Non-Gaussian Hardy Equation: Critical Insights

• The non-Gaussian Hardy equation is a family of PDEs featuring algebraic weight functions that replace Gaussian profiles and control singularities.
• Key formulations such as the Hardy–Hénon equation and Hardy–Poincaré inequalities are analyzed through scaling, integral representations, and radial transformations.
• Advanced models incorporate fractional time derivatives and stable Lévy generators to capture memory effects and anisotropic diffusion in a non-Gaussian setting.

Searching arXiv for recent and foundational papers on the non-Gaussian Hardy equation and related Hardy–Hénon / Hardy–Poincaré formulations. The non-Gaussian Hardy equation denotes, in current arXiv usage, a family of Hardy-type PDEs and inequalities in which the singular, weighted, or diffusive structure is not Gaussian. Representative formulations include the higher-order Hardy–Hénon equation

$$(-\Delta)^m u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$

the second-order Hardy–Hénon equation

$$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$

the Hardy–Poincaré framework with polynomially weighted measures $d\mu_\alpha(x)=(1+|x|^2)^\alpha dx$, and the time-fractional Hardy-type equation

$$\partial_t^\alpha(u-u_0)+\Psi_\beta(-i\nabla)u=|x|^{-\gamma}|u|^{p-1}u \quad \text{in }(0,\infty)\times\mathbb{R}^d.$$

In these settings, the non-Gaussian character comes from algebraic weights such as $|x|^\sigma$, polynomially decaying measures, Riesz kernels, or stable Lévy generators with non-Gaussian heat kernels rather than Gaussian profiles (Ngô et al., 2020, Giga et al., 2022, Dolbeault et al., 2011, Solís et al., 15 Jul 2025).

1. Terminology and principal formulations

In the cited literature, the phrase is not restricted to a single canonical equation. It is used for Hardy-type problems in which the Hardy singularity interacts with a non-Gaussian ambient structure. For the Hardy–Hénon equation, the relevant weight is the power law $|x|^\sigma$: $\sigma>0$ is Hénon-type, $\sigma=0$ is Lane–Emden-type, and $\sigma<0$ is Hardy-type. In the Hardy–Poincaré setting, the non-Gaussian feature is the family of weights $h_\alpha(x)=(1+|x|^2)^\alpha$, which interpolate between the pure Hardy and Gaussian Poincaré regimes after rescaling. In the time-fractional setting, the non-Gaussian feature is the stable-type pseudo-differential operator $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$0, whose fundamental solution has heavy polynomial tails and anisotropic scaling rather than Gaussian decay (Ngô et al., 2020, Dolbeault et al., 2011, Solís et al., 15 Jul 2025).

Formulation	Non-Gaussian feature	Reference
$$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$1	power-law Hardy–Hénon weight	(Ngô et al., 2020)
$$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$2	algebraic weight, full parameter classification	(Giga et al., 2022)
Hardy–Poincaré inequalities	measure $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$3	(Dolbeault et al., 2011)
$$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$4	stable Lévy generator and fractional time	(Solís et al., 15 Jul 2025)

A recurrent misconception is that a Hardy equation must place the singular factor on the linear side as a term like $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$5. The Hardy–Hénon literature shows that the Hardy weight may instead appear in the nonlinear source $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$6, while retaining the same analytical role of controlling singularity at the origin and modifying scaling (Ngô et al., 2020). This suggests that “Hardy” is best understood structurally rather than syntactically.

2. Higher-order Hardy–Hénon equations

For the higher-order equation

$$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$7

the basic parameters are the polyharmonic order $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$8, the dimension $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$9, the weight exponent $d\mu_\alpha(x)=(1+|x|^2)^\alpha dx$0, and the nonlinearity exponent $d\mu_\alpha(x)=(1+|x|^2)^\alpha dx$1. A central quantity is

$d\mu_\alpha(x)=(1+|x|^2)^\alpha dx$2

which measures the singular behavior of the formal power profile $d\mu_\alpha(x)=(1+|x|^2)^\alpha dx$3. The sharp necessary condition for the existence of nonnegative nontrivial distributional solutions is

$d\mu_\alpha(x)=(1+|x|^2)^\alpha dx$4

If this inequality fails, then no nonnegative nontrivial distributional solution exists. When $d\mu_\alpha(x)=(1+|x|^2)^\alpha dx$5, this threshold can be rewritten as

$d\mu_\alpha(x)=(1+|x|^2)^\alpha dx$6

For $d\mu_\alpha(x)=(1+|x|^2)^\alpha dx$7, the condition is sharp for distributional solutions: if $d\mu_\alpha(x)=(1+|x|^2)^\alpha dx$8, the obstruction disappears and a distributional solution of power type can be built (Ngô et al., 2020).

