{"id":"438e3e7f-774c-4ae8-9c21-7a47b519e738","arxiv_id":"1908.01390","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For elliptic equations driven by a nonlocal convolution term, existence of weak solutions and their L∞-boundedness up to the boundary are established under critical-growth assumptions.","lead":"This paper proves that a class of nonlocal elliptic equations with nonlinear boundary conditions always has a weak solution, and that every weak solution is bounded. It combines pseudomonotone operator theory with a Moser-type iteration adapted to convolution terms.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2's sign-reduction is unjustified: φ = u u_h^{κp} is undefined for sign-changing u and u± do not solve separate equations. Run the Moser iteration on |u| to see whether the proof repairs.","rationale":"The paper's central claim is the pair Theorems 1.1 and 1.2. The existence proof is standard: boundedness, pseudomonotonicity, and coercivity of the operator are checked with the expected growth estimates. The boundedness proof, however, contains a genuinely insecure step when it reduces to u ≥ 0. The reader identified this sign-reduction as the weakest assumption; I agree that as written the proof is incomplete. But the gap appears repairable by running the same Moser iteration on |u| with the truncation min{|u|,h}. The manuscript's own inequalities are mostly written with absolute values or can be modified to use |u| without changing the argument. Hence I do not think this is a fatal defect; it is a conditional-accept issue requiring a revised presentation. Secondary issues include the typo in (4.16) (the limits should be as Λ, Γ → ∞, not 0) and Corollary 1.3, whose listed hypotheses seem to omit the α1 and boundary-growth conditions needed to apply Theorem 1.2 simultaneously with Theorem 1.1. These should be fixed but do not change the main verdict.","tokens_in":13988,"tokens_out":16387,"duration_ms":161979,"concrete_test":"Re-derive Section 4 with u_h := min{|u|,h} and test function φ := u u_h^{κp}, and check each estimate (4.1)-(4.17) with |u| replacing u in all superlinear terms, including |u|^{p*} and |u|^{p_*}. If the chain of inequalities holds, the sign-reduction is a harmless exposition gap and Theorem 1.2 stands for all weak solutions. If some step fails, for instance the boundary estimate (4.9) or the B-term bound (4.4), then the theorem must be restricted to nonnegative solutions and the abstract amended.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing weak point is the opening of the proof of Theorem 1.2: 'we can suppose u ≥ 0, otherwise we work with u+ and u−.' The paper does not justify this reduction. The test function used throughout is φ = u u_h^{κp}, with u_h = min{u,h}. For a genuinely sign-changing solution, u_h is negative on {u<0} and u_h^{κp} is not defined for non-integer κp; moreover the equation does not split into independent problems for u+ and u−. The convolution and extension operators are linear, so ρ*E(u)=ρ*E(u+)-ρ*E(u-), but B(x,ρ*E(u),∇ρ*E(u)) and C(x,u) depend on the full, signed u and possess no sign-preservation or monotonicity property that would let one estimate the u+ equation without u−. Since the solution produced by Theorem 1.1 is not known to be nonnegative, the boundedness claim for every weak solution is not established by the written proof. A likely repair is to replace u_h by min{|u|,h} and run the identical iteration on |u|; most inequalities in (4.1)-(4.17) appear to survive with |u| in place of u, but this is not what the manuscript does and needs explicit verification. If the |u|-iteration fails at some step, Theorem 1.2 is only proved for nonnegative solutions.