{"id":"881e1e3f-7244-4fb8-9318-9c2f67dc1881","arxiv_id":"1908.07134","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper uses standard variational methods to claim infinitely many solutions for a p-Laplacian corner degenerate equation, but the strong equation and the weak form used in the proofs are inconsistent.","lead":"This paper claims to prove that a class of degenerate quasi-linear elliptic equations on corner manifolds has infinitely many solutions. The proofs solve a different weak equation than the strong form stated in the abstract and introduction.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1's proof requires q < p(p+1), but the theorem claims q < p*; for N ≤ p+1 the stated range is not covered.","rationale":"The reader's primary objection, that the weak form (1.2) is not the weak form of the strong PDE (1.1), is not correct. A direct calculation shows that for A = |∇u|^{p-2}∇u, using div_M(x1A) = x1 div_M A + x1 A1 and the identity ∫ div_M B φ dσ = -∫ B·∇φ dσ + ∫ B1 φ dσ, one obtains ∫ x1 A·∇φ dσ = -∫ x1 div_M A φ dσ. Thus (1.2) is exactly the weak formulation of (1.1) after multiplying the equation by the positive weight x1(x1x2)^p. The reader's derived 'strong operator with weights x1x2^{-1}, x1x2, ...' appears to come from omitting the boundary term B1 in the integration by parts. Since the stated weak-form equivalence holds, the reader's weakest assumption is not the actual load-bearing problem. However, the paper still has a serious internal gap: the proof of Theorem 1.1 relies on a weighted Sobolev embedding that requires q < p(p+1), while the theorem claims p < q < p*. For p in (N-1,N), p* exceeds p(p+1), so the theorem's range is too broad for the given proof. This is a concrete, checkable inconsistency between the theorem statement and the proof's hypotheses. Additionally, Proposition 3.6 is explicitly omitted despite being needed for condition (I5), and Lemma 4.1 uses a finite binomial expansion for non-integer p, which is invalid. These reinforce that the paper's central claims are not established as written. Since the overall disposition remains rejection, the verdict is unchanged, but the rationale differs from the reader's.","tokens_in":20757,"tokens_out":20864,"duration_ms":182010,"concrete_test":"Re-derive the estimate in Proposition 3.5 using only the hypotheses of Theorem 1.1, for the explicit parameter choice N=3, p=5/2, q=10. Here p < q < p*=15 but q > p(p+1)=8.75. Check the embedding condition of Lemma 3.1 for the nonlinear term: with γ1'=(N-p-1)/q=-0.05 and γ2'=(N-p)/q=0.05, we get N/q-γ1'=0.35 < N/p-γ1=0.4, so the embedding used to control the nonlinear term is not continuous. Verify that the estimate J(u) ≥ ||u||^p(1/p - C||u||^{q-p}) cannot be obtained in this regime, and if possible exhibit a function in H^{1,(N-1/p,N/p)}_{p,0}(M) for which ∫ x1^{3.5}x2^{2.5}|u|^{10} dσ diverges. This analytical check settles whether the theorem's range must be narrowed to q < p(p+1).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.1 uses a weighted Sobolev embedding that is only valid when q < p(p+1). In Proposition 3.5 the paper states 'according to both Lemma 3.1 and the condition q < p* < p(p+1)', but p* < p(p+1) is equivalent to N > p+1. When 2 < p < N and N ≤ p+1, i.e. p ≥ N-1, we have p* ≥ p(p+1), so the theorem's range p < q < p* includes values q ≥ p(p+1). For such q, the embedding H^{1,(N-1/p,N/p)}_{p,0}(M) ↪ L^q_{((N-p-1)/q,(N-p)/q)}(M) in Lemma 3.1 fails: the first inequality becomes (p+1)/q ≤ 1/p, and for q > p(p+1) it fails outright. Consequently the nonlinear term ∫ x1^{p+1}x2^p|u|^q dσ may be infinite for admissible u, so the energy functional J is not well-defined on the claimed solution space. The mountain-pass/PS argument in Section 3 therefore does not establish existence for the full parameter range stated in Theorem 1.1. This is a load-bearing gap independent of the weak-form question: even if (1.2) is accepted as the correct weak form, the variational proof only covers q < p(p+1), not all p < q < p*. The paper also omits the proof of Proposition 3.6 (condition I5) and uses an invalid finite binomial expansion for real p in Lemma 4.1, but the q-range gap alone is sufficient to prevent the theorem from being proved as stated.