For classical solutions, the same exponent separates the subcritical and critical/supercritical regimes. If $d\mu_\alpha(x)=(1+|x|^2)^\alpha dx$9, $$\partial_t^\alpha(u-u_0)+\Psi_\beta(-i\nabla)u=|x|^{-\gamma}|u|^{p-1}u \quad \text{in }(0,\infty)\times\mathbb{R}^d.$$0, and

$$\partial_t^\alpha(u-u_0)+\Psi_\beta(-i\nabla)u=|x|^{-\gamma}|u|^{p-1}u \quad \text{in }(0,\infty)\times\mathbb{R}^d.$$1

then there is no nonnegative nontrivial classical solution. When $$\partial_t^\alpha(u-u_0)+\Psi_\beta(-i\nabla)u=|x|^{-\gamma}|u|^{p-1}u \quad \text{in }(0,\infty)\times\mathbb{R}^d.$$2, the denominator vanishes, and the paper proves separately that no classical solution exists for any $$\partial_t^\alpha(u-u_0)+\Psi_\beta(-i\nabla)u=|x|^{-\gamma}|u|^{p-1}u \quad \text{in }(0,\infty)\times\mathbb{R}^d.$$3. By contrast, if $$\partial_t^\alpha(u-u_0)+\Psi_\beta(-i\nabla)u=|x|^{-\gamma}|u|^{p-1}u \quad \text{in }(0,\infty)\times\mathbb{R}^d.$$4, $$\partial_t^\alpha(u-u_0)+\Psi_\beta(-i\nabla)u=|x|^{-\gamma}|u|^{p-1}u \quad \text{in }(0,\infty)\times\mathbb{R}^d.$$5, and

$$\partial_t^\alpha(u-u_0)+\Psi_\beta(-i\nabla)u=|x|^{-\gamma}|u|^{p-1}u \quad \text{in }(0,\infty)\times\mathbb{R}^d.$$6

then positive radially symmetric classical solutions exist (Ngô et al., 2020).

The exponent

$$\partial_t^\alpha(u-u_0)+\Psi_\beta(-i\nabla)u=|x|^{-\gamma}|u|^{p-1}u \quad \text{in }(0,\infty)\times\mathbb{R}^d.$$7

is the higher-order weighted Hardy–Hénon critical exponent. Its origin is scaling: under $$\partial_t^\alpha(u-u_0)+\Psi_\beta(-i\nabla)u=|x|^{-\gamma}|u|^{p-1}u \quad \text{in }(0,\infty)\times\mathbb{R}^d.$$8, invariance forces

$$\partial_t^\alpha(u-u_0)+\Psi_\beta(-i\nabla)u=|x|^{-\gamma}|u|^{p-1}u \quad \text{in }(0,\infty)\times\mathbb{R}^d.$$9

The weight $|x|^\sigma$0 changes the balance between the operator and the nonlinearity. For $|x|^\sigma$1, the theory is largely complete; $|x|^\sigma$2 is a Hardy borderline, and the data specifically note that Mitidieri–Pohozaev show even punctured supersolutions fail to exist there (Ngô et al., 2020).

3. Distributional formulation, integral representation, and super polyharmonicity

The higher-order theory distinguishes classical, punctured, and distributional solutions. For $|x|^\sigma$3, a classical solution belongs to $|x|^\sigma$4; for $|x|^\sigma$5, it belongs to $|x|^\sigma$6. A punctured solution is $|x|^\sigma$7 on $|x|^\sigma$8. A distributional solution satisfies

$|x|^\sigma$9

and

$|x|^\sigma$0

Throughout that theory, “solution” means nonnegative and nontrivial (Ngô et al., 2020).

A decisive structural result is that, for $|x|^\sigma$1 and $|x|^\sigma$2, every distributional solution satisfies a Riesz-potential integral equation. With

$|x|^\sigma$3

one has

$|x|^\sigma$4

The proof uses a ring condition at infinity, a representation theorem of Caristi–D’Ambrosio–Mitidieri, and testing against radial kernels (Ngô et al., 2020).

The same paper proves weak and strong super polyharmonicity. The weak form states that for $|x|^\sigma$5 and every nonnegative $|x|^\sigma$6,

$|x|^\sigma$7

For classical solutions in the critical and supercritical range, this bootstraps to

$|x|^\sigma$8

The super polyharmonic property restores a positivity mechanism that plays the role ordinarily filled by a maximum principle in second-order theory (Ngô et al., 2020).

