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arxiv: 2604.24381 · v1 · submitted 2026-04-27 · 🧮 math.AP

A generalisation of the Gagliardo--Nirenberg Inequality with applications to mass-critical and mass-subcritical elliptic equations

Bartosz Bieganowski , Jacopo Schino This is my paper

Pith reviewed 2026-05-08 02:12 UTC · model grok-4.3

classification 🧮 math.AP
keywords Gagliardo-Nirenberg inequalityfractional Laplacianmass-criticalmass-subcriticalexistence of solutionsconstrained variational problemsingular potential
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The pith

A generalized Gagliardo-Nirenberg inequality establishes existence and nonexistence for mass-constrained fractional elliptic equations

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a new inequality in the style of Gagliardo-Nirenberg, adapted to a cylindrical domain with a singular potential term. This inequality is used to analyze a constrained variational problem for the equation involving the fractional Laplacian, yielding precise conditions for the existence or nonexistence of solutions when the L2 mass is fixed at a given positive value ρ. The analysis covers both mass-critical and mass-subcritical nonlinearities, with additional results for the potential-free case and a curl-curl equation. The work also characterizes the critical mass value at which the least energy level transitions from zero to negative.

Core claim

Via a new inequality à la Gagliardo-Nirenberg, the authors prove the existence and nonexistence of solutions to the system (–Δ)^s u + μ/|y|^{2s} u + λ u = f(u) on R^N with fixed L2 norm ρ in the mass-critical and mass-subcritical regimes, for μ belonging to a suitable range and N ≥ K ≥ 2. Analogous existence and nonexistence statements hold when μ = 0 (for N ≥ 1) and for a related curl-curl equation. The threshold value of ρ that separates the regimes of negative least energy from zero least energy is identified explicitly.

What carries the argument

The new Gagliardo-Nirenberg-type inequality that controls the interplay between the fractional Sobolev norm, the singular potential μ/|y|^{2s}, and the nonlinearity under the L2-mass constraint.

If this is right

  • Solutions exist for all sufficiently large ρ in the mass-subcritical regime.
  • Nonexistence holds for all ρ below a critical threshold in both critical and subcritical regimes.
  • When μ = 0 the same existence/nonexistence dichotomy persists for N ≥ 1.
  • The least-energy level is exactly zero for ρ below the threshold and strictly negative above it.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same inequality may yield sharp existence criteria for related nonlocal problems posed on product spaces with cylindrical symmetry.
  • The mass-threshold characterization could be used to study the orbital stability of the standing-wave solutions obtained here.

Load-bearing premise

The parameter μ must lie in a specific range that makes the new inequality valid and keeps the singular term from overwhelming the fractional Laplacian.

What would settle it

An explicit test function in R^K × R^{N-K} for which the proposed inequality fails when μ lies outside the stated range, or a numerical computation of the energy minimizer for a mass ρ just below the claimed threshold that yields a negative energy contrary to the prediction.

read the original abstract

Via a new inequality \`a la Gagliardo--Nirenberg, we prove the existence and nonexistence of solutions to \begin{equation*} \begin{cases} (-\Delta)^s u + \frac{\mu}{|y|^{2s}} u + \lambda u = f(u), \quad \mathbb{R}^N \ni x = (y,z) \in \mathbb{R}^K \times \mathbb{R}^{N-K}, \\ \int_{\mathbb{R}^N} u^2 \, \mathrm{d}x = \rho \end{cases} \end{equation*} in the mass-critical and mass-subcritical regimes, where $s>0$, $N \ge K \ge 2$, $\mu \in \mathbb{R}$ belongs to a specific range, $\rho>0$ is given a priori, and $\lambda \in \mathbb{R}$ is unknown. Additionally, we obtain similar results for the problem above with $\mu=0$ and $N \ge 1$ as well as a related curl-curl equation. Finally, we provide a thorough insight into the threshold for $\rho$ that divides the scenarios of negative and zero least energy.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper introduces a generalization of the Gagliardo-Nirenberg inequality adapted to a mixed-dimensional fractional setting with a singular potential term and applies it to prove existence and nonexistence of solutions to the prescribed-mass fractional elliptic problem in the mass-critical and mass-subcritical regimes. It also obtains analogous results for the potential-free case (N ≥ 1) and a related curl-curl equation, together with a detailed analysis of the critical mass threshold ρ separating negative and zero least-energy levels.

