{"id":"c8f73a9e-df49-489d-bc95-c84304e85d55","arxiv_id":"2507.22552","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Existence of positive and ground state solutions for the discrete fractional p-Laplacian Choquard equation on Z^d is established.","lead":"This paper proves that a fractional p-Laplacian Choquard equation on an infinite lattice graph has a strictly positive solution, and under an extra monotonicity condition, a positive ground state solution. It extends known variational techniques to a new combination of nonlocal operators on graphs.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.2's (PS)_c proof uses a false convexity inequality: the RHS is the sum of the two inner-product terms, not the difference; the displayed form is already contradicted by constant functions for p=2.","rationale":"The paper's central existence theorem (Theorem 1.1) depends on a mountain-pass argument whose compactness step, Lemma 3.2, is the only place strong convergence in H^{s,p} is proved. Inside that step, the load-bearing object is Lemma 15 of the unpublished preprint [39]. The manuscript's displayed version of that lemma is not a valid inequality: the factor must act on the difference of the two inner-product terms, not their sum. This is not a matter of convention; the scalar p=2 example with constant functions on a finite set makes the printed inequality plainly false. The theorem may still be true because the standard difference inequality, together with the already-established estimates Term1=o(1) and Term2=o(1), immediately yields ∥un−u∥_H→0. So the flaw is likely a sign error in transcription rather than a fatal mathematical defect. However, because the paper cites an unpublished preprint and does not reproduce the lemma, the reader cannot determine whether the error is local to this manuscript or originates in [39]. This is the most load-bearing concern: if Lemma 15 is the sum form, the (PS)_c proof collapses and Theorem 1.1 is unproven; if it is the difference form, the fix is trivial and the argument is sound modulo the other cited machinery. The reader's weakest_assumption already identified Lemma 15 of [39] as a key reliance; the present critique sharpens that to a specific sign error, hence partial agreement. The verdict remains CONDITIONAL: the central claim is plausible and likely repairable, but the proof as written is not self-contained and contains a false inequality that must be corrected before Theorem 1.1 can be accepted as proven.","tokens_in":15900,"tokens_out":20795,"duration_ms":255627,"concrete_test":"Check the exact statement of Lemma 15 in arXiv:2506.07694. If it is the difference version, (|x|^{p−2}x−|y|^{p−2}y)(x−y) ≥ c|x−y|^p, re-run the last display of §3 with Term1 − Term2 in place of Term1 + Term2; since both terms are o(1), the proof closes. If it is the sum version, evaluate it with p=2, h=1 on a finite set S, u=1_S, v=3_S: the claimed inequality gives 4|S| ≤ −16|S|, so the lemma is false and Theorem 1.1's (PS)_c proof fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3, proof of Lemma 3.2. After deriving Term1 := ∫(|∇s un|^{p−2}∇s un∇s(un−u) + h|un|^{p−2}un(un−u)) = o(1) and Term2 := ∫(|∇s u|^{p−2}∇s u∇s(un−u) + h|u|^{p−2}u(un−u)) = o(1), the proof invokes 'Lemma 15 in [39]' in the form ∥un−u∥_H^p ≤ 2^{p−2}p(Term1 + Term2). This inequality is false. For p=2, h=1 on a finite set S, u=1_S, v=3_S, the left-hand side is 4|S| while Term1+Term2 = ∫h(u+v)(u−v) = −8|S|, so the claimed right-hand side is negative. The valid strong-convexity inequality for the p-Laplacian is ∥u−v∥_H^p ≤ C⟨A(u)−A(v),u−v⟩ = C(Term1 − Term2), because z↦|z|^{p−2}z is monotone for p≥2. Since the proof has already shown both Term1 and Term2 are o(1), their difference is o(1), and the intended conclusion ∥un−u∥_H→0 follows if the minus sign is restored. But as printed, the (PS)_c condition rests on a false inequality and on an unpublished lemma ([39]) whose exact statement is not reproduced. If Lemma 15 of [39] genuinely contains the sum form, it is false and Theorem 1.1 has no valid (PS)_c proof.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":16265,"tokens_out":20670,"duration_ms":226750,"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":[{"comment":"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.","section":"Section 3, Lemma 3.2"},{"comment":"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.","section":"Section 4, Lemma 4.4"},{"comment":"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.","section":"Sections 2 and 3"}],"minor_comments":[{"comment":"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.","section":"Section 4, Lemma 4.3"},{"comment":"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.","section":"Section 2, Lemma 2.9"},{"comment":"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.","section":"Section 2, paragraph before Lemma 2.5"},{"comment":"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.","section":"Section 2, Lemma 2.8"},{"comment":"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.","section":"Throughout"},{"comment":"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.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The manuscript depends on two unpublished arXiv preprints for foundational lemmas, and one of those lemmas is quoted with a sign error that makes the displayed inequality false. This suggests that the editor may want to verify the status of [39] or require the author to include self-contained proofs of the compact embedding and the convexity inequality. The remaining issues, in particular the u_0≠0 step in Lemma 4.4, are fixable with short additional arguments, so I do not see a reason to reject the paper outright."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read on Wang's arXiv:2507.22552. It's the first treatment of the fractional p-Laplacian Choquard equation on Z^d, and the two main theorems—positive solution by mountain pass, ground state by Nehari—are plausible. The paper is competent: the reduction to the positive part is handled well, Lemma 2.9's strict positivity argument is clean, and the discrete HLS estimates are applied correctly. Credit for that.