The threshold

$|x|^\sigma$9

has an additional interpretation: it is exactly the condition under which a punctured or classical solution is automatically a distributional solution. In the Hardy regime $\sigma>0$0, explicit punctured power profiles may satisfy the PDE pointwise while failing to satisfy the distributional formulation when this integrability condition is violated (Ngô et al., 2020).

4. Exhaustive second-order classification

For the second-order Hardy–Hénon equation

$\sigma>0$1

the 2022 classification treats arbitrary $\sigma>0$2, arbitrary $\sigma>0$3, and arbitrary $\sigma>0$4. For $\sigma>0$5, the existence theorem is complete: there exists a nontrivial nonnegative classical solution in $\sigma>0$6 if and only if

$\sigma>0$7

Equivalently, there is no nontrivial nonnegative classical solution in dimension $\sigma>0$8 for any $\sigma>0$9 and $\sigma=0$0, and for $\sigma=0$1 the supercritical Hardy–Sobolev regime is exactly

$\sigma=0$2

This recovers the classical Lane–Emden picture at $\sigma=0$3, the Hénon picture at $\sigma=0$4, and the Hardy regime at $\sigma=0$5 (Giga et al., 2022).

The one-dimensional classification is qualitatively different. On the full line $\sigma=0$6, the ODE

$\sigma=0$7

admits a nontrivial nonnegative classical solution if and only if one of the following holds: $\sigma=0$8 On the half-line $\sigma=0$9, the existence criterion becomes

$\sigma<0$0

In the one-dimensional existence regimes, there are explicit power-type solutions

$\sigma<0$1

with $\sigma<0$2 precisely in those regimes (Giga et al., 2022).

The same paper develops a detailed one-dimensional nonuniqueness theory. In the regime $\sigma<0$3 and $\sigma<0$4, there exists a one-parameter family $\sigma<0$5 of positive classical solutions on $\sigma<0$6, all satisfying

$\sigma<0$7

and strictly ordered by the parameter. A concrete example is the case $\sigma<0$8, $\sigma<0$9, where

$h_\alpha(x)=(1+|x|^2)^\alpha$0

all solve $h_\alpha(x)=(1+|x|^2)^\alpha$1 on $h_\alpha(x)=(1+|x|^2)^\alpha$2. In the regime $h_\alpha(x)=(1+|x|^2)^\alpha$3, $h_\alpha(x)=(1+|x|^2)^\alpha$4, additional asymptotic behaviors occur, and when $h_\alpha(x)=(1+|x|^2)^\alpha$5 the paper proves the existence of oscillatory positive solutions for which $h_\alpha(x)=(1+|x|^2)^\alpha$6 remains bounded but has no limit as $h_\alpha(x)=(1+|x|^2)^\alpha$7 (Giga et al., 2022).

Methodologically, the classification combines spherical averaging, monotonicity, integral estimates, and a one-dimensional Kelvin transform

$h_\alpha(x)=(1+|x|^2)^\alpha$8

which maps

$h_\alpha(x)=(1+|x|^2)^\alpha$9

to the same equation with transformed parameter

$$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$00

This duality connects distinct Hardy-type regimes and is one reason the classification is exhaustive (Giga et al., 2022).

5. Hardy–Poincaré inequalities and the non-Gaussian Hardy equation as an ODE

A complementary use of the term appears in the theory of improved Hardy, Gaussian Poincaré, and Hardy–Poincaré inequalities. For $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$01, define

$$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$02

The Hardy–Poincaré inequality takes the form

$$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$03

with explicit sharp spectral constant $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$04. This family interpolates between the classical Hardy inequality as $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$05 and the Gaussian Poincaré inequality as $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$06 after rescaling (Dolbeault et al., 2011).

The paper then studies improved inequalities by recursive “expansion of the square.” In the Hardy setting, this yields the Filippas–Tertikas asymptotic expansion with positive remainder terms $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$07. In the Gaussian setting, the quadratic form

$$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$08

admits an improved inequality

$$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$09

with asymptotically optimal coefficients. The same recursive strategy is carried into the Hardy–Poincaré family (Dolbeault et al., 2011).

