{"id":"6b6a2e3c-56bb-4ebf-b87e-2929af5f60f0","arxiv_id":"2507.20464","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Ground states of p-Laplacian systems with Choquard-type nonlinearity on Z^N exist for large λ and converge, as λ → ∞, to a ground state of the limit problem on the potential wells.","lead":"This paper proves that certain nonlinear equations on infinite lattice graphs have ground state solutions that, as an external parameter grows, concentrate onto the region where the potential vanishes. The result extends an established existence-and-concentration program from single equations to systems with a nonlocal Choquard-type interaction.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1's proof never justifies that the mountain-pass sequence is at the Nehari level mλ; without the equality c_MP = mλ, the recovered critical point is not shown to be a ground state.","rationale":"I read the paper as an extension of the single-equation graph Choquard theory to systems under the weak potential condition (A'2). The variational architecture is standard and most lemmas are plausible. The weakest point, in my view, is the unproved identification of the mountain-pass level with mλ in Theorem 1.1; without it the existence proof does not establish a ground state. This is a genuine gap, but it is a repairable omission rather than a contradiction, and the reader's CONDITIONAL verdict already reflects the need for revision. I do not see a reason to strengthen to REJECT: the fiber argument should close the gap, and the exponent-2 typo in Lemma 4.1 is similarly repairable. The deeper issue flagged by the reader, the reliance on (A'2) for compactness, is an assumption of the theorem, not an inconsistency. Hence my assessment leaves the verdict unchanged.","tokens_in":21392,"tokens_out":38375,"duration_ms":391592,"concrete_test":"Add a proof of the equality c_MP = mλ using the fiber map from Lemma 2.8: define t(u,v)>0 by the unique solution of ⟨J'λ(t(u,v)(u,v)), t(u,v)(u,v)⟩=0, show Nλ is a topological sphere separating 0 from {Jλ<0}, and conclude every path from 0 to a point of negative energy attains sup Jλ ≥ mλ; the path t↦t(u,v) gives the reverse inequality. If the separation property fails, Theorem 1.1 as written does not produce a (PS) sequence at mλ.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 1.1 (after Lemma 2.9) the author writes that the mountain-pass geometry 'hence' gives a sequence with Jλ(u_k,v_k)→mλ and J'λ(u_k,v_k)→0. The mountain-pass theorem only yields a (PS)_c sequence, where c is the minimax level over paths from 0 to a point of negative energy. The equality c = mλ is neither stated nor proved in the paper. This equality is load-bearing: the final step identifies the recovered critical point as a ground state by computing Jλ(uλ,vλ) = (1/p−1/(2γ))||(uλ,vλ)||^p = lim (1/p−1/(2γ))||(u_k,v_k)||^p = mλ; if the PS level were a different c, this computation would give c rather than mλ, and minimality on Nλ would be unproved. The gap is repairable by a standard fiber argument: for each nonzero (u,v), Lemma 2.8 gives a unique t with t(u,v)∈Nλ, so Nλ is a radial graph over the unit sphere and separates 0 from the set {Jλ<0}; every admissible path must therefore cross Nλ, giving c_MP ≥ mλ, while radial paths give c_MP ≤ mλ. But as written the paper omits this and asserts the stronger conclusion.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a p-Laplacian Choquard-type system on the lattice graph Z^N with potentials λa+1 and λb+1. Under assumptions (F1), (A1), and (A'2), it claims two results: Theorem 1.1 asserts existence of a ground state solution for all sufficiently large λ, and Theorem 1.2 asserts that along any sequence λ_k→∞ the ground states converge, up to subsequence, in W^{1,p}(V)×W^{1,p}(V) to a ground state of a limit system posed on the potential wells Ω_a and Ω_b with zero Dirichlet data on the vertex boundary. The proofs are based on the Nehari manifold method, a mountain-pass geometry, tail estimates exploiting the finiteness of the sublevel sets in (A'2), a (PS)_c compactness argument, and a comparison of the Nehari levels m_λ and m_Ω.","tokens_in":21550,"tokens_out":16193,"duration_ms":155296,"significance":"If the proof gaps identified below are repaired, the paper would be a meaningful extension of recent single-equation