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REVIEW 3 major objections 6 minor 42 references

Positive solutions to fractional $p$-Laplacian Choquard equation on lattice graphs

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Under mild growth conditions and a coercive potential, the fractional p-Laplacian Choquard equation on Z^d has a strictly positive solution; adding a monotonicity condition gives a positive ground state.

desk verdict Plausible first combination of fractional p-Laplacian and Choquard on lattice graphs, but the (PS)_c proof rests on a false convexity inequality. read the letter →

arxiv 2507.22552 v1 pith:EDIQ7IID submitted 2025-07-30 math.AP

Lidan Wang This is my paper · ORCID
classification math.AP MSC 35A1535R0235R11
keywords fractionalp-LaplacianChoquardequationlatticegraphspositivesolutiongroundstatemountain-passtheoremNeharimanifolddiscreteHardy-Littlewood-Sobolevinequality
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves existence of strictly positive solutions to a discrete fractional p-Laplacian Choquard equation on the integer lattice Z^d, where the Riesz-potential term is built from the Green's function of the discrete fractional Laplacian and behaves like |x-y|^{-(d-\$\alpha$)}. The result matters because it transplants a well-studied nonlocal Choquard problem from continuous Euclidean space to a discrete lattice setting, where compactness must come from the potential rather than from decay at infinity. Under hypotheses (h1)-(h2) and (f1)-(f3), the mountain-pass theorem yields a nontrivial weak solution, and a truncation argument shows it is positive at every vertex. Adding the monotonicity assumption (f4) upgrades this to a ground state solution minimizing the energy on the Nehari manifold.

What carries the argument

The object carrying the argument is the energy functional $J_{s,p}(u)=\frac{1}{p}\int(|\nabla^s u|^p + h(x)|u|^p)\,d\mu - \frac{1}{2}\int (R_\alpha * F(u))F(u)\,d\mu$ on the fractional Sobolev space $H^{s,p}$, the completion of compactly supported functions in the norm $(\int(|\nabla^s u|^p + h(x)|u|^p)\,d\mu)^{1/p}$. Three tools make the variational proof work: the compact embedding $H^{s,p}\hookrightarrow\ell^q$ for every $q\ge p$, imported from the fractional Sobolev framework for locally finite graphs; the discrete Hardy-Littlewood-Sobolev inequality controlling the convolution term; and a convexity inequality for the fractional $p$-Laplacian that converts weak convergence into strong convergence in the compactness step. The positivity mechanism is the truncation $\tilde{f}(t)=0$ for $t<0$, with a lemma showing that any nontrivial weak solution of the truncated equation is strictly positive.

What would settle it

Find a bounded sequence in $H^{s,p}$ for a coercive potential $h$ that escapes to infinity and has no strongly convergent subsequence in $\ell^q$ for some $q\ge p$; such a sequence would falsify the compact embedding lemma and collapse the compactness argument behind Theorems 1.1 and 1.2.

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Extended reading notes

Core claim

The central claim is that the equation $(-\Delta)_p^s u + h(x)|u|^{p-2}u = (R_\alpha * F(u))f(u)$ on $\mathbb{Z}^d$ has a strictly positive solution whenever the potential $h$ is bounded below by $h_0>0$ and tends to infinity as $|x|\to\infty$, and $f$ satisfies the stated growth and superlinearity conditions; if in addition the ratio defining (f4) is strictly increasing in $t$, a positive ground state solution exists. The proof constructs the energy functional on the weighted fractional Sobolev space $H^{s,p}$, shows it has mountain-pass geometry and satisfies the compactness condition via the compact embedding $H^{s,p}\hookrightarrow\ell^q$, then uses the Nehari manifold to select a minimizer. Positivity is obtained by truncating $f$ to vanish for negative arguments; a weak solution of the truncated equation is shown to satisfy $u>0$ everywhere on $\mathbb{Z}^d$.