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the quasi-linear degenerate elliptic Dirichlet problem (1.1) on a model corner manifold M=(0,δ)^2×X, where the degenerate operator is built from ∇M=(x1∂x1, x1x2∂x2, ∂x') and divM=∇M·. The paper defines weak solutions through (1.2) and an energy functional J, then claims in Theorem 1.1 that for 2<p<N and p<q<p*, λ>0, there are infinitely many nontrivial weak solutions, with critical values tending to infinity in Theorem 1.2. For the case p=q, Theorems 1.3 and 1.4 claim an infinite sequence of eigenvalue-eigenvector pairs (u_k, λ_k) with λ_k→∞. Section 3 uses abstract critical-point theorems from [2], establishing (PS) and conditions (I1)–(I5), while Section 4 employs a Lusternik-Schnirelman argument on a constraint manifold.","tokens_in":21108,"tokens_out":11250,"duration_ms":99209,"significance":"If the results were correct, they would extend known multiplicity theorems for the p-Laplacian to a class of corner-degenerate operators, which is a genuinely useful direction given the applications to corner singularities. The paper has clear strengths: it works in a weighted Sobolev framework, gives a detailed (PS) verification using a Brezis-Lieb-type lemma, and follows explicit abstract minimax schemes for both the superlinear and the eigenvalue cases. However, several load-bearing gaps prevent the claims from being accepted as proved: the weak form does not correspond to the stated PDE under the paper's own definitions, the q-range in Theorem 1.1 is not covered by the embedding used, a required condition (I5) is left unproved, and the p=q argument relies on an invalid binomial expansion for non-integer p.","major_comments":[{"comment":"The weak form (1.2) is not equivalent to the PDE (1.1) under the definitions given in the introduction. With dσ=dx1/x1 dx2/(x1x2)dx', direct integration by parts converts the left side of (1.2) into −∫ φ x1x2[∂_{x1}(A_1 x2^{-1}) + ∂_{x2}(A_2 x1) + ∂_{x'}(A' (x1x2)^{-1})] dσ, where A=|∇_M u|^{p−2}∇_M u. The strong operator in (1.1) is (x1x2)^{−p}div_M A = (x1x2)^{−p}(x1∂_{x1}A_1+x1x2∂_{x2}A_2+∂_{x'}A'), which has different weights and different derivative structure. Since the paper neither proves nor derives the equivalence asserted in the sentence after (1.2), the theorems prove existence of solutions to a different equation than the one stated.","section":"§1, Eq. (1.1) vs. Eq. (1.2)"},{"comment":"Theorem 1.1 claims p<q<p*, but Proposition 3.5 uses the condition q<p*<p(p+1). This condition holds only when N>p+1; for 2<p<N with N≤p+1 one has p*≥p(p+1), so the theorem's range includes q≥p(p+1). The embedding in Lemma 3.1 with the weights γ'=((N−p−1)/q,(N−p)/q) used in the energy estimates requires (p+1)/q>1/p, i.e. q<p(p+1). For q≥p(p+1) the nonlinear term ∫ x1^{p+1}x2^p|u|^q dσ need not be finite on H^{1,((N−1)/p,N/p)}_{p,0}(M), so J is not defined on the claimed solution space. The proof as written therefore covers only q<p(p+1), not the full range stated in Theorem 1.1.","section":"§3.1, Proposition 3.5 and Lemma 3.1"},{"comment":"Condition (I5) is a hypothesis of Lemmas 2.9 and 2.10 and is verified by Proposition 3.6, but the proposition is not proved; the text says only \"We omit the easy proof of Proposition 3.6 here for the limit length of writing.\" Since the conclusion of Theorem 1.1 depends on these lemmas, an omitted proof of a required condition leaves a gap in the derivation of the critical-value sequence.","section":"§3.1, Proposition 3.6"},{"comment":"The proof of Lemma 4.1 applies the binomial theorem to |u0+δ|^{p−2}(u0+δ) and expands it in powers of δ up to p−2. Here p is a real number in (2,N), not an integer, so the finite binomial expansion with combinatorial coefficients is not valid. The same invalid expansion is used in Lemma 4.2 and in the estimates for D(u). Since uniform continuity and compactness of b underpin the deformation and minimax argument, the proof of Theorem 1.3 is not