Generalized Energy-Critical Hartree Equation

• The generalized energy-critical Hartree equation is a class of nonlocal dispersive and elliptic PDEs where the energy norm remains invariant under scaling.
• It incorporates variable kernels, singular weights, and inverse-square potentials to model diverse threshold dynamics, ground state behavior, and scattering phenomena.
• Analytical methods such as concentration-compactness, profile decomposition, and variational inequalities are used to study well-posedness, nondegeneracy, and stability.

The generalized energy-critical Hartree equation is a class of nonlocal nonlinear dispersive and elliptic equations in which the Hartree or Choquard interaction is tuned so that the natural energy space is invariant under the scaling symmetry of the model. A representative time-dependent form is

$$i\partial_tu+\Delta u+\bigl(I_\lambda*|u|^p\bigr)|u|^{p-2}u=0, \qquad p=\frac{2N-\lambda}{N-2},$$

while the $3D$ inhomogeneous model studied by Guzmán–Xu is

$$i\partial_tu+\Delta u+|x|^{-b}\bigl(I_\alpha*|x|^{-b}|u|^{p}\bigr)|u|^{p-2}u=0, \qquad p=3+\alpha-2b.$$

In both settings, the designation “energy-critical” refers to the invariance of the $\dot H^1$ norm under the scaling of the equation (Li et al., 2023, Guzmán et al., 2023).

1. Equation class and critical scaling

In the cited literature, “generalized” refers to several distinct but closely related extensions of the Hartree model: replacement of the Newton kernel by a general Riesz kernel, introduction of a power $p\ge2$, insertion of singular inhomogeneous weights such as $|x|^{-b}$ or $|x|^{-\tau}$, addition of an inverse-square potential, and reformulation on non-Euclidean geometries such as the Heisenberg group and the torus (Kim, 2022, Kim et al., 2023, Yang et al., 11 Aug 2025, Babb et al., 24 Mar 2025).

For the homogeneous generalized Hartree equation in $\mathbb R^N$,

$$i\partial_tu+\Delta u+\bigl(I_\lambda*|u|^p\bigr)|u|^{p-2}u=0,$$

the critical exponent is

$p=\frac{2N-\lambda}{N-2},$

and the scaling

$3D$0

leaves both the equation and the energy invariant (Li et al., 2023). In the $3D$1 inhomogeneous equation,

$3D$2

and one checks that $3D$3, so the problem is $3D$4-critical (Guzmán et al., 2023).

A broader critical-index formulation appears in the inhomogeneous Hartree Cauchy problem

$3D$5

for which the critical power is

$3D$6

When $3D$7, this becomes

$3D$8

which is the energy-critical regime (Kim, 2022). With inverse-square potential,

$3D$9

energy-criticality occurs when

$$i\partial_tu+\Delta u+|x|^{-b}\bigl(I_\alpha*|x|^{-b}|u|^{p}\bigr)|u|^{p-2}u=0, \qquad p=3+\alpha-2b.$$0

so that the $$i\partial_tu+\Delta u+|x|^{-b}\bigl(I_\alpha*|x|^{-b}|u|^{p}\bigr)|u|^{p-2}u=0, \qquad p=3+\alpha-2b.$$1 norm is scaling-invariant (Kim et al., 2023).

Analogous critical exponents persist outside Euclidean flat space. On the Heisenberg group, the upper critical exponent is

$$i\partial_tu+\Delta u+|x|^{-b}\bigl(I_\alpha*|x|^{-b}|u|^{p}\bigr)|u|^{p-2}u=0, \qquad p=3+\alpha-2b.$$2

with $$i\partial_tu+\Delta u+|x|^{-b}\bigl(I_\alpha*|x|^{-b}|u|^{p}\bigr)|u|^{p-2}u=0, \qquad p=3+\alpha-2b.$$3, while on $$i\partial_tu+\Delta u+|x|^{-b}\bigl(I_\alpha*|x|^{-b}|u|^{p}\bigr)|u|^{p-2}u=0, \qquad p=3+\alpha-2b.$$4 for $$i\partial_tu+\Delta u+|x|^{-b}\bigl(I_\alpha*|x|^{-b}|u|^{p}\bigr)|u|^{p-2}u=0, \qquad p=3+\alpha-2b.$$5-order Hartree nonlinearities the scaling-critical regularity is