Load-bearing premise

The argument rests on the fractional Sobolev framework of an unpublished preprint, specifically the compact embedding $H^{s,p}$ into $\ell^q$ and a convexity inequality for the fractional $p$-Laplacian, plus the discrete Hardy-Littlewood-Sobolev inequality; if those tools fail for the integer lattice, the existence theorems do not follow.

Editorial extensions

If this is right

  • For every integer lattice $\mathbb{Z}^d$ with any $s\in(0,1)$, $p\ge 2$, and $\alpha\in(0,d)$, the equation admits a strictly positive solution under hypotheses (h1), (h2), (f1)-(f3).
  • When (f4) is added, that solution is a ground state: it minimizes $J_{s,p}$ over the Nehari manifold and hence among all nontrivial weak solutions.
  • The critical value produced by the mountain-pass theorem is positive, so the solution obtained is genuinely nonzero.
  • The compactness tools used are stated for locally finite graphs, so the existence mechanism is not tied to the specific geometry of $\mathbb{Z}^d$ and can be expected to transfer to other coercive lattice graphs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Testing (f4) on pure power nonlinearities $f(t)=t^{q-1}$ would show exactly which powers $q$ admit the ground-state conclusion; this check is left implicit in the paper.
  • The same mountain-pass construction should work on other locally finite graphs with a coercive potential and the same kernel growth, provided the discrete Hardy-Littlewood-Sobolev inequality and the convexity inequality hold; this is an extension, not a paper claim.
  • Because every nontrivial solution is shown to be strictly positive, the method cannot generate sign-changing or nodal solutions; a different minimax scheme would be needed for those.
  • A finite-box numerical experiment on $\mathbb{Z}^d$ with periodic boundary conditions could test whether approximate mountain-pass solutions remain positive and whether their ground-state energy converges as the box grows.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper studies the fractional p-Laplacian Choquard equation (-Δ)_p^s u + h(x)|u|^{p-2}u = (R_α * F(u)) f(u) on the lattice graph Z^d, with s∈(0,1), p≥2, α∈(0,d). Under growth and decay assumptions on the potential h and the nonlinearity f, Theorem 1.1 asserts existence of a strictly positive solution via the mountain-pass theorem, and Theorem 1.2 asserts existence of a positive ground state solution via the Nehari manifold method. The arguments are standard variational ones: compact embedding of the fractional Sobolev space H^{s,p} into ℓ^q, the discrete Hardy-Littlewood-Sobolev inequality, verification of the (PS)_c condition, mountain-pass geometry, Nehari minimization, and a positivity argument for nonnegative weak solutions.

Significance. If the gaps identified below are repaired, the paper would provide a reasonable first extension of Choquard-type existence results to the discrete fractional p-Laplacian setting on lattice graphs. The variational structure is appropriate, the assumptions are explicit, and the two main theorems give concrete existence statements with a clear separation between the mountain-pass result and the Nehari ground-state result. The main mathematical content is a direct adaptation of known continuous counterparts, so the novelty is moderate; the paper does not ship machine-checked proofs or parameter-free derivations, but the proof strategy is standard and, after correction, likely sound.