supported.","section":"§4.1, Eq. (4.3)"}],"minor_comments":[{"comment":"There are typos: \"degenera te\" in the abstract and \"processes\" should be \"possesses\" in Theorems 1.1 and 1.3.","section":"Abstract and Theorems 1.1, 1.3"},{"comment":"The proof of Proposition 2.4 simply says \"Follow the same process of Proposition 3.2 in [5]\"; at a minimum the relevant hypotheses and the role of the cut-off functions should be stated so the Poincaré-type inequality is verifiable from the text.","section":"§2, Proposition 2.4"},{"comment":"In the (PS) verification, the line J(u_k)−(1/q)<J'(u_k),u_k> uses notation that is not fully defined; in particular <J'(u_k),u_k> should be the pairing between H^{-1,(-(N−1)/p,−N/p)}_p and H^{1,((N−1)/p,N/p)}_{p,0}(M), and the displayed formula for the difference <J'(u_k)−J'(u),u_k−u> should include the weight x1 explicitly for consistency.","section":"§3.1, Lemma 3.4"},{"comment":"In Lemma 4.8 the dual space is written H^{-1,(-N−1/p,−N/p)}_p(B), where B is not defined in this subsection; this appears to be a relic of the earlier notation for the unit ball and should be M.","section":"§4.1, Lemma 4.8"}],"recommendation":"reject","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What you should know up front: this paper claims infinitely many solutions for a quasi-linear degenerate p-Laplacian on a stretched corner, but the weak form it actually studies is not the weak form of the stated strong equation. That mismatch alone undermines the central claim as written. There is also an independent gap in the proof of Theorem 1.1: the variational argument only covers q < p(p+1), not the full stated range p < q < p*.\n\nThe problem itself is new and reasonable. The p>2 extension of the p=2 case from [5] is a legitimate question, and the corner-degenerate weighted Sobolev setup is appropriate. The variational machinery — Ambrosetti–Rabinowitz for p<q and Lusternik–Schnirelman for p=q — is standard, but it is carefully adapted to the weighted spaces, and Lemma 3.1 gives a usable embedding if the weights are chosen correctly. The author also cites the singular-analysis literature fairly.\n\nThe soft spots are serious. First, equation (1.1) has -(x1x2)^{-p} div_M(...) on the left; integrating by parts against dσ = dx1/x1 dx2/(x1x2) dx' gives a term with weight (x1x2)^{-p}, not x1. The functional J defines critical points for (1.2), not for (1.1). So the theorems solve a different equation. This is not a one-line typo; the weights propagate through the entire argument. Second, Proposition 3.5 asserts q < p* < p(p+1), but that inequality requires N > p+1. For N ≤ p+1, the theorem's range includes q ≥ p(p+1), where the embedding in Lemma 3.1 fails and J may not even be well-defined. The stress-test note is correct about this. Third, Proposition 3.6 is simply omitted, though it is needed for condition I5. Fourth, Lemma 4.1 uses a finite binomial expansion for real p, which is invalid; the map z ↦ |z|^{p-2}z needs a mean-value or Hölder-type estimate instead.\n\nNone of this is fatal to the underlying idea, but as written the theorems are not proved. I would not cite this version. A serious referee could still be useful, because the author has identified a real problem and the gaps are identifiable and likely fixable, but the paper needs substantial revision before it can be accepted.","headline":"The paper targets a real p>2 extension of a corner-degenerate p-Laplacian result, but the weak form it analyzes is not the weak form of the stated strong equation, and the proof of Theorem 1.1 does not cover the full claimed q-range.","tokens_in":21623,"tokens_out":7125,"would_cite":false,"duration_ms":65520,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J92","35J70","35P30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Corner-degenerate p-Laplacian: infinitely many weak solutions, and an infinite eigenvalue sequence in the resonant case.","keywords":["degenerate elliptic equations","corner singularity","weighted Sobolev spaces","p-Laplacian","variational methods","genus","nonlinear eigenvalue problem","multiple weak solutions"],"falsifier":"Compute the