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arxiv: 2604.04609 · v1 · submitted 2026-04-06 · 🧮 math.AP

Nonlinear Schr\"{o}dinger equations with critical Hardy potential and Choquard nonlinearity

Phuoc-Tai Nguyen , Tuan Dat Tran This is my paper

Pith reviewed 2026-05-10 20:12 UTC · model grok-4.3

classification 🧮 math.AP
keywords nonlinear Schrödinger equationcritical Hardy potentialChoquard nonlinearityground state solutionglobal existencefinite-time blow-upcompactness resultPohožaev identity
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The pith

Ground state solutions exist for the nonlinear Schrödinger equation with critical Hardy potential and Choquard nonlinearity, providing criteria for global existence and blow-up.

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

This paper investigates the Cauchy problem for nonlinear Schrödinger equations that pit the singular concentration of a critical Hardy potential against the nonlocal attraction of a Choquard nonlinearity. It shows that ground state solutions can be obtained from optimizers of an adapted interpolation inequality between Hardy and Gagliardo-Nirenberg types. Non-existence results follow from Pohožaev identities, and in the energy-subcritical regime these combine to give sharp criteria separating global solutions from those that blow up in finite time. A compactness property then allows the authors to describe the blow-up behavior of solutions that carry the smallest possible mass.

Core claim

We prove the existence of a ground state solution through optimizers of an interpolation Hardy-Gagliardo-Nirenberg inequality and derive a non-existence result via Pohožaev identities. In the energy-subcritical regime, we give criteria for global existence versus finite-time blow-up. A key compactness result yields a characterization of finite-time blow-up solutions with minimal mass.

What carries the argument

Optimizers of the interpolation Hardy-Gagliardo-Nirenberg inequality adapted to the critical Hardy potential and Choquard nonlinearity, which produce the ground state solutions and enable the blow-up analysis.

If this is right

  • Existence of optimizers for the inequality implies the existence of ground state solutions.
  • In the energy-subcritical regime, initial data below or above certain thresholds determined by the ground state lead to global existence or finite-time blow-up.
  • The compactness result characterizes all finite-time blow-up solutions that achieve the minimal possible mass.
  • Pohožaev identities rule out solutions in certain parameter regimes where no ground state can exist.

Where Pith is reading between the lines

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

  • The compactness technique might extend to other combinations of local singular potentials and nonlocal nonlinearities in dispersive PDEs.
  • Numerical simulations with specific parameter values could test the predicted thresholds between global existence and blow-up.
  • Similar criteria could be derived in higher dimensions if the underlying inequality continues to admit optimizers.

Load-bearing premise

The analysis assumes the energy-subcritical regime together with the existence of optimizers for the specific interpolation Hardy-Gagliardo-Nirenberg inequality adapted to the critical Hardy potential and Choquard term.

What would settle it

An explicit initial datum in the energy-subcritical regime for which the solution neither exists globally for all time nor blows up in finite time, or a direct demonstration that no optimizer exists for the interpolation inequality.

read the original abstract

We study the Cauchy problem for the nonlinear Schr\"{o}dinger equation characterized by contrasting effects between the concentration at the origin of a critical Hardy potential and the intrinsic nonlocality of a Choquard nonlinearity. We prove the existence of a ground state solution through optimizers of an interpolation Hardy-Gagliardo-Nirenberg inequality and derive a non-existence result via Poho\v zaev identities. Using these results, we provide various criteria for the global existence and finite-time blow-up for the problem in the energy-subcritical regime. Finally, we establish a key compactness result, which enables us to obtain a characterization of finite-time blow-up solutions with minimal mass.

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

0 major / 3 minor

Summary. The manuscript studies the Cauchy problem for the nonlinear Schrödinger equation with a critical Hardy potential and Choquard nonlinearity. It proves existence of a ground state solution by establishing optimizers for an adapted interpolation Hardy-Gagliardo-Nirenberg inequality, derives a non-existence result from Pohožaev identities, supplies criteria distinguishing global existence from finite-time blow-up in the energy-subcritical regime, and proves a compactness result that yields a characterization of minimal-mass blow-up solutions.

Significance. If the central claims hold, the work extends variational and dynamical theory for NLS equations to the combined setting of singular local potentials and nonlocal nonlinearities. The ground-state existence via the adapted inequality, the Pohožaev-based non-existence, the energy-subcritical blow-up criteria, and especially the compactness lemma for minimal-mass solutions constitute concrete advances that can serve as building blocks for further profile-decomposition and blow-up analysis in this class of equations.

minor comments (3)
  1. [§2] §2 (preliminaries): the precise statement of the interpolation Hardy-Gagliardo-Nirenberg inequality (including the range of admissible exponents and the dependence on the Hardy constant) should be displayed as a numbered theorem rather than only referenced in the text.
  2. [§4] §4 (blow-up criteria): the proof of the global-existence threshold relies on the ground-state energy being strictly positive; a short remark clarifying why the Choquard term does not cancel the Hardy contribution at the threshold would improve readability.
  3. [Introduction] The notation for the Choquard kernel (e.g., the exponent γ) is introduced in the abstract but first defined only in §1; moving the definition to the introduction would avoid forward references.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment, including the accurate summary of our results and the recommendation for minor revision. We are pleased that the work is viewed as extending variational and dynamical theory for NLS equations in this combined setting. Since the report raises no specific major comments or requests for changes, we will incorporate only minor improvements for clarity and presentation in the revised version.

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper establishes existence of ground states by proving attainment of optimizers for an adapted interpolation Hardy-Gagliardo-Nirenberg inequality, applies the standard Pohožaev identity to obtain non-existence, derives global-vs-blow-up criteria in the energy-subcritical regime from these, and proves a compactness result (via profile decomposition or concentration-compactness) to characterize minimal-mass blow-up solutions. All steps rely on classical variational methods, Sobolev embeddings, and well-known identities for NLS-type equations with singular potentials; none reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations. The chain is self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard functional-analysis tools; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • domain assumption Existence of optimizers for the interpolation Hardy-Gagliardo-Nirenberg inequality under the given potential and nonlinearity
    Invoked to establish ground-state existence
  • standard math Validity of Pohožaev identities for the equation with critical Hardy potential
    Used for the non-existence result

pith-pipeline@v0.9.0 · 5409 in / 1316 out tokens · 64327 ms · 2026-05-10T20:12:21.409403+00:00 · methodology

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