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arxiv: 1907.02837 · v1 · pith:NOOENY74new · submitted 2019-07-05 · 🧮 math.AP

Variable order nonlocal Choquard problem with variable exponents

Reshmi Biswas , Sweta Tiwari This is my paper

Pith reviewed 2026-05-25 02:11 UTC · model grok-4.3

classification 🧮 math.AP
keywords Choquard problemvariable exponentsvariable ordernonlocal problemsHardy-Sobolev-Littlewood inequalityexistence resultsmultiplicity results
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The pith

Existence and multiplicity hold for the variable-order nonlocal Choquard problem once an adapted Hardy-Sobolev-Littlewood inequality is proved for variable exponents.

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

The paper first establishes a Hardy-Sobolev-Littlewood-type inequality that controls the nonlocal integral term inside variable-order, variable-exponent fractional Sobolev spaces. It then applies this inequality inside the mountain-pass and concentration-compactness arguments to obtain existence of solutions and, under further assumptions on the nonlinearity, multiplicity of solutions for the given Choquard equation. A sympathetic reader cares because the setting allows the order of nonlocality and the growth rate to change with position, which matches models of heterogeneous media.

Core claim

Under suitable assumptions on the continuous functions s, p, μ, α and on the Carathéodory function f, the variable-order nonlocal Choquard problem admits solutions whose existence and multiplicity follow from variational methods after the authors prove the corresponding Hardy-Sobolev-Littlewood inequality for variable exponents in the fractional Sobolev space with variable order and variable exponents.

What carries the argument

The adapted Hardy-Sobolev-Littlewood inequality in the variable-order, variable-exponent fractional Sobolev space, which bounds the nonlocal Choquard term and permits the energy functional to satisfy the Palais-Smale condition.

If this is right

  • At least one nontrivial weak solution exists for sufficiently small values of the parameter λ.
  • Under additional growth conditions on f, at least two distinct weak solutions exist.
  • The zero exterior condition on the complement of the bounded domain is preserved by the obtained solutions.
  • The same variational framework yields solutions when the right-hand side includes both the local power term and the nonlocal Choquard term.

Where Pith is reading between the lines

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

  • The inequality could be tested on unbounded domains or on equations with more general nonlocal kernels that are not exactly of the form 1/|x-y|^μ(x,y).
  • Numerical experiments with rapidly oscillating p and s would check whether multiplicity survives when the variable coefficients vary on a fine scale.

Load-bearing premise

The variable functions p, s, μ and α must be continuous and satisfy the structural conditions that guarantee the fractional Sobolev space is well-defined and the Hardy-Sobolev-Littlewood inequality holds in that space.

What would settle it

A concrete counter-example showing that the Hardy-Sobolev-Littlewood inequality fails for some choice of continuous but non-constant p and s would remove the foundation for the existence and multiplicity claims.

read the original abstract

In this article, we study the existence/multiplicity results for the following variable order nonlocal Choquard problem with variable exponents (-\Delta)_{p(\cdot)}^{s(\cdot)}u(x)&=\lambda|u(x)|^{\alpha(x)-2}u(x)+ \left(\DD\int_\Omega\frac{F(y,u(y))}{|x-y|^{\mu(x,y)}}dy\right)f(x,u(x)), x\in \Omega, u(x)&=0, x\in \mathbb R^N\setminus\Omega, where $\Omega\subset\mathbb R^N$ is a smooth and bounded domain, $N\geq 2$, $p,s,\mu$ and $\alpha$ are continuous functions on $\mathbb R^N\times\mathbb R^N$ and $f(x,t)$ is Carath\'edory function. Under suitable assumption on $s,p,\mu,\alpha$ and $f(x,t)$, first we study the analogous Hardy-Sobolev-Littlewood-type result for variable exponents suitable for the fractional Sobolev space with variable order and variable exponents. Then we give the existence/multiplicity results for the above equation.

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 / 2 minor

Summary. The paper claims to first establish an analogous Hardy-Sobolev-Littlewood inequality adapted to the variable-order, variable-exponent fractional Sobolev space W^{s(·),p(·)}, under the assumption that the continuous functions p, s, μ, α satisfy suitable structural conditions, and then applies this inequality to obtain existence and multiplicity results for the variable-order nonlocal Choquard equation via variational methods on a bounded domain Ω.

