Existence and $L^{\infty}$-estimates for elliptic equations involving convolution

Dumitru Motreanu; Greta Marino

REVIEW 3 major objections 5 minor 7 references

Existence and $L^{\infty}$-estimates for elliptic equations involving convolution

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict A solid existence proof for a new nonlocal boundary problem, with an L∞ theorem that has a repairable sign-reduction gap; worth refereeing. read the letter →

arxiv 1908.01390 v3 pith:IUC2TV3N submitted 2019-08-04 math.AP

Greta Marino , Dumitru Motreanu This is my paper · ORCID

classification math.AP MSC 35J6035B4535J2544A35

keywords Moseriterationboundednessofsolutionsellipticoperatorsdivergencetypecriticalgrowthontheboundaryconvolutionpseudomonotonenonlocalvalueproblemaprioriestimates

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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 $\gamma u \in L^{\infty}(\partial\Omega)$.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The 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.

The 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.

The 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.

The 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.

Minor 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.

This 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.

Referee Report

3 major / 5 minor

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.

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 (3)

[Section 4, first paragraph of the proof of Theorem 1.2] 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 3, Eq. (3.15)] 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 4, after Eq. (4.17)] 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.

minor comments (5)

[Section 3, Eq. (3.6)] 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 4, Eq. (4.16)] 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 4, Eq. (4.17)] 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.
[Abstract and Introduction] The abstract states that ∂Ω is C^1, while the theorems assume Lipschitz continuity; this should be harmonized.
[Section 3, estimate following Eq. (3.11)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: boundedness is derived from the equation via a Moser iteration, not assumed or fitted.

full rationale

The derivation chain is not circular. Theorem 1.1 establishes existence by verifying boundedness, pseudomonotonicity and coercivity of the operator T directly from hypotheses (A), with all estimates (3.3)-(3.9) coming from the stated growth conditions, Sobolev embedding, convolution Young estimates, and the fixed extension operator; the surjectivity theorem [3] is standard external input. Theorem 1.2 derives L^r and then L^∞ bounds from the weak formulation (1.4) through Moser iteration; the estimates (4.1)-(4.17) are generated in the paper, and the dependence of constants on the solution (4.8) is an a priori bound, not a fitted parameter disguised as a prediction. The citations to [5,6] supply the iteration template ('proceeding as in [5, Theorem 3.1]'), which is prior technical machinery, not the target conclusion, so the self-citations are not load-bearing in a circular sense. The sign-reduction assertion ('we can suppose u ≥ 0, otherwise we work with u+ and u−') is a potential correctness gap for sign-changing solutions, but it is a missing justification, not a reduction of the conclusion to an input; it does not constitute circularity under the stated criteria.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new entities and fits no parameters. It relies on standard Sobolev, trace, convolution and pseudomonotone operator theory, plus the specific Moser iteration scheme from the authors' earlier work. The only ad hoc element is the unproven reduction to nonnegative solutions.

assumptions (6)

standard math Sobolev embedding and trace theorems on bounded Lipschitz domains
Used throughout Sections 3 and 4 for L^r and boundary estimates.
standard math Existence of a (1,p)-extension operator E with uniform W^{1,p} bounds
Section 1, based on Adams [1] and Brezis [2]; required for the convolution ρ*E(u) to be well-defined.
standard math Surjectivity theorem for bounded, coercive, pseudomonotone operators
Theorem 2.3, cited from [3]; the engine of the existence proof.
standard math Convolution estimates ‖ρ*f‖_{L^r} ≤ ‖ρ‖_{L^1}‖f‖_{L^r}
Section 2, equation (2.3); controls the nonlocal term.
domain assumption Modified Moser iteration from [5,6] with critical Sobolev growth on the boundary
The boundedness proof in Section 4 invokes 'proceeding as in [5, Theorem 3.1]'; the technical smallness condition (1.5) ensures ilde r < p*/p.
ad hoc to paper Sign reduction to nonnegative solutions
Section 4 states 'we can suppose u ≥ 0, otherwise we work with u+ and u-'; this is not justified for the non-monotone nonlocal equation.

