{"id":"55500180-abf3-4a96-adc8-fd87fb3fe3dc","arxiv_id":"2510.15694","paper_version":5,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"All least-energy solutions of the vectorial p-Laplacian Lane–Emden system are of the form (c^1 ω, …, c^m ω), where c is a unit vector and ω solves the scalar p-Laplacian equation.","lead":"This paper proves existence, regularity, and a structural classification for systems driven by the vectorial p-Laplacian with power-type nonlinearities. Its main result says every lowest-energy solution is a single scalar ground state copied into each component, weighted by a unit vector.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Scale invariance of Q invalidates Step 2's final inference: arbitrary minimizers of l_{p,q} need not solve (P_{p,q}), so Proposition 5.2 cannot be applied to them.","rationale":"The reader's weakest_assumption correctly identifies the scale-normalization issue in Proposition 5.1 as the load-bearing point. My reading of the proof confirms this: the final step of Theorem 1.7 applies Proposition 5.2 to an arbitrary minimizer of l_{p,q}, but an arbitrary minimizer is not a solution of (P_{p,q}); only the specially scaled function in Proposition 5.1 is. Because Q_{p,q} is homogeneous of degree zero, the family of minimizers contains many scalings, and the scalar factor ω obtained from an arbitrary minimizer does not satisfy (5.2) unless the minimizer was already chosen at the least-energy scale. This is a genuine gap in the written argument, not merely a missing reference. However, it is repairable by starting from a least energy solution rather than an arbitrary minimizer and tracking the scaling through Proposition 5.1. The central classification claim therefore appears correct for q≠p, but the proof as written is incomplete. This does not change the reader's conditional verdict; it sharpens the specific repair needed. I am not treating q=p as the primary concern because the intended Lane-Emden range and the q≠p denominators in Proposition 5.1 suggest the authors meant to exclude it, though the statement of Theorem 1.7 currently does not; this should also be fixed in revision.","tokens_in":16996,"tokens_out":22334,"duration_ms":179851,"concrete_test":"Let ω0 be a positive scalar solution of (5.2), fix c∈S^{m-1}, and set u_s=s c ω0 for s>0. Check: (i) Q_{p,q}(u_s)=Q_{p,q}(cω0)=l_{p,q}, so every u_s is a minimizer of l_{p,q}; (ii) u_s solves (P_{p,q}) if and only if s=1, since substitution yields (s^{p-1}-s^{q-1})ω0^{q-1}≠0 for s≠1, q≠p. This directly exposes the invalid inference from 'u is a minimizer' to 'ω solves (5.2)'. Then test the repair: for a least energy solution u, set v=u/|u|_q; verify v is a Q-minimizer, apply the Lagrange identity to obtain v=cη, and define ω=|u|_q η. Verify that -Δ_p ω = λω^{p-1}+ω^{q-1}. If this verification succeeds, Theorem 1.7 remains true despite the gap in the written proof.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The weakest point is the final inference in Step 2 of the proof of Theorem 1.7, immediately after the Lagrange identity (5.3). The functional Q_{p,q} is scale-invariant: Q_{p,q}(tu)=Q_{p,q}(u) for every t>0. Therefore, if v minimizes l_{p,q}, then every scalar multiple tv also minimizes l_{p,q}. Proposition 5.1 guarantees only that one specific scaling, l^{1/(q-p)}v (after normalizing |v|_q=1), is a least energy solution. An arbitrary minimizer u satisfies the constrained Euler equation -Δ_p u - λ|u|^{p-2}u = l |u|^{q-2}u, not the unconstrained system (P_{p,q}) unless l=1. The proof takes an arbitrary minimizer u, shows u=cω, and then states 'ω solves (5.2) by Proposition 5.2'. But Proposition 5.2 applies only to solutions of (P_{p,q}), and u is not a solution. Concretely, if ω0 solves (5.2) and u_s=s c ω0 with s≠1, then u_s is still a Q-minimizer, but ω=sω0 does not solve (5.2) because the powers p and q scale differently (s^{p-1}≠s^{q-1} for q≠p). The theorem remains plausibly true: one should start from a least energy solution u, set v=u/|u|_q, show v is a Q-minimizer, decompose v=cη, and then use the fact that u=c(|u|_q η) solves (P_{p,q}) to identify the scalar factor ω=|u|_q η solving (5.2). The written proof, however, does not perform this repair and instead overclaims the classification for all minimizers of l_{p,q}.