\n\nThe soft spot is real. In Lemma 3.2, the (PS)_c proof invokes Lemma 15 of [39] in the form ||un-u||_H^p ≤ 2^{p-2}p (Term1 + Term2), where Term1 and Term2 are the two inner-product integrals. That inequality is false. For p=2, h=1 on a finite set S, take u=1_S and un=3_S: the left side is 4|S|, while Term1+Term2 = -8|S|. The valid strong-convexity inequality uses Term1 − Term2, not the sum. Since the proof has already shown that both terms are o(1), their difference is also o(1), and the intended convergence follows—but only with the minus sign. As printed, the (PS)_c condition rests on an invalid inequality. If Lemma 15 of [39] genuinely states the sum form, then it is false and Theorem 1.1 has no valid proof as written. The author should fix this by writing the difference and citing the correct statement.\n\nThe other concern is secondary but relevant: the paper leans heavily on the unpublished preprint [39] for the compact embedding (Lemma 2.7) and for Lemma 15. That is not disqualifying, but the author should reproduce the needed statements or make them publicly verifiable.\n\nThe Nehari section (Section 4) does not use the faulty inequality, so Theorem 1.2's proof might survive a correction of Lemma 3.2. Still, the present version cannot be accepted without the fix. The paper is aimed at specialists in discrete variational PDEs; the result is plausible and likely salvageable. I'd send it to a referee—it deserves a careful check, not a desk reject—but I'd warn the referee to focus on the (PS)_c step and the exact statement of Lemma 15. After the sign correction, this could appear in a specialist journal.","headline":"Plausible first combination of fractional p-Laplacian and Choquard on lattice graphs, but the (PS)_c proof rests on a false convexity inequality.","tokens_in":16776,"tokens_out":4133,"would_cite":false,"duration_ms":43306,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35A15","35R02","35R11"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["fractional p-Laplacian","Choquard equation","lattice graphs","positive solution","ground state solution","mountain-pass theorem","Nehari manifold","discrete Hardy-Littlewood-Sobolev inequality"],"falsifier":"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.","tokens_in":15706,"feed_emoji":"🧩","tokens_out":9673,"duration_ms":105897,"temperature":0.7,"pith_summary":"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.","feed_headline":"Positive and ground state solutions proven for lattice Choquard","feed_subtitle":"Strict positivity and a least-energy ground state follow from mountain-pass and Nehari-manifold arguments on Z^d.","key_machinery":"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.","core_discovery":"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 $\\mathbb{Z}^d$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the fractional Sobolev space $H^{s,p}$, the compact embedding into $\\ell^q$ for all $q\\ge p$, and the convexity inequality used in the compactness step.","marker":"[39]"},{"why":"Provides the discrete Hardy-Littlewood-Sobolev inequality that controls the Riesz-potential convolution term.","marker":"[26]"},{"why":"Is the prior lattice-graph $p$-Laplacian Choquard result whose estimates are adapted for the general nonlinearity.","marker":"[33]"},{"why":"Gives the asymptotic behavior of the Green's function $R_\\alpha$, showing it behaves like the Riesz potential $|x-y|^{-(d-\\alpha)}$.","marker":"[18]"},{"why":"Provides the two-sided estimate for the kernel $K_s(x,y)$ that defines the fractional gradient and $p$-Laplacian.","marker":"[25]"}],"fun_headline_variants":["Positive and ground state solutions for lattice Choquard","Lattice Choquard: existence of positive and ground states","Fractional p-Laplacian Choquard on Z^d: positive solutions","Mountain-pass and Nehari yield positive ground state on lattice","Existence proved for positive and ground state Choquard on Z^d"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Positive and ground state solutions for lattice Choquard","Lattice Choquard: existence of positive and ground states","Fractional p-Laplacian Choquard on Z^d: positive solutions","Mountain-pass and Nehari yield positive ground state on lattice","Existence proved for positive and ground state Choquard on Z^d"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000786,"raw_usage":{"total_tokens":3450,"prompt_tokens":912,"completion_tokens":2538,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":528,"completion_tokens_details":{"reasoning_tokens":2447}},"tokens_in":528,"tokens_out":2538,"duration_ms":19276,"temperature":1.0,"reasoning_tokens":2447,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T11:32:44.738125+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Fractional Sobolev spaces and fractional $p$-Laplace equations on locally finite graphs","cited_arxiv_id":"2506.07694","evidence_quote":"Supplies the fractional Sobolev space $H^{s,p}$, the compact embedding into $\\ell^q$ for all $q\\ge p$, and the convexity inequality used in the compactness step."},{"cited_title":"Wang, The ground state solutions to discrete nonlinear Choquard equations with Hardy weights","cited_arxiv_id":null,"evidence_quote":"Provides the discrete Hardy-Littlewood-Sobolev inequality that controls the Riesz-potential convolution term."},{"cited_title":"p-Laplacian equations with general Choquard nonlinearity on lattice graphs","cited_arxiv_id":"2408.10584","evidence_quote":"Is the prior lattice-graph $p$-Laplacian Choquard result whose estimates are adapted for the general nonlinearity."},{"cited_title":"Michelitsch, B","cited_arxiv_id":null,"evidence_quote":"Gives the asymptotic behavior of the Green's function $R_\\alpha$, showing it behaves like the Riesz potential $|x-y|^{-(d-\\alpha)}$."}],"review_version":1}