Choquard results on graphs to systems, and it would weaken the compactness assumptions used in prior work from decay or summability conditions to the finiteness condition (A'2). The compactness mechanism built on tail estimates and finite sublevel sets is a useful contribution, and the paper carefully presents the discrete Hardy-Littlewood-Sobolev inequality and a Brezis-Lieb-type splitting lemma adapted to the graph setting. However, the central claims are not fully established as written: the identification of the mountain-pass level with the Nehari level is missing, the non-degeneracy of F is not imposed, and one lower-bound estimate in the convergence proof is not justified. These issues are local and repairable, but they are load-bearing for the main theorems.","major_comments":[{"comment":"The proof asserts that the mountain-pass geometry of Lemma 2.9 'hence' yields a (PS)_{m_λ} sequence. The mountain-pass theorem only provides a (PS)_c sequence at the minimax level c over paths from 0 to a point of negative energy, and the equality c = m_λ is neither stated nor proved. This equality is load-bearing because the final identification J_λ(u_λ,v_λ)=m_λ uses the (PS) level in place of m_λ. The gap is repairable by the standard radial-graph argument: every admissible path from 0 to a point of negative energy must cross N_λ, giving c ≥ m_λ, while the ray t(u,v) through a point of N_λ gives c ≤ m_λ; however, as written the manuscript omits this argument.","section":"Section 3, proof of Theorem 1.1"},{"comment":"The statement that for any nonzero (u,v), J_λ(t(u,v))→−∞ as t→∞ requires ∫_V (R_α*F(u,v))F(u,v) dμ > 0. Under (F1) alone this need not hold: F may vanish on a nontrivial cone, for example F(u,v)=|u|^γ, for which the nonlocal term is zero whenever u≡0, and then J_λ(t(0,v))→+∞ along the v-direction. The mountain-pass geometry only needs one direction with positive nonlocal energy, so the proof can likely be repaired by adding a non-degeneracy assumption such as F(u,v)>0 for all (u,v)≠(0,0), or by explicitly choosing a suitable test direction; but as written Lemma 2.9 and hence Theorem 1.1 rely on an unjustified assertion.","section":"Lemma 2.9(ii)"},{"comment":"The displayed lower bound m_{λ_k} ≥ (1/p−1/(2γ)) ∫_{B_r(x_k)∩{a≥M_1}} λ_k a |u_k−u|^2 dμ is not justified: on {a≥M_1} one has u=0, so |u_k|=|u_k−u|, but for p≥2 the pointwise inequality |u_k−u|^p ≥ |u_k−u|^2 is false on points where |u_k−u|<1. Consequently the chain leading to λ_k M_1(δ²/4+o_k(1)) does not follow. The contradiction argument is still recoverable because |u_k(x_k)−u(x_k)|≥δ/2 and a(x_k)≥M_1 for large k imply ∥u_k∥^p_{λ_k} ≥ λ_k M_1(δ/2)^p; however, the proof as written needs this correction.","section":"Lemma 4.1"}],"minor_comments":[{"comment":"In the proof, the terms F(w_k,v_k) should be F(w_k,z_k) in the displayed estimates.","section":"Lemma 2.5"},{"comment":"The constant M is introduced although (A'2) quantifies M_1 and M_2; the tail estimate for u should use M_1 and that for v should use M_2.","section":"Lemma 3.2"},{"comment":"The references to 'the proof of Lemma 3.1' and 'By Lemma 3.1' should refer to Lemma 4.1 for the convergence m_{λ_k}→m_Ω and the ℓ^q convergence.","section":"Proof of Theorem 1.2"},{"comment":"There is a typo: 'respecct' should be 'respect'.","section":"Section 1"},{"comment":"The application of Lemma 3.5 requires a uniform bound c≤c* for the (PS) level; since m_λ≤m_Ω, one should explicitly take c*=m_Ω so that λ_0 is independent of the sequence.","section":"Proof of Theorem 1.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a plausible incremental contribution, and the proof strategy is standard and likely repairable. The main issues are the missing mountain-pass/Nehari level identification, the missing non-degeneracy condition on F, and a local inequality error in Lemma 4.1; all appear fixable within the scope of the paper. I do not see concerns about the citation pattern or the journal fit."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper fills the last obvious slot in the graph-variational program: p-Laplacian systems with Choquard-type nonlocal coupling on Z^N, under the weaker potential condition (A'2) rather than (A2) or (Ã2). The existence and concentration results are plausible, and the architecture (Nehari manifold, tail estimates, PS compactness, limiting problem on the wells) is standard. The technical core is the tail estimate under (A'2), and it looks right.