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies variational quasilinear elliptic systems driven by the vectorial p-Laplacian on a bounded domain, with Dirichlet boundary conditions. After establishing differentiability of the energy functional, the authors prove existence of one global minimizer in the p-sublinear case, infinitely many solutions in the p-superlinear case (with and without a linear eigenvalue perturbation), and an L∞/C^{1,β} regularity result for p≥2. The main new contribution is Theorem 1.7, which asserts that for the Lane–Emden type system (P_{p,q}) with q∈(1,p*) and λ<λ1, there exists a least energy solution and every such solution has the factorized form u=(c^1ω,…,c^mω) with c∈S^{m−1} and ω>0 solving the scalar equation. The proof uses a scale-invariant Rayleigh quotient l_{p,q} and a Lagrange identity to reduce the vectorial problem to the scalar one.","tokens_in":17415,"tokens_out":14752,"duration_ms":113747,"significance":"If the classification theorem were correct, it would give a complete and elegant description of least energy ground states for the vectorial Lane–Emden system: all ground states are obtained from a scalar ground state by a constant unit vector of coefficients. This would extend the scalar and cooperative-system results of Saldaña–Tavares and Correia–Oliveira–Tavares to the vectorial p-Laplacian setting. The existence and regularity results are largely standard adaptations, but the paper collects them in a convenient unified framework. The main novelty is the classification, and the proof strategy—passing through the quotient l_{p,q} and using the identity of Hynd–Kawohl–Lindqvist—is natural and potentially effective. However, as written, the proof of the classification contains a gap that is load-bearing for the theorem, and the theorem statement itself appears to be false in the included case q=p. There is also a gap in the Palais–Smale boundedness argument for Theorem 1.4 when λ>0. These issues are fixable, but they require substantive revision.","major_comments":[{"comment":"The boundedness of Palais–Smale sequences for λ>0 is not established. The proof uses the weighted Young inequality to write ∫(|Du|^p − λ|u|^p) ≥ ‖u‖^p − Cδ − Cδ^{q/p}∫|u|^q, and then claims that for δ small enough this is ≥ c‖u‖^p − C_2(δ). This absorption is invalid because the term Cδ^{q/p}∫|u|^q is not controlled by ‖u‖^p when q>p: for a Palais–Smale sequence, ∫|u|^q may grow like ‖u‖^q, and no a priori bound is available. Thus the conclusion that C(1+‖u_n‖) ≥ (1/p−1/μ)‖u_n‖^p − C_2(δ) does not follow. This gap also affects Step 1 of Theorem 1.7 for q>p, which explicitly relies on “Step 3 of proof of Theorem 1.4” to obtain a converging subsequence of a minimizing sequence.","section":"Proof of Theorem 1.4, Step 3"},{"comment":"The final inference is invalid. The proof starts with an arbitrary minimizer u of the quotient l_{p,q}. Such a minimizer satisfies the constrained Euler equation −Δ_p u − λ|u|^{p−2}u = l_{p,q}|u|^{q−2}u, not the unconstrained system (P_{p,q}). Therefore, after showing u=cω, the statement “ω solves (5.2) by Proposition 5.2” does not follow, because Proposition 5.2 applies only to solutions of (P_{p,q}). The theorem remains plausibly true, but the proof must start from a least energy solution w, decompose the normalized function v=w/|w|_q into cη, and then use that w=c(|w|_q η) solves (P_{p,q}) to conclude that |w|_q η solves (5.2). The Harnack positivity assertion for ω at this stage is also unsupported, since u is not yet known to solve an elliptic equation.","section":"Proof of Theorem 1.7, Step 2 (p. 16–17)"},{"comment":"The statement includes q=p, but for q=p the problem reduces to the eigenvalue equation −Δ_p u = (λ+1)|u|^{p−2}u. Nontrivial solutions exist only if λ+1 is an eigenvalue of the Dirichlet p-Laplacian. For generic λ<λ1, e.g. λ=0 when λ1≠1, there is no nontrivial solution, so the asserted existence of a least energy solution is false. The theorem should exclude q=p or impose an additional condition such as λ+1=λ1. The proof itself only treats q

