Critical Hardy Potential

• Critical Hardy potential is a singular potential that saturates Hardy inequalities, marking the threshold where energy coercivity fails in PDE settings.
• It plays a crucial role in spectral analysis and boundary behavior, influencing existence, multiplicity, and blow-up phenomenon in diverse equations.
• Its optimal constant (e.g., (N-2)^2/4 for the Laplacian) dictates transitions in variational methods and underpins rigorous trace and capacity theories.

A critical Hardy potential is a singular potential in elliptic and parabolic partial differential equations (PDEs)—notably Schrödinger-type, $p$-Laplace, and fractional Laplacian operators—whose coefficient saturates the sharp Hardy inequality in the relevant functional space. At this threshold, the coercivity or compactness properties of the associated energy forms sharply change, producing a delicate interplay between singularity, spectral theory, existence/multiplicity of solutions, and blow-up. The critical Hardy constant, typically denoted $\mu_c$, is given by the largest value such that the Hardy-type inequality remains valid. For the Laplacian in $\mathbb{R}^N$, this is $\mu_c = (N-2)^2/4$.

The critical Hardy potential has profound consequences in spectral theory, fine regularity of solutions, blow-up analysis, and sharp capacity or trace descriptions of boundary behavior. Its appearance is universal across classical, quasilinear, fractional, nonlocal, and geometric PDE frameworks.

1. Hardy’s Inequality and Definition of the Critical Potential

The classical Hardy inequality in $\mathbb{R}^N$, $N \geq 3$, states

$$\int_{\mathbb{R}^N} |\nabla u(x)|^2 dx \geq \frac{(N-2)^2}{4} \int_{\mathbb{R}^N} \frac{u(x)^2}{|x|^2} dx, \qquad u \in C_c^\infty(\mathbb{R}^N).$$

The constant $(N-2)^2/4$ is optimal and unattainable in $H^1(\mathbb{R}^N)$; equality is never achieved by any nontrivial $H^1$-function (Bieganowski et al., 19 Nov 2025, Gkikas et al., 2014).

The corresponding potential

$V_c(x) = -\frac{(N-2)^2}{4 |x|^2}$

is called the critical Hardy potential; its coefficient exactly saturates the inequality. In broader contexts (domains with boundary, Dirichlet/Neumann/Fractional Laplacian, metrics, higher order, or weights), alternative forms and critical constants are derived; e.g., for the fractional Laplacian ($-\Delta$^α/2\alpha/2), the sharp constant is

$$c_*(d, \alpha) = 2^\alpha \frac{\Gamma^2\left(\frac{d+\alpha}{4}\right)}{\Gamma^2\left(\frac{d-\alpha}{4}\right)}.$$

Here, the critical Hardy potential is $V(x) = -c_*/|x|^\alpha$ (Bogdan et al., 2017, Jakubowski et al., 2022).

2. Rigidity, Optimality, and Spectral Analysis at Criticality

At the critical constant:

• The quadratic form associated to the operator

$$Q[u] := \int_\Omega |\nabla u|^2 dx - \mu_c \int_\Omega \frac{u^2}{|x|^2} dx$$

is nonnegative but fails to be coercive in $H_0^1(\Omega)$.

• No nontrivial functions in $H^1$ (or appropriate Sobolev spaces) saturate the inequality; extremals for the inequality are not in the energy space but rather of the form $|x|^{-\frac{N-2}{2}}$ up to lower order terms.
• The operator’s spectrum behavior changes: the resolvent may become non-compact, and the corresponding semigroup and heat kernel acquire anomalous singular behavior at the singularity (Bieganowski et al., 19 Nov 2025, Bogdan et al., 2017).

For general domains, or for the boundary distance $d(x, \partial\Omega)$, the operator $L_\kappa = -\Delta - \kappa / d(x)^2$ is critical at $\kappa = 1/4$ (Gkikas et al., 2014, Gkikas et al., 2014).

3. Existence, Nonexistence, and Multiplicity of Solutions

The critical Hardy potential exerts a bifurcating effect on solution space topology and compactness:

• Existence below criticality: For $\mu < \mu_c$, coercivity holds and classical variational methods yield ground states and possibly multiple solutions for critical/critical-conformal growth problems in $H^1$ or related spaces (Oliva et al., 2018, You et al., 18 Jul 2024, Li et al., 2022, Wang et al., 2015, Le, 28 Aug 2024).
• Loss of existence at criticality: For $\mu \geq \mu_c$, coercivity fails, variational minimization may yield only trivial or distributional solutions, and spectral theory may degenerate.
• Multiplicity phenomena: Blow-up and profile-decomposition show that at or near criticality, Palais–Smale sequences may concentrate at the singularity, resulting in stratified compactness below critical energy levels (Bieganowski et al., 19 Nov 2025, Maliki et al., 2019, Gkikas et al., 2014, Ghoussoub et al., 2018).
• Critical boundary conditions: In boundary value problems, especially with measure data, the critical Hardy operator precisely marks when unique solvability (with arbitrary boundary measure) fails, and further Besov-capacity conditions become necessary for admissible traces (Gkikas et al., 2014, Gkikas et al., 2014).

4. Eigenvalue Problems, Optimal Constants, and Asymptotics

The critical Hardy potential enters sharply into eigenvalue problems:

• Neumann and Robin eigenvalues: In two dimensions, the optimal Hardy potential involves additional logarithmic weights, e.g., $V_a(x) = |x|^{-2} (\log(a/|x|))^{-2}$, yielding optimal constant 1/4 and precise asymptotic behavior for eigenfunctions near the singularity of the form

$$u_a(x) \sim |x|^{\mu(a)} (\log(a/|x|))^{-1+\mu(a)} \text{ as } x \to 0$$

with exponent $\mu(a) = \sqrt{1 - \lambda_a / 4}$ (Sano et al., 2022).