Euler-Lagrange equation of the functional $J$ with respect to the measure $d\\sigma=x_1^{-2}x_2^{-1}dx_1dx_2dx'$ in ordinary coordinates. The three derivative terms acquire coefficients $x_1x_2^{-1}$, $x_1x_2$, and $x_1^{-1}x_2^{-1}$; equation (1.1) has the single coefficient $(x_1x_2)^{-p}$ in front of $\\mathrm{div}_M(\\nabla_Mu)$. If the two expressions are not equal, then (1.2) is not the weak form of (1.1), and the infinitely many critical points belong to a different boundary-value problem.","tokens_in":1732,"feed_emoji":"♾️","tokens_out":5541,"duration_ms":153068,"temperature":0.7,"pith_summary":"This paper claims that a quasilinear elliptic equation built from the corner-degenerate gradient operator has infinitely many nontrivial weak solutions. For exponents $2<p<N$ and $p<q<p^*$, the main theorem states that for every $\\lambda>0$ the Dirichlet problem has infinitely many nontrivial weak solutions, and their energy levels tend to infinity. For the resonant case $q=p$, the paper states an infinite sequence of eigenpairs $(u_k,\\lambda_k)$ with $\\lambda_k\\to\\infty$. If correct, the results show that the degenerate operator inherits the same multiplicity picture as the classical $p$-Laplacian in a corner-geometry setting.","feed_headline":"Infinitely many solutions for a degenerate corner equation","feed_subtitle":"A variational proof finds infinitely many nonzero weak solutions; when p=q the eigenvalue sequence tends to infinity.","key_machinery":"The central object is the corner-degenerate gradient $\\nabla_M=(x_1\\partial_{x_1},x_1x_2\\partial_{x_2},\\partial_{x'})$ with divergence $\\mathrm{div}_M=\\nabla_M\\cdot$, acting on the weighted Sobolev space $H^{1,((N-1)/p,N/p)}_{p,0}(M)$ with measure $d\\sigma=x_1^{-2}x_2^{-1}dx_1dx_2dx'$. What carries the argument is the compact embedding of this space into weighted $L^q$ spaces (Lemma 3.1). The even functional is then analyzed by genus of symmetric sets: abstract critical-point lemmas yield infinitely many critical values in the $p<q$ case, while for $q=p$ a tangent flow on the level manifold $\\mathcal{M}=\\{u:\\frac1p\\int_M x_1|\\nabla_Mu|^p\\,d\\sigma=\\alpha\\}$ runs a symmetric min-max scheme with a duality map in place of the gradient.","core_discovery":"On the weighted Sobolev space $H^{1,((N-1)/p,N/p)}_{p,0}(M)$, the energy functional $J(u)=\\frac{1}{p}\\int_M x_1|\\nabla_M u|^p\\,d\\sigma-\\frac{\\lambda}{q}\\int_M x_1(x_1x_2)^p|u|^q\\,d\\sigma$ has infinitely many critical points. For $2<p<N$ and $p<q<p^*$, Theorem 1.1 asserts infinitely many nontrivial weak solutions in the sense of (1.2), with critical values $c_m\\to\\infty$; Theorem 1.3 asserts that when $q=p$ there are infinitely many pairs $(u_k,\\lambda_k)$ satisfying the weak equation, with $\\lambda_k\\to\\infty$. The proof uses the compact embedding of the weighted Sobolev space into a weighted Lebesgue space, together with symmetric genus-based min-max arguments.","pith_inferences":["Editorial extension: The paper does not say whether the infinitely many solutions are sign-changing; because the functional is even, applying the same genus count to subsets of fixed sign would be a natural further test, but that conclusion is not in the paper.","Editorial extension: The compact embedding is the main place the corner geometry enters; if the same embedding holds for more general weights or domains, the multiplicity theorem should carry over, though this is not demonstrated.","Editorial extension: If the weak/strong equivalence check described in the falsifier reveals a mismatch, the correct strong equation behind (1.2) can be computed by integrating by parts; the theorems would then describe that corrected problem, not necessarily equation (1.1)."],"forward_implications":["For every fixed $\\lambda>0$ and $p<q<p^*$, the Dirichlet problem has infinitely many nontrivial weak solutions, with energy levels $c_m\\to\\infty$.","When $q=p$, there is an infinite sequence of eigenpairs $(u_k,\\lambda_k)$ solving the weak equation, with $\\lambda_k\\to\\infty$; the degenerate operator therefore has an unbounded spectrum.","The weighted