major comments (3)
  1. [Section 3, Lemma 3.2] The displayed inequality after "By Lemma 15 in [39]" is false as written. The proof claims ∥u_n−u∥_{H^{s,p}}^p ≤ 2^{p−2}p(Term1+Term2), where Term1 and Term2 are the two inner-product expressions just defined. For the monotone operator A(w)=(−Δ)_p^s w + h|w|^{p−2}w, the correct convexity inequality is ∥u_n−u∥^p ≤ C⟨A(u_n)−A(u), u_n−u⟩ = C(Term1−Term2). A concrete counterexample to the printed form is obtained for p=2, h≡1, taking u=3φ and u_n=φ, where φ is the indicator function of a finite set S in Z^d. Then ∥u_n−u∥_{H^{s,p}}^2 = 4∥φ∥_{H^1}^2 > 0, while Term1+Term2 = ⟨A(φ)+A(3φ), φ−3φ⟩ = −8∥φ∥_{H^1}^2, so the right-hand side is negative. The intended conclusion ∥u_n−u∥→0 still follows if the sign is corrected, because both Term1 and Term2 have already been shown to be o(1). The author must correct the sign and either prove the needed inequality or give the exact statement of Lemma 15 of [39], which is currently an unpublished preprint.
  2. [Section 4, Lemma 4.4] The sentence "by Lemma 4.1, we have ∥u_n∥_{H^{s,p}} ≥ η > 0, which implies that u_0 ≠ 0" is not justified by the cited lemma. A bounded sequence in a Banach space can have norms bounded below and still converge weakly to zero, so the implication requires an additional argument. In this setting it can be obtained from the Nehari relation: since u_n ∈ M_{s,p}, ∥u_n∥_{H^{s,p}}^p = ∫(R_α*F(u_n^+))f(u_n^+)u_n^+ ≤ C(∥u_n∥_p^p + ∥u_n∥_p^τ), while Lemma 4.1 gives ∥u_n∥_{H^{s,p}} ≥ η; these inequalities force ∥u_n∥_p ≥ c>0, and then the strong convergence u_n→u_0 in ℓ^p from Lemma 2.7 yields u_0≠0. This step is load-bearing for the Nehari minimization, so the proof should be completed explicitly.
  3. [Sections 2 and 3] Two load-bearing results are taken from the unpublished preprint [39]: the compact embedding H^{s,p}↪ℓ^q for q≥p (Lemma 2.7) and the convexity inequality used in Lemma 3.2. Because [39] is not publicly available in a peer-reviewed venue and because the quotation of the latter inequality is demonstrably wrong as written, the paper is not self-contained at its critical point. The author should either prove these results (the convexity inequality is a short standard argument from the monotonicity of t↦|t|^{p−2}t) or reproduce the precise statements and provide a verifiable reference. This is necessary for the (PS)_c condition and hence for Theorem 1.1.
minor comments (6)
  1. [Section 4, Lemma 4.3] The statement "for any u ∈ H^{s,p}\{0}, there exists a unique t_u > 0 such that t_u u ∈ M_{s,p}" is false after the reduction f(t)=0 for t<0: if u≤0 and u≠0, then u^+=0, so J_{s,p}(tu) = (t^p/p)∥u∥^p has no maximum on (0,∞). The lemma should be restricted to u with u^+≠0; the later applications in Lemma 4.4 and Lemma 4.5 only require that case, since u_0∈M_{s,p} implies u_0^+≠0.
  2. [Section 2, Lemma 2.9] In the proof of strict positivity, the assertion "we have (−Δ)_p^s u(x_0)=0" is not immediate and needs a short justification: since u≥0 and u(x_0)=0 is a minimum, the left-hand side is nonpositive, while the right-hand side of the equation is nonnegative, so both must vanish.
  3. [Section 2, paragraph before Lemma 2.5] The displayed definition of R_α contains a stray expression "2π (d_j/ℓ_j) → k_j, ℓ_j = 0,1,...,d_j−1, as d_j → ∞" that is not defined or used and appears to be a leftover from a discretization argument; it should be removed or clarified.
  4. [Section 2, Lemma 2.8] The proof of Lemma 2.8 uses the notation u_0 for the weak limit, while the statement of the lemma uses u; this is confusing and should be made consistent.
  5. [Throughout] There are several typos and formatting issues, including "p-Laplacain" in the introduction, "forth inequality" instead of "fourth inequality" in the proof of Lemma 3.1(i), and missing diacritics in "Hölder" and "Laplacian" in some places. These should be corrected in a revision.
  6. [References] Reference [39] is an arXiv preprint; if it has been accepted for publication by the time of revision, the citation should be updated. The same applies to other arXiv-only references that may have appeared in journals.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the existence proofs are a standard variational argument whose cited background lemmas are independent of the target theorems.