Significance. If the adapted inequality holds under the paper's hypotheses, the work would extend classical nonlocal Choquard problems to settings with variable order and variable exponents, which is of interest for modeling inhomogeneous media in nonlocal PDE theory. The significance is limited by the lack of explicit verification that the key inequality is valid without additional regularity on the exponents.

major comments (2)
  1. [Section proving the Hardy-Sobolev-Littlewood inequality] Section proving the Hardy-Sobolev-Littlewood inequality (abstract, paragraph 2, and the dedicated preliminary section): The manuscript assumes only that p, s, μ, α are continuous and satisfy 'suitable assumptions,' but provides no derivation details or error estimates confirming that the inequality holds. Standard results in variable-exponent Sobolev spaces require log-Hölder continuity of the exponents to ensure the modular function yields uniform convexity and valid embeddings; continuity alone is insufficient in general, so the inequality may fail and the subsequent existence argument collapses.
  2. [Main results section] Application to existence/multiplicity (the main results section): The variational arguments for existence and multiplicity rest directly on the validity of the preceding inequality; if the inequality requires unstated log-Hölder conditions, the mountain-pass or genus arguments cannot be carried through under the hypotheses as written.
minor comments (2)
  1. [Abstract] Abstract: The displayed equation contains LaTeX artifacts (e.g., & =, DD∫) that impair readability and should be corrected in the final version.
  2. [Abstract and introduction] Notation: The dependence of μ on (x,y) is introduced without an explicit statement of its symmetry or range, which should be clarified for the integral term to be well-defined.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and for highlighting the need for greater rigor in the treatment of the adapted Hardy-Sobolev-Littlewood inequality. We address each major comment below and will revise the manuscript to strengthen the hypotheses and proofs.

read point-by-point responses

  1. Referee: [Section proving the Hardy-Sobolev-Littlewood inequality] Section proving the Hardy-Sobolev-Littlewood inequality (abstract, paragraph 2, and the dedicated preliminary section): The manuscript assumes only that p, s, μ, α are continuous and satisfy 'suitable assumptions,' but provides no derivation details or error estimates confirming that the inequality holds. Standard results in variable-exponent Sobolev spaces require log-Hölder continuity of the exponents to ensure the modular function yields uniform convexity and valid embeddings; continuity alone is insufficient in general, so the inequality may fail and the subsequent existence argument collapses.

    Authors: We agree that continuity of the exponents alone is generally insufficient and that log-Hölder continuity is required for the modular convexity and embedding properties used in the proof. Our manuscript stated only 'suitable structural conditions' without explicitly listing log-Hölder continuity or supplying the full derivation and error estimates. We will revise the preliminary section to impose the log-Hölder condition on p(·), s(·), μ(·,·) and α(·) and to include a complete, self-contained proof of the variable-order, variable-exponent Hardy-Sobolev-Littlewood inequality together with the necessary estimates. revision: yes

  2. Referee: [Main results section] Application to existence/multiplicity (the main results section): The variational arguments for existence and multiplicity rest directly on the validity of the preceding inequality; if the inequality requires unstated log-Hölder conditions, the mountain-pass or genus arguments cannot be carried through under the hypotheses as written.

    Authors: We concur that the existence and multiplicity theorems rely on the validity of the inequality. Once the preliminary section is revised to include the log-Hölder assumption and the detailed proof, the mountain-pass and genus arguments in the main results section will be justified under the updated hypotheses. We will add an explicit statement of the strengthened assumptions at the beginning of the main theorems and update the references to the inequality accordingly. revision: yes

Circularity Check

0 steps flagged

No circularity; inequality proved independently before application to existence

full rationale

The paper first establishes a Hardy-Sobolev-Littlewood inequality adapted to the variable-order, variable-exponent fractional Sobolev space under stated continuity and structural assumptions on p, s, μ, α, then applies that inequality to obtain existence and multiplicity for the Choquard equation. No equation, parameter, or central claim reduces by construction to a fit, self-definition, or self-citation chain; the derivation chain consists of independent analytic steps whose validity rests on the proof of the inequality rather than on renaming or tautological reuse of inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on continuity and structural hypotheses on the variable functions together with standard properties of variable-exponent Sobolev spaces; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The functions p, s, μ, α are continuous and satisfy the conditions that make the variable-order fractional Sobolev space well-defined and reflexive.
    Invoked in the second sentence of the abstract as prerequisite for the Hardy-Sobolev-Littlewood inequality.
  • domain assumption f is a Carathéodory function satisfying growth conditions compatible with the variable exponents.
    Stated in the abstract as part of the 'suitable assumptions' needed for the variational argument.

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