In this framework, the “non-Gaussian Hardy equation” is the radial ODE governing the critical potential obtained from expansion of a square. For the Hardy–Poincaré family, the key equation is

$$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$10

It is presented as the non-Gaussian analogue of the Gaussian equation

$$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$11

and the Hardy equation

$$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$12

The improved Hardy–Poincaré inequalities are then obtained by solving this equation asymptotically and iterating suitable changes of variables, producing weights $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$13, $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$14, $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$15, and $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$16 with sharp coefficient $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$17 at each order (Dolbeault et al., 2011).

The conceptual significance is that a Hardy equation need not be a semilinear PDE at all. It may also be the Euler–Lagrange or ground-state ODE underlying a weighted quadratic inequality. This broadens the scope of the term while preserving its central analytic themes: singularity, scale, optimal remainder terms, and asymptotic sharpness (Dolbeault et al., 2011).

6. Fractional time and non-Gaussian stable generators

The time-fractional non-Gaussian Hardy-type equation introduces both memory and nonlocal spatial diffusion: $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$18 with

$$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$19

The time derivative is the Riemann–Liouville derivative

$$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$20

and the spatial operator $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$21 is a pseudo-differential operator of order $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$22, identified as the generator of a stable, possibly anisotropic, Lévy process. When $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$23 and the symbol is isotropic, one recovers the classical Laplacian; for $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$24, the heat kernel is non-Gaussian (Solís et al., 15 Jul 2025).

The linear problem is governed by kernels $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$25 and $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$26, with $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$27 represented by subordination in terms of the stable heat kernel $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$28 and a time-fractional kernel $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$29. The scaling laws are

$$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$30

and the pointwise bounds exhibit algebraic tails together with a logarithmic correction at the critical dimension $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$31. A mild solution is defined by

$$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$32

where $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$33 and $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$34 (Solís et al., 15 Jul 2025).

The critical exponent for this theory is

$$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$35

If

$$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$36

then the equation is locally well posed in $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$37 for sufficiently small $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$38. For global small-data theory, the paper assumes

$$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$39

and proves global mild solutions for sufficiently small $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$40 data or for data satisfying

$$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$41

with sufficiently small $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$42. In both cases, $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$43 as $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$44 (Solís et al., 15 Jul 2025).

The same paper proves nonexistence of local positive mild solutions under a complementary largeness condition. If

$$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$45

then there exists positive $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$46 such that no local positive mild solution exists, and any fixed point fails to belong to $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$47 for any $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$48. A corollary shows the same instantaneous blow-up mechanism when the initial datum dominates the critical Hardy profile

$$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$49

Under additional moment and integrability hypotheses, global solutions satisfy the large-time asymptotic expansion

$$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$50

with

$$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$51

This is a fractional-time, non-Gaussian analogue of classical Hardy and Fujita threshold phenomena (Solís et al., 15 Jul 2025).

7. Hardy–Sobolev singular potentials, radial transforms, and structural synthesis

A related Euclidean formulation places the Hardy singularity on the linear side: $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$52 For radial functions, the map

$$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$53

removes the Hardy singularity and transforms the equation into a semilinear equation without the singular potential, in an effective dimension $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$54 and with modified coefficient $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$55. In the critical whole-space case $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$56, this reduces the radial Hardy–Sobolev equation to the standard critical equation

$$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$57

so the full family of radial positive solutions follows from the Caffarelli–Gidas–Spruck classification. In the subcritical ball case $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$58, the transform yields uniqueness of the positive radial solution for all $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$59 (Dancer et al., 2015).

The same framework gives a sharp radial Hardy–Sobolev inequality and a detailed bifurcation theory. In the critical whole-space problem, degeneracy of the radial branch occurs at the explicit values

$$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$60

At these values, nonradial directions appear in the kernel of the linearized operator, and the paper proves the existence of continua of nonradial weak solutions bifurcating from the radial branch, with symmetry classes such as $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$61 and $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$62. An analogous symmetry-breaking picture is developed in the ball, where bifurcation occurs at parameter values detected by a weighted eigenvalue condition involving spherical harmonics (Dancer et al., 2015).

Across these formulations, the same organizing quantities recur: critical exponents

$$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$63

borderline singularities $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$64, $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$65, or $$-\Delta u = |x|^\sigma u^p \quad \text{in }\mathbb{R}^n,$$66, and structural reductions to integral equations, radial ODEs, or transformed nonsingular problems (Ngô et al., 2020, Giga et al., 2022, Solís et al., 15 Jul 2025, Dancer et al., 2015). This suggests a unifying principle: in non-Gaussian Hardy problems, the decisive balance is between the singular Hardy profile, the scaling of the operator, and the integrability or decay allowed by the ambient non-Gaussian structure.