p, consistent with this correction.","section":"Theorem 1.7 statement (p. 4)"}],"minor_comments":[{"comment":"In the converse direction, the displayed equation has a typo: the left-hand side should be ∫|∇ω|^{p−2}∇ω·∇φ_j, not ∫|ω|^{p−2}ω φ_j. The intended equation is clear from context, but the typo should be corrected.","section":"Proposition 5.2 (p. 16)"},{"comment":"For q∈(1,p), the coercivity bound explicitly uses λ<λ1. It would be helpful to state that λ<λ1 is used here; the proof is correct but the role of the eigenvalue is implicit.","section":"Proof of Theorem 1.7, Step 1"},{"comment":"The notation “t:=|v|^{-1}_q” is ambiguous; it should read t=1/|v|_q. Also, after substituting tv, the denominator (∫|v|^q)^{p/q} should be written explicitly to avoid confusion with the definition of Q_{p,q}(tv).","section":"Proof of Theorem 1.7, Step 2"},{"comment":"The statement says “if and only if”, but the proof only shows one direction and then an equality chain. The logical structure is correct, but the wording could be clarified to indicate that the converse follows from the equality l_{p,q}=(pq/(q−p)c_{p,q})^{(q−p)/q}.","section":"Proposition 5.1"}],"recommendation":"major_revision","confidential_remarks":"The classification theorem is the main advertised contribution, and the stress-test concern about Step 2 of Theorem 1.7 is real: the proof conflates minimizers of the quotient l_{p,q} with solutions of (P_{p,q}). The gap is repairable with a rescaling argument, but the written proof needs substantive revision. The same applies to the Palais–Smale boundedness in Theorem 1.4, where the Young inequality step is algebraically flawed. Additionally, the q=p case in Theorem 1.7 is genuinely false as stated. These are all fixable within the scope of the paper, so I recommend major revision rather than rejection. The identity from Proposition 5.1 is imported from [16] and appears sound; the main novelty is the factorized form, which should be reproved carefully."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the genuinely new thing is Theorem 1.7, the classification of least-energy solutions for the vectorial p-Laplacian Lane–Emden system. I think it is probably true and worth publishing after repair, but the written proof has a real gap at the last step. The proof starts with an arbitrary minimizer u of the scale-invariant quotient l_{p,q}, shows u=cω, and then says “ω solves (5.2) by Proposition 5.2.” That does not follow. Proposition 5.2 applies to solutions of (P_{p,q}), and an arbitrary minimizer of l_{p,q} is not a solution: because l_{p,q} is scale-invariant, any scalar multiple of a minimizer is also a minimizer, and only the specific normalization l^{1/(q-p)}v satisfies the unconstrained system. The fix is straightforward: start from a least-energy solution u, normalize it to v=u/|u|_q, show v minimizes l, run the same pointwise argument to get v=cη with |c|=1, and then use the fact that u solves (P) to identify ω=|u|_q η as the solution of (5.2). That repair makes the theorem hold. The stress-test note is correct on this point.\n\nThe rest of the paper is solid but mostly standard, and the authors say so. The existence results 1.1–1.4 are classical. The boundedness step in Theorem 1.4 has a small omission: the negative λ|u|^p term should be absorbed into the positive (1/μ−1/q)|u|^q term, which requires choosing μ|∇|u|| on a set of positive measure. Either observation would contradict Theorem 1.7.","tokens_in":16850,"feed_emoji":"📐","tokens_out":6991,"duration_ms":57803,"temperature":0.7,"pith_summary":"The paper studies systems where a vector-valued unknown u=(u^1,...,u^m) solves a quasilinear elliptic equation driven by the vectorial p-Laplacian. For the power-type Lane–Emden nonlinearity, it proves that every least energy solution has the factored form u=(c^1 ω,...,c^m ω), where c lies on the unit sphere in R^m and ω is a positive solution of the corresponding scalar equation. This matters because it says the multicomponent ground state is completely described by a scalar profile and a fixed direction in the target space; the vector structure does not create genuinely vectorial ground states. The paper also establishes existence of solutions in sublinear and superlinear regimes and a regularity result for weak solutions.","feed_headline":"Vector p-Laplacian ground states are scalar solutions in disguise","feed_subtitle":"The multicomponent Lane–Emden system's least-energy solutions always factor into one positive profile and a fixed unit direction.","key_machinery":"The scale-invariant quotient Q_{p,q} and its infimum l_{p,q} are the central object. Proposition 5.1 establishes an equivalence between minimizers of l_{p,q} and least energy solutions, including the identity