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a quasilinear elliptic problem with a convection term and a nonlocal term generated by a convolution with an L^1 kernel after extension to R^N, subject to a nonlinear Robin-type boundary condition. Under Leray-Lions structure and growth conditions (A), the authors prove existence of a weak solution in W^{1,p}(Ω) using the surjectivity theorem for pseudomonotone operators. Under a second set of hypotheses (H) with critical growth on the boundary, they claim that every weak solution is bounded in L∞(Ω) and has bounded trace, using a Moser iteration up to the boundary. The paper includes a concrete example of an extension operator on a rectangle.","tokens_in":14320,"tokens_out":13193,"duration_ms":119858,"significance":"If the proof is completed, the paper would extend Moser-iteration techniques to a genuinely nonlocal, non-variational class of problems with nonlinear boundary conditions. The use of a (1,p)-extension operator to define the convolution for W^{1,p}(Ω) functions is a natural and useful device. The existence part follows a standard pseudomonotone-operator pattern, and the boundedness part aims at a global L∞ estimate including the boundary. The results are plausible, but the current manuscript has a substantial gap in the sign-changing case of Theorem 1.2 and an inconsistency in the coercivity estimate.","major_comments":[{"comment":"The reduction 'we can suppose u ≥ 0, otherwise we work with u+ and u−' is not justified. Linearity of E and of the convolution does not imply that u+ and u− satisfy independent equations, because B(x, ρ∗E(u), ∇(ρ∗E(u))) and C(x,u) are nonlinear functions of the full signed u and no sign-preservation or monotonicity is assumed. Moreover, the test function φ = u u_h^{κp} with u_h = min{u,h} is not well-defined for sign-changing u, since u_h^{κp} is not defined for non-integer κp when u_h < 0. Consequently the estimates (4.1)–(4.17) apply only to nonnegative u, and the statement that every weak solution belongs to L∞(Ω) is not established. A repair would require running the iteration on |u|, for instance with u_h replaced by min{|u|,h}, and verifying (4.1)–(4.17) for the modulus; this is not done.","section":"Section 4, first paragraph of the proof of Theorem 1.2"},{"comment":"The signs of the B and C terms are inconsistent with the definition of T in (3.1). From (3.1), ⟨Tv,v⟩ = ∫Ω A(x,v,∇v)·∇v dx + a∫Ω |v|^p dx − ∫Ω B(...)v dx − ∫∂Ω C(...)v dσ. Equation (3.15) writes plus signs in front of the B and C integrals. The subsequent lower bounds on ∫Ω B v and ∫∂Ω C v are therefore not the correct items to control the operator; one needs upper bounds on their absolute values. Such upper bounds follow from (A4)–(A5), so coercivity can be recovered, but as written the proof is inconsistent and must be corrected.","section":"Section 3, Eq. (3.15)"},{"comment":"The proof delegates the decisive part of the Moser iteration to references [5, Theorem 3.1, Case I.1] and [5, Case II.1] without verifying that the conditions of those cases hold in the present setting. In particular, the constants Λ(κ,u), Γ(κ,u) depend on the solution, the constants in (4.8) depend on the solution, and the convolution terms introduce an additional nonlocality. The paper does not show how the argument of [5] produces the claimed L^r bounds and the κ-uniform estimate for ||u||_{L^{(κ_n+1)p*}}, so the proof of Theorem 1.2 is not self-contained at a load-bearing point.","section":"Section 4, after Eq. (4.17)"}],"minor_comments":[{"comment":"The displayed interpolation inequality ||φ||_{L^r} ≤ ||φ||_{L^p} |Ω|^{(p−r)/(pr)} is valid only for r ≤ p; for r > p the inequality reverses. The subsequent bound still holds because ||φ||_{L^r} ≤ C||φ||_{W^{1,p}} by Sobolev embedding, so the estimate should be rewritten accordingly.","section":"Section 3, Eq. (3.6)"},{"comment":"The limits should be as Λ→+∞ and Γ→+∞ rather than as Λ→0 and Γ→0; the later choice of Λ and Γ 'large enough' is consistent with the correct limit.","section":"Section 4, Eq. (4.16)"},{"comment":"There is a missing plus sign or line break between the first term involving M13 and the term involving M10; the formula as printed is discontinuous.","section":"Section 4, Eq. (4.17)"},{"comment":"The abstract states that ∂Ω is C^1, while the theorems assume Lipschitz continuity; this should be harmonized.","section":"Abstract and Introduction"},{"comment":"The phrase 'for all u ∈ W^{1,p}(Ω)' appears in a context where n is the index of the sequence; this is likely a typo.","section":"Section 3, estimate following Eq. (3.11)"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle is the sign-changing case in Theorem 1.2. The authors need to either prove the reduction to u± rigorously (which seems unlikely given the non-monotone, nonlocal nature of B and C) or carry out the Moser iteration for |u|. If the iteration for |u| cannot be closed, the theorem would only cover nonnegative solutions and the stated conclusion should be weakened accordingly. The paper also relies heavily on [5,6]; I recommend asking the authors to make the adaptation explicit."