{"id":"2dc7e055-db5a-4627-a7cf-1e097b5a2920","arxiv_id":"2606.05990","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Establishes a conditional Harnack inequality for the critical p-Laplace equation via Pohozaev-neck analysis that upgrades preliminary singular decay to the sharp p-harmonic rate, conditional on bubble classification and neck control.","lead":"The paper proves a conditional Schoen-type Harnack inequality for positive weak solutions of the critical p-Laplace equation under Sobolev growth and monotonicity assumptions, using a Pohozaev-neck argument to upgrade decay rates. A smart generalist might read it to see how new techniques handle nonlinear PDEs where standard conformal methods fail for p not equal to 2.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Central claim holds only under two external hypotheses whose validity is not established in the paper.","rationale":"The reader's weakest_assumption already isolates the dependence on the two external inputs. Because the paper is explicitly conditional and supplies no new evidence for those inputs, the load-bearing concern is precisely the one identified; the UNVERDICTED verdict therefore requires no adjustment.","tokens_in":1780,"tokens_out":328,"duration_ms":18055,"concrete_test":"Take a sequence of positive solutions u_k to the critical p-Laplace equation on expanding balls that satisfy the monotonicity and Sobolev-growth hypotheses; extract a blow-up limit at a point where the preliminary neck control is known to hold; check whether the limit is necessarily an Aubin-Talenti bubble. If a non-bubble limit appears, the conditional statement cannot be invoked for that sequence.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The inequality (sup_{B_R} u)(inf_{B_{2R}} u)^{p-1} ≤ C R^{p-n} is stated to follow from the Pohozaev-neck upgrade argument only after assuming (i) every bounded positive entire blow-up limit is an Aubin-Talenti p-bubble and (ii) a preliminary upper bound of order |x|^{-(n-p)/p} on normalized necks. The manuscript supplies neither a proof nor an independent verification of these two inputs; the abstract explicitly flags both as prerequisites. Without them the neck argument cannot be applied and the claimed Harnack bound does not follow.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript claims a conditional Schoen-type Harnack inequality for positive weak solutions of -Δ_p u = g(u) (1 < p < n) under global critical Sobolev growth and the monotonicity condition that s^{-(p^*-1)}g(s) is nonincreasing. Under the two explicit hypotheses that bounded positive entire blow-up limits are Aubin-Talenti p-bubbles and that normalized necks satisfy a preliminary upper bound of order |x|^{-(n-p)/p}, solutions in B_{3R} obey (sup_{B_R} u)(inf_{B_{2R}} u)^{p-1} ≤ C R^{p-n}. The proof consists of a Pohozaev-neck argument that upgrades the preliminary singular decay to the sharp p-harmonic fundamental solution rate |x|^{-(n-p)/(p-1)}.","tokens_in":1923,"tokens_out":489,"duration_ms":27632,"significance":"If the two stated hypotheses hold, the Pohozaev-neck upgrade supplies a method for obtaining sharp Harnack control in the critical p-Laplacian setting where Kelvin-transform and moving-sphere techniques are unavailable. The argument is logically self-contained once the inputs are granted and isolates the precise rate-improvement step needed for the inequality.","major_comments":[{"comment":"Abstract (paragraph beginning 'The result is conditional on two inputs'): the claimed inequality follows from the Pohozaev-neck upgrade only after the two external hypotheses are granted; the manuscript supplies neither a proof nor a citation establishing the bubble classification or the preliminary neck bound, so the central claim remains conditional on work outside the present scope.","section":"Abstract"}],"minor_comments":[{"comment":"The introduction should contain a short paragraph summarizing the logical dependence on the two inputs and indicating where in the literature (or in forthcoming work) each hypothesis is expected to be verified.","section":"Introduction"},{"comment":"Notation for the normalized neck quantities and the precise statement of the preliminary rate |x|^{-(n-p)/p} should be introduced before the Pohozaev identity is applied, to make the upgrade step easier to follow.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and the recommendation of minor revision. The manuscript is explicitly conditional on the two stated hypotheses, as already indicated in the abstract and introduction; we address the comment below.","responses":[{"response":"We agree that the result is conditional on the bubble classification and the preliminary neck bound, which are stated as explicit hypotheses both in the abstract and in the body of the paper. The contribution of the manuscript is the Pohozaev-neck upgrade step that improves the decay rate once these inputs are granted; the paper does not claim to prove the classification or the preliminary bound. To address the concern we will revise the abstract to make the conditional nature even more prominent and will add citations to existing literature where the classification of bounded entire solutions as p-bubbles and preliminary neck estimates have been established (or are treated as standard assumptions) for the critical p-Laplacian. A self-contained proof of the full classification lies outside the scope of this work.","revision_made":"partial","referee_comment":"[Abstract] Abstract (paragraph beginning 'The result is conditional on two inputs'): the claimed inequality follows from the Pohozaev-neck upgrade only after the two external hypotheses are granted; the manuscript supplies neither a proof nor a citation establishing the bubble classification or the preliminary neck bound, so the central claim remains conditional on work outside the present scope."}],"tokens_in":1399,"tokens_out":303,"duration_ms":16642,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core contribution is a Pohozaev-type neck argument that upgrades a preliminary singular decay |x|^{-(n-p)/p} to the sharp rate |x|^{-(n-p)/(p-1)} for positive weak solutions of -Δ_p u = g(u) under the given growth and monotonicity assumptions on g. This replaces Kelvin-transform or moving-sphere methods that do not extend to p ≠ 2. Under the two stated hypotheses (blow-up limits classified as Aubin-Talenti p-bubbles, plus initial neck control), the paper derives the Harnack bound (sup_{B_R} u) (inf_{B_{2R}} u)^{p-1} ≤ C R^{p-n} in B_{3R}.