l_{p,q} = (pq/(q-p)c_{p,q})^{(q-p)/q}. The factorization step uses Lagrange's identity on the vectors u_i and ∇u_i to convert the equality |Du|=|∇|u|| into the conclusion that the ratios u_i/|u| are constant a.e., giving u=cω.","core_discovery":"The central claim is Theorem 1.7: for q in (1,p*) and λ below the first eigenvalue λ1 of the scalar p-Laplacian, the system -Δ_p u = λ|u|^{p-2}u + |u|^{q-2}u has a least energy solution, and every such solution is of the form (c^1 ω,...,c^m ω) with c=(c^1,...,c^m) in S^{m-1} and ω>0 solving -Δ_p ω = λ ω^{p-1} + ω^{q-1} with zero Dirichlet boundary conditions. The proof shows that the least energy level of the vector system is determined by the infimum of the Rayleigh-type quotient Q_{p,q}(u) = (∫|Du|^p - λ∫|u|^p)/(∫|u|^q)^{p/q}, and that any minimizer of this quotient has |Du|=|∇|u||, forcing all components u^i to be constant multiples of one positive scalar function.","pith_inferences":["If the classification is correct, it suggests that for a whole family of homogeneous vector problems driven by the vectorial p-Laplacian, the ground-state manifold is a copy of the scalar ground-state set times S^{m-1}, so symmetry breaking appears only through the choice of direction in target space.","A natural extension would be to test whether the same factored form persists under small anisotropic perturbations of the nonlinearity; the proof's reliance on exact scale invariance indicates that such perturbations could produce non-polarized ground states.","The threshold λ<λ1 is probably where the quotient Q_{p,q} remains bounded below; at or above λ1, the classification likely fails or requires a different normalization.","One could numerically minimize Q_{p,q} for a two-component system on a disk and check whether the ratio u^1/u^2 is constant for all computed minimizers, providing a concrete test of the factorized-structure claim."],"forward_implications":["All least energy solutions of the vector Lane–Emden system share the same scalar spatial profile, so no ground state exists whose components vary independently.","The ground state energy of the vector system is directly computable from the scalar ground state: c_{p,q} = ((q-p)/(pq)) l_{p,q}^{q/(q-p)}.","Some components of a least energy solution may vanish, and signs may differ, since only the unit vector c is constrained, not its entries.","Existence is assured for every subcritical q and every λ<λ1, including both p-sublinear and p-superlinear cases.","Under p≥2 and subcritical growth, weak solutions are bounded and, on C^{1,α} domains, are C^{1,β}, so the classified ground states are classical."],"fun_headline_variants":["Vector p-Laplacian ground states are always scalar multiples","Multicomponent p-Laplacian solutions reduce to one scalar profile","Every least-energy vector solution is a scalar times a fixed direction","Vectorial p-Laplacian minimizers collapse onto a single scalar"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"Everything rests on Proposition 5.1, the exact equivalence between minimizers of the scale-invariant quotient Q_{p,q} and least-energy solutions; if that equivalence fails for some subcritical q or the minimizer is not achieved at the least-energy scale, the factored form does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Vector p-Laplacian ground states are always scalar multiples","Multicomponent p-Laplacian solutions reduce to one scalar profile","Every least-energy vector solution is a scalar times a fixed direction","Vectorial p-Laplacian minimizers collapse onto a single scalar"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000291,"raw_usage":{"total_tokens":1573,"prompt_tokens":819,"completion_tokens":754,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":563,"completion_tokens_details":{"reasoning_tokens":694}},"tokens_in":563,"tokens_out":754,"duration_ms":7047,"temperature":1.0,"reasoning_tokens":694,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T09:21:47.533965+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a least-energy solution of the two-component system in a ball with q just below p* whose two components are not constant multiples of a single positive function; equivalently, exhibit a minimizer u of Q_{p,q} for which |Du|>|∇|u|| on a set of positive measure. Either observation would contradict Theorem 1.7.","supporting_citations":[],"review_version":1}