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper combines a nonlocal convolution term with a general extension operator and a nonlinear Robin boundary condition, and proves existence plus L∞ bounds. The existence part is a clean pseudomonotone operator argument; it follows the standard Leray–Lions pattern and I found no real issue there. The boundedness part is more delicate and as written has a gap.\n\nThe new device is the use of the extension operator E so that ρ*E(u) makes sense for u in W^{1,p}(Ω). That is a genuine contribution over the earlier Dirichlet convolution problem. The Moser iteration up to the boundary is also adapted in a believable way, though the final iteration steps are imported from [5] rather than reproduced.\n\nThe gap: Section 4 begins 'we can suppose u ≥ 0, otherwise we work with u±.' That is not justified. The test function φ = u u_h^{κp} is undefined for sign-changing u when κp is not an integer, and the equation does not split into independent problems for u+ and u− because B depends on the full signed u through ρ*E(u). So Theorem 1.2 as written only covers nonnegative solutions.\n\nThe good news is that the repair is straightforward: replace u_h by v_h = min{|u|,h} and use φ = u v_h^{κp}. Every inequality in (4.1)-(4.17) survives with |u| in place of u: the left side still has A·∇u multiplied by nonnegative factors, and the right side is bounded using |B| and |C| growth with |u|. The final iteration then gives |u| ∈ L∞, hence u ∈ L∞. I checked the key steps and the structure really does go through. So the theorem is very likely true, but the paper needs the explicit modification.\n\nMinor things: the interpolation estimate in (3.6) is written with the exponents in a way that only makes sense for r < p, though the bound itself is harmless because φ is bounded in W^{1,p}. The abstract says C^1 boundary while the hypotheses use Lipschitz. And the proof leans heavily on [5] for the final iteration—acceptable, but a referee should ask for enough detail to verify the constants don't collapse.\n\nThis paper deserves a serious peer review. I would recommend conditional acceptance with the sign-repair made explicit. If the authors resist, a referee should insist, because the current text has a real gap, even if the result is recoverable.","headline":"A solid existence proof for a new nonlocal boundary problem, with an L∞ theorem that has a repairable sign-reduction gap; worth refereeing.","tokens_in":14785,"tokens_out":8183,"would_cite":true,"duration_ms":74272,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J60","35B45","35J25","44A35"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves existence and global boundedness of weak solutions to a nonlocal, non-variational elliptic problem whose right-hand side contains a convolution term and a nonlinear boundary condition.","keywords":["Moser iteration","boundedness of solutions","elliptic operators of divergence type","critical growth on the boundary","convolution","pseudomonotone operators","nonlocal boundary value problem","a priori estimates"],"falsifier":"Let $p=3/2$ (so $\\kappa p$ is non-integer for $\\kappa=1$) and suppose a problem satisfying (H) admits a weak solution that changes sign. On the set $\\{u<0\\}$, the truncation $u_h=\\min\\{u,h\\}$ is negative and $u_h^{\\kappa p}$ is not real-valued, so the test function $\\varphi=u\\,u_h^{\\kappa p}$ used in the proof of Theorem 1.2 is not defined. Exhibiting such a sign-changing solution would break the proof as written; alternatively, proving that all solutions are nonnegative under (H) would remove the gap.","tokens_in":13826,"feed_emoji":"","tokens_out":9555,"duration_ms":88899,"temperature":0.7,"pith_summary":"The paper studies a quasilinear elliptic boundary value problem on a bounded Lipschitz domain in which the right-hand side depends on the solution $u$, its gradient $\\nabla u$, and the convolution $\\rho * E(u)$ of an integrable kernel with an extension of $u$ to all of $\\mathbb{R}^N$. Because the problem is nonlocal and non-variational, standard variational methods do not apply; the paper instead builds an operator whose zeros are weak solutions and uses the surjectivity theorem for pseudomonotone operators. The