radial and radially decreasing up to translation, and in the planar Riesz setting the ground-state existence range is $u$5 (Moroz et al., 2016).

5. Logarithmic, constrained, and mixed-diffusion planar models

For the planar logarithmic Choquard equation

$u$6

a positive solution in the standard Sobolev space $u$7 has been obtained by asymptotically approximating the logarithmic kernel by

$u$8

solving the corresponding regularized problems, and passing to the limit $u$9. This yields a positive radial solution under Moser–Trudinger-critical assumptions on $F$0, and extends earlier results that required log-weighted spaces or more restrictive hypotheses near the origin (Cassani et al., 2023).

A different logarithmic framework appears in the strongly indefinite equation

$F$1

with $F$2 and no sign condition on $F$3. In that setting the energy is defined on

$F$4

whose norm is not translation invariant. A new $F$5-equivariant Cerami condition, combined with deformation arguments built from a family of scalar products $F$6, yields a sequence of high-energy solutions when $F$7 is $F$8-invariant, and more generally in several $F$9-equivariant settings (Cabrera et al., 25 Feb 2025).

Normalized solutions with prescribed mass have also been established for the planar Riesz-kernel equation

$0<\alpha<2$00

Under Trudinger–Moser-critical assumptions on $0<\alpha<2$01, there exists for every $0<\alpha<2$02 a positive radial solution of mountain-pass type. Under an additional monotonicity condition on

$0<\alpha<2$03

the fiber map has a unique maximum on each $0<\alpha<2$04-preserving scaling orbit, and the normalized mountain-pass solution is also a ground state in $0<\alpha<2$05 (Huang et al., 2024).

The planar theory also includes mixed local–nonlocal diffusion. For

$0<\alpha<2$06

with $0<\alpha<2$07, $0<\alpha<2$08, a coercive potential, and Trudinger–Moser critical exponential growth, there exists a least energy positive solution. The proof combines Nehari manifold minimization, compactness below a critical Trudinger–Moser threshold, local regularity, and a strong maximum principle (Chen et al., 29 Jun 2026).

A fractional logarithmic counterpart arises from planar Schrödinger–Poisson systems. There the two-dimensional Green kernel again produces a Choquard term with $0<\alpha<2$09, and the resulting fractional planar Choquard equation admits a positive radial solution under exponential critical growth, together with radial symmetry, monotonicity, polynomial decay of $0<\alpha<2$10, and logarithmic asymptotics for the Poisson potential (Cassani et al., 2023).

6. Nodal patterns, bubble theory, and asymptotic regimes

Sign-changing planar Choquard solutions have been constructed in a Coxeter-symmetric framework. For

$0<\alpha<2$11

finite Coxeter groups generate saddle-type nodal solutions with conical nodal domains. In the planar case $0<\alpha<2$12, the relevant groups are essentially dihedral reflection groups, so the nodal domains are sectors. If $0<\alpha<2$13 is odd and has constant sign on $0<\alpha<2$14, the solution has fixed sign on a fundamental sector and opposite signs on adjacent sectors (Xia, 2021).

A distinct planar regime is the exponential Choquard equation

$0<\alpha<2$15

under the finite-mass condition

$0<\alpha<2$16

All such solutions are explicitly classified: $0<\alpha<2$17 with

$0<\alpha<2$18

For the linearized operator at the standard bubble $0<\alpha<2$19, the kernel in $0<\alpha<2$20 is exactly the span of the three symmetry generators given by the two translations and the scaling mode. This is the planar nondegeneracy statement for Choquard bubbles (Gao et al., 4 Aug 2025).

Bounded-domain asymptotics supply another genuinely planar phenomenon. For

$0<\alpha<2$21

with $0<\alpha<2$22 smooth and bounded, least energy solutions $0<\alpha<2$23 as $0<\alpha<2$24 neither blow up nor vanish: $0<\alpha<2$25 Under suitable assumptions, they develop exactly one peak, while the modified solutions $0<\alpha<2$26 do blow up. After rescaling around the maximum point, the profile converges to the explicit bubble solving

$0<\alpha<2$27

Moreover,

$0<\alpha<2$28

and the blow-up point $0<\alpha<2$29 is a critical point of the Robin function (Gao et al., 4 Aug 2025).

These developments show that the planar Choquard equation is not a single model but a family of two-dimensional nonlocal elliptic problems with several distinct critical structures. The Riesz-kernel whole-space equation admits a Berestycki–Lions-type groundstate theory in $0<\alpha<2$30; logarithmic kernels require either weighted spaces or asymptotic approximation; fractional and mixed-diffusion operators introduce additional scales; and planar asymptotics reveal Liouville-type bubbles, sectorial nodal patterns, Robin-function selection of concentration points, and nondegenerate three-parameter bubble manifolds (Battaglia et al., 2016, Cassani et al., 2023, Gao et al., 4 Aug 2025).