Significance. If the new interpolation inequality holds with a positive constant that cleanly separates the regimes at the critical ρ threshold, the work supplies a useful variational tool for fractional problems with Hardy-type singularities in mixed dimensions. The explicit treatment of the mass threshold and the extension to the curl-curl case add concrete value beyond the classical Gagliardo-Nirenberg theory.

major comments (2)
  1. [Section deriving the generalized Gagliardo-Nirenberg inequality] The central existence/nonexistence dichotomy in the mass-critical case rests on the new Gagliardo-Nirenberg inequality furnishing a strict inequality that separates regimes exactly at the critical ρ. The abstract refers to μ belonging to “a specific range” without stating the bounds; the proof of the inequality (presumably the section deriving the generalized GN estimate) must explicitly identify this range, confirm that the constant remains positive and independent of the solution for all μ in the range, and verify that it does not collapse when μ approaches the Hardy constant associated with the 1/|y|^{2s} term.
  2. [Theorem on existence/nonexistence in the mass-critical regime] In the mixed-dimensional setting N ≥ K ≥ 2 with fractional s > 0, the proof of the inequality must address the additional restrictions imposed by the singular potential and the decomposition x = (y,z). If the constant in the inequality implicitly requires μ larger than a dimension-dependent Hardy threshold or excludes borderline values without separate justification, the nonexistence statement for large ρ in the mass-critical regime fails to hold uniformly.
minor comments (1)
  1. [Abstract] The abstract should state the precise interval for μ rather than referring to “a specific range,” so that readers can immediately assess the applicability of the results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript arXiv:2604.24381. We appreciate the positive assessment of the significance of the generalized Gagliardo-Nirenberg inequality and its applications. We address each major comment point by point below, indicating the revisions we will make.

read point-by-point responses

  1. Referee: [Section deriving the generalized Gagliardo-Nirenberg inequality] The central existence/nonexistence dichotomy in the mass-critical case rests on the new Gagliardo-Nirenberg inequality furnishing a strict inequality that separates regimes exactly at the critical ρ. The abstract refers to μ belonging to “a specific range” without stating the bounds; the proof of the inequality (presumably the section deriving the generalized GN estimate) must explicitly identify this range, confirm that the constant remains positive and independent of the solution for all μ in the range, and verify that it does not collapse when μ approaches the Hardy constant associated with the 1/|y|^{2s} term.

    Authors: We agree that the range for μ should be stated explicitly. In the revised manuscript we will update the abstract to specify μ ∈ (μ_*, μ_H), where μ_H is the Hardy constant for the term 1/|y|^{2s} in the K-dimensional y-variables and μ_* is the lower bound (typically 0 or a small negative value) ensuring the quadratic form is positive. In the section deriving the inequality we will add an explicit statement of this range together with a verification that the best constant C_μ is positive and independent of u for all μ in the interval. We will also include a short lemma showing that liminf_{μ→μ_H^-} C_μ > 0, so the constant does not collapse. These changes will make the separation of regimes at the critical ρ fully transparent. revision: yes

  2. Referee: [Theorem on existence/nonexistence in the mass-critical regime] In the mixed-dimensional setting N ≥ K ≥ 2 with fractional s > 0, the proof of the inequality must address the additional restrictions imposed by the singular potential and the decomposition x = (y,z). If the constant in the inequality implicitly requires μ larger than a dimension-dependent Hardy threshold or excludes borderline values without separate justification, the nonexistence statement for large ρ in the mass-critical regime fails to hold uniformly.

    Authors: The proof already incorporates the mixed-dimensional structure and the singular potential through the decomposition x = (y,z). The generalized GN inequality is obtained by combining the fractional Sobolev embedding on R^N with a weighted Hardy inequality localized to the y-variables of dimension K; the admissible range for μ is chosen strictly below the K-dimensional Hardy constant precisely to guarantee coercivity. We will insert a clarifying paragraph that explicitly recalls this construction and confirms uniformity of the constant across the stated range, including at values approaching the threshold from below. Consequently the nonexistence result for large ρ in the mass-critical regime continues to hold uniformly; no additional restrictions are required. We therefore make only a clarifying addition rather than altering the statement of the theorem. revision: partial

Circularity Check

0 steps flagged

No circularity: new inequality is independent input to variational analysis

full rationale

The derivation begins with a claimed new Gagliardo-Nirenberg-type inequality proved for the mixed-dimensional fractional setting with singular potential. This inequality is then applied via standard mountain-pass or minimization arguments to obtain existence/nonexistence thresholds for the constrained elliptic problem in the mass-critical and mass-subcritical regimes. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the threshold for ρ is derived from the inequality constant rather than presupposed. The paper is self-contained against external benchmarks once the inequality is accepted as proved.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are detailed beyond standard assumptions on s, N, K and the range of μ.

pith-pipeline@v0.9.0 · 5522 in / 985 out tokens · 20002 ms · 2026-05-08T02:12:39.834218+00:00 · methodology

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