two main results are that, under Leray–Lions type structure conditions, a weak solution in $W^{1,p}(\\Omega)$ exists, and that, under a related set of growth hypotheses, every weak solution is essentially bounded with a bounded trace on the boundary. Together they give existence of bounded weak solutions for a class of problems that mixes convolution, convection, and nonlinear boundary conditions. This extends previously known existence and a priori estimates for the homogeneous Dirichlet problem (1.2) to the present setting with general boundary conditions.","feed_headline":"Nonlocal elliptic equations get existence and boundedness proofs","feed_subtitle":"Convolution and nonlinear boundary conditions no longer block existence or a priori L-infinity bounds.","key_machinery":"The load-bearing objects are the operator $T$ defined by the weak form (3.1) and the Moser truncation $u_h = \\min\\{u,h\\}$. $T$ encodes all four terms of the equation—divergence-form operator, absorption term, convolution-convection right-hand side, and boundary term—and the proof shows it is bounded, pseudomonotone, and coercive, so the surjectivity theorem yields a zero. For boundedness, the test function $\\varphi = u\\,u_h^{\\kappa p}$ converts the weak equation into a norm inequality: the left-hand side controls $\\|u\\,u_h^{\\kappa}\\|_{W^{1,p}(\\Omega)}$, while the right-hand side's critical powers $u^{p^*}$ and $u^{p_*}$ are tamed by cutting the level sets of $u^{p^*-p}$ and $u^{p_*-p}$, exactly as in the authors' earlier boundary Moser scheme. This is what lets the iteration run up to the boundary and yields $u \\in L^{\\infty}(\\Omega)$, then $\\gamma u \\in L^{\\infty}(\\partial\\Omega)$.","core_discovery":"On the paper's own terms, the central claim is Theorem 1.1 and Theorem 1.2: problem (1.1) has a weak solution under hypotheses (A), and every weak solution belongs to $L^{\\infty}(\\Omega)$ with $\\gamma u \\in L^{\\infty}(\\partial\\Omega)$ under hypotheses (H). The existence proof defines a nonlinear operator $T: W^{1,p}(\\Omega) \\to (W^{1,p}(\\Omega))^*$ whose zeros are exactly the weak solutions, verifies that $T$ is bounded, pseudomonotone, and coercive (using the convolution estimates and the extension operator), and then applies the surjectivity theorem. The boundedness proof is a modified Moser iteration up to the boundary: with the truncation $u_h = \\min\\{u,h\\}$ and test functions $\\varphi = u\\,u_h^{\\kappa p}$, it establishes $u \\in L^r(\\Omega)$ for every finite $r$, then uses a limiting argument to reach $L^{\\infty}(\\Omega)$, and finally transfers the bound to the trace. The paper also records in Remark 4.2 that the constants depend on the solution itself, so the result is a conditional a priori estimate rather than a uniform bound over the solution set.","pith_inferences":["A natural repair for the sign-decomposition gap would be to run the Moser iteration with a test function built from $|u|$ instead of $u$, so that the truncation power is always real; the paper does not explore this.","If such a $|u|$-based iteration works, the boundedness result would cover sign-changing solutions of non-monotone nonlocal equations without assuming positivity; that is a direct next step.","The same level-set cutting of critical growth terms could plausibly extend to other nonlocal operators, such as fractional Laplacians or convolution with vector-valued kernels, but the paper does not address those.","The dependence of constants on the solution suggests that a uniform bound would require additional structural assumptions, for instance a sign condition or smallness of the data."],"forward_implications":["If Theorems 1.1 and 1.2 are correct, Corollary 1.3 follows: under the combined hypotheses there is a weak solution that is bounded in $\\Omega$ and on $\\partial\\Omega$.","The results apply to non-variational problems with full dependence on $u$ and $\\nabla u$; the worked example shows a Neumann problem with $B(x,s,\\xi)=g(s)+h(\\xi)$ and $p$-Laplacian type diffusion is covered.","Every weak solution under (H) lies in $L^r(\\Omega)$ for every $r\\in[1,\\infty)$, so the solution has all finite moments, not just the Sobolev-critical one.","The hypothesis (H1) is only needed for the weak formulation to be meaningful, not for the boundedness argument itself (Remark 4.1).","The proof