$$i\partial_tu+\Delta u+|x|^{-b}\bigl(I_\alpha*|x|^{-b}|u|^{p}\bigr)|u|^{p-2}u=0, \qquad p=3+\alpha-2b.$$6

and the energy-critical case is exactly $$i\partial_tu+\Delta u+|x|^{-b}\bigl(I_\alpha*|x|^{-b}|u|^{p}\bigr)|u|^{p-2}u=0, \qquad p=3+\alpha-2b.$$7 (Yang et al., 11 Aug 2025, Babb et al., 24 Mar 2025). A recurring source of confusion is therefore that “energy-critical Hartree equation” does not designate a single PDE; it designates a scaling relation between the dispersive part, the convolution kernel, and the nonlinear power.

2. Conserved quantities and variational structure

For sufficiently regular solutions, the central conserved quantities are mass and energy. In the $$i\partial_tu+\Delta u+|x|^{-b}\bigl(I_\alpha*|x|^{-b}|u|^{p}\bigr)|u|^{p-2}u=0, \qquad p=3+\alpha-2b.$$8 focusing inhomogeneous generalized Hartree equation, the formal conservation laws are

$$i\partial_tu+\Delta u+|x|^{-b}\bigl(I_\alpha*|x|^{-b}|u|^{p}\bigr)|u|^{p-2}u=0, \qquad p=3+\alpha-2b.$$9

and

$\dot H^1$0

Guzmán–Xu also establish a sharp Gagliardo–Nirenberg-type inequality,

$\dot H^1$1

whose optimizers are nonnegative radial ground states $\dot H^1$2 solving

$\dot H^1$3

(Guzmán et al., 2023).

For the homogeneous generalized energy-critical Hartree equation, the conserved energy takes the form

$\dot H^1$4

and the positive radial ground state $\dot H^1$5 solves

$\dot H^1$6

The ground state is variationally characterized through the sharp Hardy–Littlewood–Sobolev and Sobolev inequalities (Li et al., 2023).

In the classical energy-critical Hartree equation in dimension $\dot H^1$7,

$\dot H^1$8

the conserved energy is

$\dot H^1$9

and the stationary profile $p\ge2$0 is the optimizer in the sharp Sobolev–Hardy–Littlewood–Sobolev inequality (Miao et al., 2011).

Several variants retain the same variational architecture. With inverse-square potential there is a best constant $p\ge2$1 such that

$p\ge2$2

and equality is attained by a positive radial ground state $p\ge2$3 solving

$p\ge2$4

(Kim et al., 2023). Slightly above threshold, the variational functional

$p\ge2$5

controls the sign-based dichotomy near the ground-state manifold (Li et al., 4 Jun 2025).

3. Ground states, bubbles, and nondegeneracy

A central structural problem is the nondegeneracy of the positive bubble. For the elliptic generalized energy-critical Hartree equation

$p\ge2$6

the standard bubble

$p\ge2$7

solves the equation after normalization of $p\ge2$8, and the linearized operator $p\ge2$9 satisfies

$|x|^{-b}$0

More precisely,

$|x|^{-b}$1

The proof uses stereographic projection, a weighted pushforward to the sphere, spherical harmonic decomposition, and the Funk–Hecke formula (Li et al., 2023).

For the time-dependent generalized energy-critical Hartree equation, Li–Liu–Tang–Xu state the corresponding nondegeneracy in both $|x|^{-b}$2 and $|x|^{-b}$3: any solution $|x|^{-b}$4 of the linearized homogeneous equation belongs to

$|x|^{-b}$5

and there are no additional bounded solutions in $|x|^{-b}$6. Their argument combines the nondegeneracy result for bounded solutions with a Moser iteration that upgrades $|x|^{-b}$7 kernel modes to $|x|^{-b}$8 (Li et al., 2023).