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Cite this review

Pith. "Pith review of Existence and $L^{\infty}$-estimates for elliptic equations involving convolution." pith.science (2026). https://pith.science/paper/IUC2TV3N

@misc{pith2026190801390,
  author       = {Pith},
  title        = {Pith review of: Existence and $L^\infty$-estimates for elliptic equations involving convolution},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IUC2TV3N}},
  note         = {Machine review of arXiv:1908.01390}
}

abstract

In this paper, with a fixed $p\in (1,+\infty)$ and a bounded domain $\Omega \subset \mathbb{R}^N$ whose boundary $\partial\Omega$ fulfills the $C^1$ regularity, we study a boundary value problem involving a nonlocal operator assigning to $u$ the convolution $\rho \ast E(u)$ of $\rho$ with $E(u)$, where $\rho$ is an integrable function on $\mathbb{R}^N$ and $E$ is an extension operator related to $\Omega$. Under verifiable conditions, we prove the existence of a (weak) solution to our problem by using the surjectivity theorem for pseudomonotone operators. Moreover, through a modified version of Moser iteration up to the boundary, we show that (any) weak solution to our problem is bounded.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

[5]

Marino and P

G. Marino and P. Winkert, Moser iteration applied to elliptic equations with critica l growth on the boundary , Nonlinear Anal. 180 (2019), 154–169. 14 G. MARINO AND D. MOTREANU

work page 2019
[1]

Sobolev spaces

R.A. Adams, “Sobolev spaces”, Academic Press, New York- London, 1975

work page 1975
[2]

Functional analysis, Sobolev spaces and par tial diﬀerential equations

H. Brezis, “Functional analysis, Sobolev spaces and par tial diﬀerential equations”, Universitext, Springer, New York, 2011

work page 2011
[3]

Nonsmooth variationa l problems and their inequalities. Comparison principles and applications

S. Carl, V.K. Le, and D. Motreanu, “Nonsmooth variationa l problems and their inequalities. Comparison principles and applications”, Springer Monographs in Mathematics, Sp ringer, New York, 2007

work page 2007
[4]

Marano and P

S.A. Marano and P. Winkert, On a quasilinear elliptic problem with convection term and n onlinear boundary condition, Nonlinear Anal. 187 (2019), 159–169

work page 2019
[6]

Marino and P

G. Marino and P. Winkert, Global a priori bounds for weak solutions of quasilinear ell iptic systems with nonlinear boundary condition, J. Math. Anal. Appl. 482 (2020), no. 2, 123555, 19pp

work page 2020
[7]

Motreanu and V.V

D. Motreanu and V.V. Motreanu, Non-variational elliptic equations involving (p, q )-Laplacian, convection and convolution, Pure Appl. Funct. Anal. (2019), to appear. (G. Marino) Technische Universit ¨at Chemnitz, F akult¨at f ¨ur Mathematik, Reichenhainer Straße 41, 09126 Chemnitz, Germany E-mail address : greta.marino@mathematik.tu-chemnitz.de (D. Motrea...

work page 2019

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [5]

Marino and P

G. Marino and P. Winkert, Moser iteration applied to elliptic equations with critica l growth on the boundary , Nonlinear Anal. 180 (2019), 154–169. 14 G. MARINO AND D. MOTREANU

work page 2019

[2] [1]

Sobolev spaces

R.A. Adams, “Sobolev spaces”, Academic Press, New York- London, 1975

work page 1975

[3] [2]

Functional analysis, Sobolev spaces and par tial diﬀerential equations

H. Brezis, “Functional analysis, Sobolev spaces and par tial diﬀerential equations”, Universitext, Springer, New York, 2011

work page 2011

[4] [3]

Nonsmooth variationa l problems and their inequalities. Comparison principles and applications

S. Carl, V.K. Le, and D. Motreanu, “Nonsmooth variationa l problems and their inequalities. Comparison principles and applications”, Springer Monographs in Mathematics, Sp ringer, New York, 2007

work page 2007

[5] [4]

Marano and P

S.A. Marano and P. Winkert, On a quasilinear elliptic problem with convection term and n onlinear boundary condition, Nonlinear Anal. 187 (2019), 159–169

work page 2019

[6] [6]

Marino and P

G. Marino and P. Winkert, Global a priori bounds for weak solutions of quasilinear ell iptic systems with nonlinear boundary condition, J. Math. Anal. Appl. 482 (2020), no. 2, 123555, 19pp

work page 2020

[7] [7]

Motreanu and V.V

D. Motreanu and V.V. Motreanu, Non-variational elliptic equations involving (p, q )-Laplacian, convection and convolution, Pure Appl. Funct. Anal. (2019), to appear. (G. Marino) Technische Universit ¨at Chemnitz, F akult¨at f ¨ur Mathematik, Reichenhainer Straße 41, 09126 Chemnitz, Germany E-mail address : greta.marino@mathematik.tu-chemnitz.de (D. Motrea...

work page 2019