Sobolev framework transfers the classical $p$-Laplacian multiplicity picture to a model corner domain, so analogous statements should hold on finite stretched corner domains rather than only on $(0,\\delta)^2\\times X$.","Each weak solution satisfies (1.2), so numerical or regularity analysis can proceed from the variational formulation without treating the strong PDE directly."],"supporting_citations":[{"why":"Supplies the symmetric eigenvalue min-max scheme adapted for the p=q case.","marker":"[1]"},{"why":"Provides the abstract critical-point theorems used to obtain infinitely many critical values in the p<q case.","marker":"[2]"},{"why":"Provides the duality map properties used to construct the tangent flow on the level manifold.","marker":"[3]"},{"why":"Provides the weighted Poincaré inequality and the prior corner-degenerate Sobolev framework used in Proposition 2.4.","marker":"[5]"},{"why":"Supplies the genus properties used to define the symmetric min-max classes.","marker":"[12]"},{"why":"Provides the deformation and minimax arguments used to extract critical points in the p=q case.","marker":"[13]"}],"fun_headline_variants":["Infinite solutions for a degenerate corner equation","Degenerate elliptic PDEs yield infinitely many solutions","Corner-degenerate operator: infinitely many critical points","Weighted Sobolev space reveals infinite solution family","Quasilinear degenerate equations: infinite multiplicity"],"cache_read_input_tokens":23680,"weakest_assumption_plain":"The load-bearing premise is that the weak equation (1.2) is equivalent to the strong equation (1.1), so the critical points produced are genuine solutions of the boundary-value problem stated in the title.","fun_headline_variants_meta":{"raw":{"variants":["Infinite solutions for a degenerate corner equation","Degenerate elliptic PDEs yield infinitely many solutions","Corner-degenerate operator: infinitely many critical points","Weighted Sobolev space reveals infinite solution family","Quasilinear degenerate equations: infinite multiplicity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000608,"raw_usage":{"total_tokens":2735,"prompt_tokens":752,"completion_tokens":1983,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":368,"completion_tokens_details":{"reasoning_tokens":1912}},"tokens_in":368,"tokens_out":1983,"duration_ms":16475,"temperature":1.0,"reasoning_tokens":1912,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:26:31.165876+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the Euler-Lagrange equation of the functional $J$ with respect to the measure $d\\sigma=x_1^{-2}x_2^{-1}dx_1dx_2dx'$ in ordinary coordinates. The three derivative terms acquire coefficients $x_1x_2^{-1}$, $x_1x_2$, and $x_1^{-1}x_2^{-1}$; equation (1.1) has the single coefficient $(x_1x_2)^{-p}$ in front of $\\mathrm{div}_M(\\nabla_Mu)$. If the two expressions are not equal, then (1.2) is not the weak form of (1.1), and the infinitely many critical points belong to a different boundary-value problem.","supporting_citations":[{"cited_title":"Amann, Lusternik-Schnirelman Theory and Nonlinear e igenvalue problems Math","cited_arxiv_id":null,"evidence_quote":"Supplies the symmetric eigenvalue min-max scheme adapted for the p=q case."},{"cited_title":"Ambrosetti, P","cited_arxiv_id":null,"evidence_quote":"Provides the abstract critical-point theorems used to obtain infinitely many critical values in the p<q case."},{"cited_title":"Brezis, Functional Analysis, Sobolev spaces, and Par tial Differential equations, Springer, 2011","cited_arxiv_id":null,"evidence_quote":"Provides the duality map properties used to construct the tangent flow on the level manifold."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the weighted Poincaré inequality and the prior corner-degenerate Sobolev framework used in Proposition 2.4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the genus properties used to define the symmetric min-max classes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the deformation and minimax arguments used to extract critical points in the p=q case."}],"review_version":1}