\n\nThe argument itself appears technically direct and uses the monotonicity condition cleanly to control the neck. The abstract states the conditional nature explicitly, which keeps the claims accurate.\n\nThe main limitation is that the paper supplies neither the classification of bounded positive entire blow-up limits nor the preliminary neck bound. Both are treated as external inputs, so the Harnack inequality is not established unconditionally. If those inputs fail or require substantial extra work, the upgrade step has limited standalone value. No other gaps are visible from the abstract and description.\n\nThis is for specialists in nonlinear elliptic equations who already work on p-Laplacian regularity or classification problems and can supply or verify the two inputs. A reader outside that niche gets little. The technique is worth referee attention because the neck method is new for this setting and the presentation is honest about its scope.","headline":"The paper gives a Pohozaev-neck upgrade from a preliminary decay rate to the sharp one for a conditional Harnack inequality in the critical p-Laplacian, but the main claim rests on two unproved inputs.","tokens_in":2445,"tokens_out":416,"would_cite":false,"duration_ms":14651,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Under blow-up classification and neck control, solutions of the critical p-Laplace equation obey the bound (sup u in B_R) times (inf u in B_{2R})^{p-1} at most C R^{p-n}.","keywords":["Pohozaev identity","neck analysis","Harnack inequality","p-Laplacian","critical exponent","blow-up limits","Aubin-Talenti bubbles","quasilinear elliptic equations"],"falsifier":"A positive weak solution in some ball B_{3R} for which (sup in B_R) times (inf in B_{2R})^{p-1} exceeds every constant times R^{p-n}, or a bounded positive entire solution that is not an Aubin-Talenti p-bubble.","tokens_in":2671,"feed_emoji":"","tokens_out":825,"duration_ms":31493,"temperature":0.7,"pith_summary":"The paper proves a conditional Harnack inequality for positive weak solutions of the critical p-Laplace equation under a global Sobolev growth condition and monotonicity of s^{-(p^*-1)} g(s). It assumes two external inputs: that every bounded positive entire blow-up limit is an Aubin-Talenti p-bubble, and that normalized necks already satisfy a preliminary singular upper bound on their rate. A Pohozaev identity applied across those necks then upgrades the decay from the weaker singular rate |x|^{-(n-p)/p} to the sharp fundamental-solution rate |x|^{-(n-p)/(p-1)}. A reader would care because the usual Kelvin-transform and moving-sphere techniques that work for the semilinear case p=2 are unavailable once p differs from 2.","feed_headline":"Pohozaev neck upgrades decay to conditional Harnack for p-Laplace","feed_subtitle":"Under bubble classification and neck control, solutions obey sup times inf^{p-1} bounded by C R^{p-n}.","key_machinery":"The Pohozaev-neck argument, which applies the Pohozaev identity in annular transition regions to upgrade the decay rate of the solution between bubbles.","core_discovery":"Under the classification that all bounded positive entire blow-up limits are Aubin-Talenti p-bubbles together with a preliminary singular-rate upper bound on normalized necks, any positive weak solution u of -Δ_p u = g(u) in the ball B_{3R} satisfies (sup_{B_R} u) (inf_{B_{2R}} u)^{p-1} ≤ C R^{p-n}. The argument uses the Pohozaev identity on annular neck regions to improve the pointwise decay rate on the necks to the sharp p-harmonic rate.","pith_inferences":["If the classification of blow-up limits can be proved independently, the Harnack inequality becomes unconditional in this setting.","The neck technique may apply to other quasilinear equations once a comparable classification result is known.","The improved decay rate supplies a new tool for analyzing isolated singularities of critical p-Laplace equations.","Similar annular Pohozaev identities could be tested on higher-order or anisotropic operators with comparable scaling."],"forward_implications":["The stated Harnack bound holds for all such solutions once the two hypotheses are granted.","The pointwise decay on the necks reaches exactly the fundamental-solution exponent (n-p)/(p-1).","The same neck argument replaces conformal methods for any p between 1 and n.","The inequality is available under the given monotonicity assumption on g without further structural restrictions."],"fun_headline_variants":["Pohozaev neck proves conditional Harnack for critical p-Laplace","Neck Pohozaev argument gives conditional Harnack p-Laplace","Conditional Harnack via Pohozaev necks for p-Laplacian","Pohozaev neck yields conditional Harnack in p-Laplace setting"],"cache_read_input_tokens":64,"weakest_assumption_plain":"That every bounded positive entire blow-up limit is an Aubin-Talenti p-bubble and that the normalized necks already obey a preliminary singular-rate upper bound.","fun_headline_variants_meta":{"raw":{"variants":["Pohozaev neck proves conditional Harnack for critical p-Laplace","Neck Pohozaev argument gives conditional Harnack p-Laplace","Conditional Harnack via Pohozaev necks for p-Laplacian","Pohozaev neck yields conditional Harnack in p-Laplace setting"]},"model":"grok-4.3","cost_usd":0.010775,"raw_usage":{"total_tokens":4784,"prompt_tokens":733,"num_sources_used":0,"completion_tokens":84,"cost_in_usd_ticks":107749500,"prompt_tokens_details":{"text_tokens":733,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3967,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":733,"tokens_out":84,"duration_ms":33179,"temperature":1.0,"reasoning_tokens":3967,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T00:41:57.041186+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A positive weak solution in some ball B_{3R} for which (sup in B_R) times (inf in B_{2R})^{p-1} exceeds every constant times R^{p-n}, or a bounded positive entire solution that is not an Aubin-Talenti p-bubble.","supporting_citations":[],"review_version":1}