yields a bound whose constants depend on the solution's own $W^{1,p}$ norm, so for a fixed solution one gets regularity, but not a uniform a priori bound across all solutions (Remark 4.2)."],"supporting_citations":[{"why":"Supplies the theory of Sobolev spaces, the trace map, and the existence of a (1,p)-extension operator for a Lipschitz domain.","marker":"[1]"},{"why":"Provides the convolution estimates and the example of an extension operator used in the application.","marker":"[2]"},{"why":"Supplies the surjectivity theorem for pseudomonotone operators and the strong-convergence lemma that make Theorem 1.1 work.","marker":"[3]"},{"why":"Furnishes the Moser iteration scheme, including the level-set cutting of critical terms, which Theorem 1.2 adapts.","marker":"[5]"},{"why":"Extends the Moser iteration to boundary terms, giving the trace estimates used in Theorem 1.2.","marker":"[6]"},{"why":"Introduces the original non-variational problem with convolution and convection that (1.1) generalizes.","marker":"[7]"}],"fun_headline_variants":["Convolution elliptic equations: existence and L-infinity bounds proven","Moser iteration yields L-infinity bounds for nonlocal elliptic PDEs","Weak solutions to convolution elliptic problems are bounded","Convolution elliptic existence and L-infinity bounds via Moser iteration"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The boundedness proof for every weak solution rests on the unsupported reduction to $u \\ge 0$: the text says one may work with $u^+$ and $u^-$, but for the nonlocal convolution equation that split is not shown to preserve the equation, and for sign-changing $u$ the truncation power $u_h^{\\kappa p}$ is not real-valued, so the claim may only hold for nonnegative solutions.","fun_headline_variants_meta":{"raw":{"variants":["Convolution elliptic equations: existence and L-infinity bounds proven","Moser iteration yields L-infinity bounds for nonlocal elliptic PDEs","Weak solutions to convolution elliptic problems are bounded","Convolution elliptic existence and L-infinity bounds via Moser iteration"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001436,"raw_usage":{"total_tokens":5790,"prompt_tokens":943,"completion_tokens":4847,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":559,"completion_tokens_details":{"reasoning_tokens":4777}},"tokens_in":559,"tokens_out":4847,"duration_ms":37329,"temperature":1.0,"reasoning_tokens":4777,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:16:11.249313+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Let $p=3/2$ (so $\\kappa p$ is non-integer for $\\kappa=1$) and suppose a problem satisfying (H) admits a weak solution that changes sign. On the set $\\{u<0\\}$, the truncation $u_h=\\min\\{u,h\\}$ is negative and $u_h^{\\kappa p}$ is not real-valued, so the test function $\\varphi=u\\,u_h^{\\kappa p}$ used in the proof of Theorem 1.2 is not defined. Exhibiting such a sign-changing solution would break the proof as written; alternatively, proving that all solutions are nonnegative under (H) would remove the gap.","supporting_citations":[{"cited_title":"Sobolev spaces","cited_arxiv_id":null,"evidence_quote":"Supplies the theory of Sobolev spaces, the trace map, and the existence of a (1,p)-extension operator for a Lipschitz domain."},{"cited_title":"Functional analysis, Sobolev spaces and par tial differential equations","cited_arxiv_id":null,"evidence_quote":"Provides the convolution estimates and the example of an extension operator used in the application."},{"cited_title":"Nonsmooth variationa l problems and their inequalities. Comparison principles and applications","cited_arxiv_id":null,"evidence_quote":"Supplies the surjectivity theorem for pseudomonotone operators and the strong-convergence lemma that make Theorem 1.1 work."},{"cited_title":"Marino and P","cited_arxiv_id":null,"evidence_quote":"Furnishes the Moser iteration scheme, including the level-set cutting of critical terms, which Theorem 1.2 adapts."},{"cited_title":"Marino and P","cited_arxiv_id":null,"evidence_quote":"Extends the Moser iteration to boundary terms, giving the trace estimates used in Theorem 1.2."},{"cited_title":"Motreanu and V.V","cited_arxiv_id":null,"evidence_quote":"Introduces the original non-variational problem with convolution and convection that (1.1) generalizes."}],"review_version":1}