• Spectral gap and ground state nonattainability: Optimal constants for Hardy–Sobolev or Hardy–Rellich inequalities (with or without boundary) cannot be achieved by functions in the natural space; minimizing sequences concentrate at the singularity. The first eigenfunction for the critical Hardy operator is not in $H_0^1$ but behaves asymptotically like $d(x)^{\alpha_+}$ with $\alpha_+ = 1$ at criticality (Gkikas et al., 2014, Gkikas et al., 2014, Ghoussoub et al., 2018).
• Fractional and nonlocal extensions: Analogous critical constants occur for the fractional Laplacian, where the ground state is $|x|^{-(d-\alpha)/2}$, and the sharp kernel estimates reflect the critical potential’s singularity (Bogdan et al., 2017, Jakubowski et al., 2022).

5. Capacity, Trace, and Removable Sets Theory

The critical Hardy potential is pivotal in the fine boundary-trace theory:

• Weak solutions with measure or distributional data are uniquely determined only below the critical Hardy constant for boundary terms (Gkikas et al., 2014).
• For nonlinearities with critical powers, explicit capacities—often of Besov or fractional Sobolev type—determine whether a Radon measure or a closed set is admissible or “removable” for a boundary trace.
• In the supercritical regime, boundary trace measures must be absolutely continuous with respect to these capacities; in the subcritical regime all finite measures qualify (Gkikas et al., 2014, Gkikas et al., 2014).

Furthermore, in subcritical and critical regimes, isolated singularities in the boundary admit a full dichotomy—between so-called “weak” (removable or moderate) and “strong” (very singular, self-similar blow-up) behaviors, depending on the criticality of the potential and the exponent of the nonlinearity (Gkikas et al., 2014).

6. Weighted Sobolev Inequalities and Logarithmic Corrections

Near or at criticality, weighted Sobolev and Hardy–Sobolev inequalities feature logarithmic corrections essential for the functional analytic structure:

• Sharp inequalities in two dimensions require logarithmic weights to compensate for the slow blow-up of the classical Hardy potential; the critical case,

$$\int_{\Omega} |u|^2 |x|^{-2} (\log (a/|x|))^{-2} dx \leq \frac{1}{4} \int_{\Omega} |\nabla u|^2 dx,$$

is not attained in the corresponding Sobolev space (Sano et al., 2022).

• More generally, in higher-order or nonlocal frameworks, critical Hardy–Sobolev inequalities combine singular weights with sharp exponents and yield optimality only below the critical constant. Test-function estimates and minimization arguments reveal the subtle scaling and concentration properties that accompany these logarithmic and weight corrections (Bieganowski et al., 19 Nov 2025, Le, 28 Aug 2024, Li et al., 2022, Malhotra et al., 28 Nov 2024, Singh, 26 Sep 2024).

7. Nonlocal, Quasilinear, Geometric, and Parabolic Generalizations

The critical Hardy potential governs existence, compactness, and qualitative spectral properties in a wide variety of contemporary PDE contexts:

• Fractional Laplacians and nonlocal PDEs: The critical Hardy constant is explicit for the fractional Laplacian, and the semigroup heat kernel near the singularity exhibits precise two-sided estimates involving ground-state weights (Bogdan et al., 2017, Jakubowski et al., 2022, Wang et al., 2015).
• Quasilinear and nonlinear systems: In $p$-Laplace and quasilinear problems, the energy functional and moving-planes symmetry arguments extend up to the critical constant, with threshold phenomena dictating uniqueness and nonexistence (Oliva et al., 2018, Le, 28 Aug 2024, Malhotra et al., 28 Nov 2024).
• Higher-order PDEs and Rellich-type operators: The critical potential becomes even more singular (e.g., $|x|^{-4}$ for the biharmonic) and sharp Rellich inequalities underpin variational existence for ground, sign-changing, and nodal solutions (Singh, 26 Sep 2024, Li et al., 2022).
• Geometry and Riemannian manifolds: Critical Hardy potentials localize around points via Riemannian distance or geometric weights, and Morse theory–driven multiplicity leverages threshold constants built from local geometry (Maliki et al., 2019, Ghoussoub et al., 2018).
• Critical parabolic and evolution equations: For time-dependent critical Hardy–Sobolev equations, thresholds for global existence, finite-time blow-up, and dichotomy in energy dissipation are established in terms of ground-state solutions and critical levels (Chikami et al., 2020).

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In summary, the critical Hardy potential is a cornerstone of analysis wherever singular, scale-invariant potentials reach the sharp bound of the corresponding Hardy-type or Sobolev-type inequalities. It delineates the regime where standard variational principles, functional inequalities, compactness, and trace theorems are valid, and introduces intricate phenomena—existence threshold, spectral singularity, removable set characterization—that shape the qualitative analysis of PDEs in both classical and modern, local and nonlocal settings (Bieganowski et al., 19 Nov 2025, Gkikas et al., 2014, Sano et al., 2022, Bogdan et al., 2017, Li et al., 2022, Le, 28 Aug 2024, Wang et al., 2015, Oliva et al., 2018, You et al., 18 Jul 2024, Maliki et al., 2019, Ghoussoub et al., 2018, Gkikas et al., 2014, Malhotra et al., 28 Nov 2024, Malhotra et al., 28 Nov 2024, Gu et al., 19 Sep 2025, Jakubowski et al., 2022, Singh, 26 Sep 2024, Chikami et al., 2020).