\n\nWhat's genuinely new is the combination. Single Choquard equations on graphs, p-Laplacian systems without Choquard terms, and (A'2)-type potential conditions all exist separately, but nobody has put all three together. The 'first work' claim in Remark 1.3(iii) is plausible from the cited literature.\n\nThe paper does a lot well. Lemma 3.2's tail estimate is the key step, and the way finiteness of {a≤M1} controls the outer tail as r→∞ while λ controls the other tail is sound. Lemma 3.5's PS argument and Lemma 4.1's m_λ→m_Ω argument interlock correctly. The integration-by-parts lemma for their definition of Δ_p is a useful detail.\n\nNow the soft spots, in order of size.\n\n1. Theorem 1.1's proof has a real gap. After Lemma 2.9, the paper says the mountain-pass geometry 'hence' gives a (PS) sequence at level m_λ. That is not what the mountain-pass theorem gives; it gives a sequence at the minimax level c. The equality c = m_λ is never stated or proved. This is load-bearing: the final step computes J_λ(u_λ,v_λ) = m_λ using that level, which is what makes the critical point a ground state. The gap is repairable by the standard radial-graph argument: N_λ is a graph over the unit sphere and separates 0 from the set {J_λ < 0}, so every admissible path crosses N_λ, giving c ≥ m_λ; radial paths give c ≤ m_λ. But the paper omits this and asserts the stronger conclusion. This needs to be fixed; it is not a typo.\n\n2. Lemma 4.1 uses |u_k−u|^2 (and δ^2) where it should be |u_k−u|^p (and δ^p). This looks like a p=2 leftover in the concentration argument. It is mechanically wrong but repairable, since u(x_k)→0 for |x_k|→∞, so the same lower bound works with exponent p.\n\n3. Lemma 3.2 has a carelessly handled constant: the bound is displayed as (2pc*/(p−1+1)) = 2c*, which doesn't match the (2γpc*/(2γ−p)) from Lemma 3.1. It doesn't change the conclusion because µ(Ω_r^-)→0 controls the limit, but the displayed constant is wrong.\n\nThe citation pattern looks fine; the paper leans on the right prior work and doesn't misrepresent it. No data or code, which is normal for this kind of paper.\n\nWho this is for: people working on variational methods on graphs. They should read it. It deserves a serious referee, not a desk reject. I'd send it out, but the referee's first question should be the c_MP = m_λ issue. Conditional accept after revision.","headline":"Solid extension of the graph variational program, but the proof of Theorem 1.1 has a genuine gap that needs a standard radial-graph argument.","tokens_in":22268,"tokens_out":6846,"would_cite":true,"duration_ms":61744,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J50","35R02"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that a Choquard-type p-Laplacian system on the integer lattice has ground state solutions for large coupling, and that these solutions converge onto the potential wells as the coupling grows.","keywords":["lattice graphs","p-Laplacian system","Choquard-type nonlinearity","ground state solutions","Nehari manifold","potential wells","concentration of solutions","discrete Hardy-Littlewood-Sobolev inequality"],"falsifier":"Take a concrete pair of potentials satisfying (A1) and (A'2)—for example $a(x)=b(x)=0$ on a finite cube and $1$ outside—and solve the variational problem on large finite boxes with $p=2$, $N=3$, $\\alpha=1$, $\\gamma=3$. If the numerically computed Nehari minimum $m_\\lambda$ does not converge to the ground-state level of the Dirichlet limit problem on the cube as $\\lambda\\to\\infty$, or if a bounded Palais-Smale sequence at a level below the threshold keeps mass in every large ball instead of converging, the central claim would be refuted.","tokens_in":21025,"feed_emoji":"🧮","tokens_out":13362,"duration_ms":119523,"temperature":0.7,"pith_summary":"This paper proves that the discrete Choquard-type system with the $p$-Laplacian on $\\mathbb{Z}^N$ has a ground state solution for every sufficiently large $\\lambda$ and every $p\\ge 2$, under assumptions (F1), (A1), and (A'2). It further proves that as $\\lambda\\to\\infty$, any sequence of ground states converges strongly in $W^{1,p}\\times W^{1,p}$ to a ground state of the limit problem posed on the potential wells $\\Omega_a,\\Omega_b$ with zero boundary data. The paper identifies this as the first treatment of a Choquard-type system on graphs, and the potential condition (A'2)—finite, nonempty sublevel sets—is weaker than the decay or summability conditions used in prior graph results, so compactness has to be recovered by a tail estimate rather than an embedding. The Nehari manifold provides the variational framework: one minimizes the energy on the set where the radial derivative vanishes, and the tail estimate converts Palais-Smale sequences into convergent ones.","feed_headline":"Ground states exist and concentrate on lattice Choquard systems","feed_subtitle":"For large λ, solutions concentrate on the potential wells, with a weaker condition than prior compactness assumptions.","key_machinery":"The load-bearing machinery is the Nehari manifold $N_\\lambda=\\{(u,v)\\ne(0,0): \\langle J'_\\lambda(u,v),(u,v)\\rangle=0\\}$, which turns the search for ground states into a constrained minimization problem. The discrete Hardy-Littlewood-Sobolev inequality controls the nonlocal term $\\int (R_\\alpha*F(u,v))F(u,v)\\,d\\mu$ by the norm $\\|(u,v)\\|_\\lambda^{2\\gamma}$, and the tail estimate of Lemma 3.2 splits $\\mathbb{Z}^N\\setminus B_r$ into the sets where $a\\ge M_1$ or $b\\ge M_2$—controlled for large $\\lambda$—and the finite sets where $a<M_1$ or $b<M_2$—controlled for large $r$. Together these give the Palais-Smale compactness condition at bounded energy levels, which converts a minimizing sequence into a true ground state and later identifies the limit profile with the Dirichlet problem on the wells.","core_discovery":"The central claim is that the parameter-dependent problem is variational and has minimal-energy solutions that do not escape to infinity. Theorem 1.1 gives, for $\\lambda\\ge\\lambda_0$ and $p\\ge 2$, a ground state $(u_\\lambda,v_\\lambda)$ at the Nehari level $m_\\lambda$, meaning a nontrivial critical point of the energy $J_\\lambda$ with minimal energy on the Nehari manifold. Theorem 1.2 says that along any sequence $\\lambda_k\\to\\infty$, a subsequence of these ground states converges in $W^{1,p}(V)\\times W^{1,p}(V)$ to a ground state of the Dirichlet problem (3) on $\\Omega_a\\times\\Omega_b$, and the convergence is accompanied by convergence of the energy levels $m_\\lambda\\to m_\\Omega$. In the paper's own framing, the nonlocal Choquard interaction does not destroy the semiclassical concentration picture familiar from local Schrödinger-type problems on graphs.","pith_inferences":["This suggests the same tail-splitting argument should transfer to more general locally finite graphs of polynomial growth, provided a discrete Hardy-Littlewood-Sobolev inequality and finite sublevel sets are available; the paper itself proves the result only on $\\mathbb{Z}^N$.","Because the compactness threshold $c_0$ in Lemma 3.1 does not depend on $\\lambda$, the proof likely yields uniform bounds on ground-state norms for all large $\\lambda$, which could support quantitative rate-of-concentration statements not written in the paper.","A natural test of whether (A'2) is close to sharp is to construct potentials whose sublevel sets are infinite but thin, such as a slowly growing sequence of wells; if those fail to concentrate, the finiteness condition is doing the load-bearing work the proof assigns to it."],"forward_implications":["For large $\\lambda$ the system has a nontrivial solution at the minimum energy level, so the nonlocal Choquard coupling does not destroy the variational ground-state structure on graphs.","As $\\lambda\\to\\infty$, ground states concentrate on the potential wells: the limiting profile vanishes outside $\\Omega_a\\times\\Omega_b$ and satisfies the Dirichlet problem (3), so the wells act as the effective domain in the strong-coupling limit.","The energy levels satisfy $m_\\lambda\\to m_\\Omega$, giving a quantitative identity linking the parameter-dependent problem to the limit problem on the wells.","The compactness threshold needed for the Palais-Smale condition is uniform in $\\lambda$ at each bounded energy level, so the existence result holds for all sufficiently large $\\lambda$ rather than only in a single parameter regime.","The weaker hypothesis (A'2) replaces the decay or summability assumptions of earlier graph results, so the concentration