On the Heisenberg group, the same theme persists. The unique positive solution is

$|x|^{-b}$9

up to Heisenberg translations, dilations, and constant phase, and the linearized operator has nullspace exactly the span of the infinitesimal generators

$|x|^{-\tau}$0

so that $|x|^{-\tau}$1 (Yang et al., 11 Aug 2025).

These nondegeneracy statements are not merely spectral refinements. The cited works identify them as crucial ingredients for Lyapunov–Schmidt constructions, spectral and dispersive stability analysis, and the construction and classification of threshold solutions (Li et al., 2023, Li et al., 2023).

4. Well-posedness and scattering theory

The local and global Cauchy theory for generalized energy-critical Hartree equations is developed in several directions. In the inhomogeneous critical case on $|x|^{-\tau}$2, Seongyeon Kim establishes local well-posedness in $|x|^{-\tau}$3, $|x|^{-\tau}$4, at the critical power

$|x|^{-\tau}$5

and proves small-data global existence and scattering. A key feature is the use of Sobolev–Lorentz spaces to control simultaneously the singular weight $|x|^{-\tau}$6 and the Riesz potential $|x|^{-\tau}$7 (Kim, 2022).

For the $|x|^{-\tau}$8 focusing inhomogeneous generalized Hartree equation,

$|x|^{-\tau}$9

Guzmán–Xu prove that if

$\mathbb R^N$0

then the solution is global and scatters in $\mathbb R^N$1, without a radiality assumption. They also prove scattering for the classical generalized Hartree equation with $\mathbb R^N$2 under radial data and, in the defocusing case, scattering with general data (Guzmán et al., 2023).

With inverse-square potential, Kim–Saanouni develop local well-posedness in the energy space

$\mathbb R^N$3

prove small-data scattering, and derive a threshold dichotomy between energy-bounded and non-global behavior below the ground-state threshold (Kim et al., 2023). On $\mathbb R^N$4, Acosta Babb–Rout prove local well-posedness in $\mathbb R^N$5 for $\mathbb R^N$6-order Hartree equations, and in the energy-critical case $\mathbb R^N$7 they obtain small-data global well-posedness in $\mathbb R^N$8 for a special three-body symmetric kernel (Babb et al., 24 Mar 2025).

The defocusing theory also has a probabilistic large-data component. Tao–Zhao prove almost sure scattering for the defocusing energy-critical Hartree equation on $\mathbb R^N$9 with randomized initial data $$i\partial_tu+\Delta u+\bigl(I_\lambda*|u|^p\bigr)|u|^{p-2}u=0,$$0 for any $$i\partial_tu+\Delta u+\bigl(I_\lambda*|u|^p\bigr)|u|^{p-2}u=0,$$1, using a modified interaction Morawetz estimate, stability theory, and the “Narrowed” Wiener randomization (Tao et al., 2023).

5. Threshold and near-threshold dynamics

The dynamical theory near the ground state follows the concentration-compactness/rigidity program and its refinements. For the focusing classical energy-critical Hartree equation in $$i\partial_tu+\Delta u+\bigl(I_\lambda*|u|^p\bigr)|u|^{p-2}u=0,$$2 with radial data, the sub-threshold dichotomy is: if $$i\partial_tu+\Delta u+\bigl(I_\lambda*|u|^p\bigr)|u|^{p-2}u=0,$$3 and $$i\partial_tu+\Delta u+\bigl(I_\lambda*|u|^p\bigr)|u|^{p-2}u=0,$$4, then the solution is global and scatters; if $$i\partial_tu+\Delta u+\bigl(I_\lambda*|u|^p\bigr)|u|^{p-2}u=0,$$5 and $$i\partial_tu+\Delta u+\bigl(I_\lambda*|u|^p\bigr)|u|^{p-2}u=0,$$6 and either $$i\partial_tu+\Delta u+\bigl(I_\lambda*|u|^p\bigr)|u|^{p-2}u=0,$$7 is radial or $$i\partial_tu+\Delta u+\bigl(I_\lambda*|u|^p\bigr)|u|^{p-2}u=0,$$8, then finite-time blow-up occurs (Miao et al., 2011).