full rationale

The paper's derivation chain is a conventional mountain-pass / Nehari-manifold argument. The target results, Theorems 1.1 and 1.2, are not built from fitted parameters, hidden definitions of the conclusion, or a self-citation chain that forces the answer. The reduction to f(t)=0 for t<0 and the use of u+ are standard truncation arguments, not a restatement of the desired positivity conclusion; Lemma 2.9 then genuinely derives strict positivity from the equation. The compact embedding (Lemma 2.7) and the convexity inequality used in Lemma 3.2 are cited from the prior work [39] by Zhang, Lin and Yang, which is not authored by the present author and is not shown to contain the target result. The discrete HLS inequality is cited to standard sources, including one self-cited paper, but it is an external classical inequality. The one omitted proof, Lemma 2.6, is attributed to the author's own [33, Lemma 2.5]; this is a minor self-citation and the lemma is a routine consequence of (f3), so it is not load-bearing in the sense of making the main claim equivalent to its input. The possible sign error in the displayed use of Lemma 15 in [39] noted by the skeptic is a correctness risk, not a circularity: a false or misquoted external lemma would invalidate the proof, but it would not make the theorem an input of the argument. Therefore, no circular step can be exhibited from the paper's own equations, and the derivation is self-contained modulo independently stated background results. Score 0 is appropriate because there is no definitional identity, fitted-input-as-prediction, or self-citation chain carrying the central claim.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no free parameters fitted to data and no new entities. Its central claim rests on established background results: the discrete HLS inequality, the fractional Sobolev space machinery from [39], and the behavior of the Green's function from [18]. The main external dependence is the unpublished preprint [39], whose lemmas are load-bearing.

assumptions (4)
  • domain assumption Discrete Hardy-Littlewood-Sobolev inequality for the kernel R_α on Z^d
    Used in Lemmas 2.8, 3.1, 3.2, 4.1, 4.4 to control the nonlocal Choquard term. Cited from [8,26] without proof.
  • domain assumption Fractional Sobolev space framework on Z^d: reflexivity of W^{s,p}, compact embedding Hs,p into ℓ^q, integration by parts formula, and the convexity inequality of Lemma 15 of [39]
    The entire variational setup rests on these results from the unpublished arXiv preprint [39]. Sections 2 and 3 invoke them repeatedly.
  • domain assumption The Green's function R_α has the stated Fourier representation and decays like |x-y|^{-(d-α)}
    Used to define the nonlocal term and to apply the HLS inequality. The decay property is cited from [18].
  • standard math Standard mountain-pass theorem and Nehari manifold method in reflexive Banach spaces
    Used to produce critical points in Theorems 1.1 and 1.2.

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Pith. "Pith review of Positive solutions to fractional $p$-Laplacian Choquard equation on lattice graphs." pith.science (2026). https://pith.science/paper/EDIQ7IID

@misc{pith2026250722552,
  author       = {Pith},
  title        = {Pith review of: Positive solutions to fractional $p$-Laplacian Choquard equation on lattice graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EDIQ7IID}},
  note         = {Machine review of arXiv:2507.22552}
}
abstract

In this paper, we study the fractional $p$-Laplacian Choquard equation $$ (-\Delta)_{p}^{s} u+h(x)|u|^{p-2} u=\left(R_{\alpha} *F(u)\right)f(u) $$ on lattice graphs $\mathbb{Z}^d$, where $s\in(0,1)$, $ p\geq 2$, $\alpha \in(0, d)$ and $R_\alpha$ represents the Green's function of the discrete fractional Laplacian that behaves as the Riesz potential. Under suitable assumptions on the potential function $h$, we first prove the existence of a strictly positive solution by the mountain-pass theorem for the nonlinearity $f$ satisfying some growth conditions. Moreover, if we add some monotonicity condition, we establish the existence of a positive ground state solution by the method of Nehari manifold.

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