phenomenon is shown for a broader class of potentials."],"supporting_citations":[{"why":"Supplies the splitting identity for p-gradient energy used to decompose Palais-Smale sequences and to prove strong convergence of differences.","marker":"[9]"},{"why":"Provide the discrete Hardy-Littlewood-Sobolev inequality and the use of finite sublevel sets for Choquard equations on graphs.","marker":"[11,23]"},{"why":"Establishes the Nehari-manifold existence and concentration framework for p-Laplacian systems with homogeneous nonlinearities that the paper extends to the Choquard coupling.","marker":"[19]"},{"why":"Treats the scalar discrete Choquard equation and supplies the kernel estimates and ground-state convergence setup for the Riesz-type term.","marker":"[22]"},{"why":"Develops the discrete Schrödinger system model with potential wells whose limit system and convergence pattern are adapted here.","marker":"[27]"},{"why":"Provides the baseline convergence result for ground states of discrete nonlinear Schrödinger equations with a decaying potential.","marker":"[28]"},{"why":"Handles the (p,q)-Laplacian system under the alternative (Ã2) summability condition, the comparison that motivates the weaker (A'2).","marker":"[29]"}],"fun_headline_variants":["Choquard ground states exist and converge on lattice graphs","Lattice Choquard ground states: existence and semiclassical limit","Choquard systems on lattices: ground states concentrate","Ground state convergence for lattice Choquard systems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The results rest on (A'2): the sets $\\{x: a(x)\\le M_1\\}$ and $\\{x: b(x)\\le M_2\\}$ must be finite and nonempty, because the proof's control of energy outside large balls and its compactness step both need those sets to be negligible at infinity.","fun_headline_variants_meta":{"raw":{"variants":["Choquard ground states exist and converge on lattice graphs","Lattice Choquard ground states: existence and semiclassical limit","Choquard systems on lattices: ground states concentrate","Ground state convergence for lattice Choquard systems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000649,"raw_usage":{"total_tokens":2979,"prompt_tokens":948,"completion_tokens":2031,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":564,"completion_tokens_details":{"reasoning_tokens":1963}},"tokens_in":564,"tokens_out":2031,"duration_ms":15035,"temperature":1.0,"reasoning_tokens":1963,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:47:24.905550+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a concrete pair of potentials satisfying (A1) and (A'2)—for example $a(x)=b(x)=0$ on a finite cube and $1$ outside—and solve the variational problem on large finite boxes with $p=2$, $N=3$, $\\alpha=1$, $\\gamma=3$. If the numerically computed Nehari minimum $m_\\lambda$ does not converge to the ground-state level of the Dirichlet limit problem on the cube as $\\lambda\\to\\infty$, or if a bounded Palais-Smale sequence at a level below the threshold keeps mass in every large ball instead of converging, the central claim would be refuted.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the splitting identity for p-gradient energy used to decompose Palais-Smale sequences and to prove strong convergence of differences."},{"cited_title":"Shao, Existence and convergence of solutions for p-Laplacian systems with homogeneous nonlinearities on graphs","cited_arxiv_id":null,"evidence_quote":"Establishes the Nehari-manifold existence and concentration framework for p-Laplacian systems with homogeneous nonlinearities that the paper extends to the Choquard coupling."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Develops the discrete Schrödinger system model with potential wells whose limit system and convergence pattern are adapted here."},{"cited_title":"Zhang, L","cited_arxiv_id":null,"evidence_quote":"Provides the baseline convergence result for ground states of discrete nonlinear Schrödinger equations with a decaying potential."},{"cited_title":"Existence and convergence of ground state solutions for a $(p,q)$-Laplacian system on weighted graphs","cited_arxiv_id":"2403.02048","evidence_quote":"Handles the (p,q)-Laplacian system under the alternative (Ã2) summability condition, the comparison that motivates the weaker (A'2)."}],"review_version":2}