At the exact threshold $$i\partial_tu+\Delta u+\bigl(I_\lambda*|u|^p\bigr)|u|^{p-2}u=0,$$9, Miao–Wu–Xu identify three distinguished radial solutions: the stationary ground state $p=\frac{2N-\lambda}{N-2},$0 and two special solutions $p=\frac{2N-\lambda}{N-2},$1, one of which scatters in one time direction and converges exponentially to $p=\frac{2N-\lambda}{N-2},$2 in the other (Miao et al., 2011). Li–Liu–Tang–Xu extend the threshold classification to the generalized energy-critical Hartree equation with Riesz kernel and radial data. Their trichotomy is formulated in terms of the comparison between $p=\frac{2N-\lambda}{N-2},$3 and $p=\frac{2N-\lambda}{N-2},$4, and again yields the alternatives of scattering, finite-time blow-up, or coincidence with the special threshold solutions $p=\frac{2N-\lambda}{N-2},$5 up to symmetries (Li et al., 2023).

Above the ground state, a further radial classification is available for the classical equation. For energies at most slightly larger than $p=\frac{2N-\lambda}{N-2},$6, the phase portrait is organized by hyperbolic dynamics near the ground-state manifold, the ejection mechanism, and the one-pass lemma. In that regime, radial solutions exhibit exactly one of scattering, finite-time blow-up, or grow-up, and each of the four combinations of forward/backward scattering or blow-up has nonempty interior in phase space (Li et al., 4 Jun 2025).

These results clarify a common misconception: threshold analysis is not exhausted by the sign of the nonlinearity. The decisive objects are the variational ground state, the spectral structure of the linearized operator, and the modulation dynamics near the static orbit.

6. Analytical methods, extensions, and open problems

The dominant methodology is the Kenig–Merle concentration-compactness/rigidity roadmap. In the inhomogeneous $p=\frac{2N-\lambda}{N-2},$7 problem, Guzmán–Xu combine linear profile decomposition, a nonlinear profile construction adapted to data localized far from the origin, extraction of a minimal non-scattering critical element, a reduced Duhamel formula, and a localized Morawetz identity. A technically distinctive feature is the two-scale cutoff argument using $p=\frac{2N-\lambda}{N-2},$8 in frequency and $p=\frac{2N-\lambda}{N-2},$9 in space, together with commutator estimates for operators such as $3D$00 (Guzmán et al., 2023).

Other settings demand different functional machinery. The critical inhomogeneous theory in $3D$01 relies on Lorentz-refined Strichartz estimates, Hardy–Littlewood–Sobolev bounds in Lorentz spaces, and fractional Leibniz and chain rules (Kim, 2022). The inverse-square model uses Caffarelli–Kohn–Nirenberg weighted interpolation inequalities and norm equivalences involving $3D$02 (Kim et al., 2023). The Heisenberg-group theory uses the Cayley transform, bi-homogeneous harmonic decomposition, and a Heisenberg analogue of Funk–Hecke diagonalization (Yang et al., 11 Aug 2025). On $3D$03, the critical theory is formulated in atomic spaces $3D$04 and dual spaces $3D$05 (Babb et al., 24 Mar 2025).

Several open directions are explicitly identified. Guzmán–Xu list the extension of global well-posedness and scattering to dimensions $3D$06 for both focusing and defocusing inhomogeneous Hartree equations, the non-radial scattering problem in the homogeneous case $3D$07 in $3D$08 and $3D$09, and the investigation of the energy-critical NLS/Hartree interplay in dimensions $3D$10 where the non-radial theory remains more subtle (Guzmán et al., 2023). Acosta Babb–Rout point to large-data global theory in the critical super-critical regime $3D$11, more singular kernels $3D$12, scattering versus blow-up dichotomies, many-body mean-field derivations on $3D$13, almost-sure well-posedness below $3D$14, and extensions to irrational tori or general manifolds (Babb et al., 24 Mar 2025).

Taken together, these developments show that the generalized energy-critical Hartree equation is less a single equation than a research program centered on scale-critical nonlocality, variational ground states, nondegenerate bubble manifolds, and the interaction between dispersive scattering and nonlocal concentration.