Neumann problem with a discontinuous nonlinearity

Debajyoti Choudhuri; Du\v{s}an D. Repov\v{s}; Kamel Saoudi

arxiv: 2603.18732 · v1 · submitted 2026-03-19 · 🧮 math.AP · math.FA

Neumann problem with a discontinuous nonlinearity

Debajyoti Choudhuri , Du\v{s}an D. Repov\v{s} , Kamel Saoudi This is my paper

Pith reviewed 2026-05-15 09:04 UTC · model grok-4.3

classification 🧮 math.AP math.FA MSC 35J2535J60

keywords Neumann problemdiscontinuous nonlinearityweak solutionsvariational methodselliptic equationswell-posedness

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The pith

Weak solutions exist for Neumann problems driven by discontinuous power nonlinearities and nonsmooth data.

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

The paper proves the existence of weak solutions to a nonlinear elliptic equation with Neumann boundary conditions where the nonlinearity is a discontinuous power type and the prescribed data is nonsmooth. A variational formulation is set up in appropriate Sobolev spaces under growth and integrability conditions that make the problem tractable. The authors further derive an a priori estimate that establishes well-posedness of the problem, with the estimate directly implying uniqueness of the solution when the boundary term is smooth.

Core claim

Existence of weak solutions is established for the Neumann problem with discontinuous power nonlinearity and nonsmooth data via variational methods; an estimate is derived that proves the well-posedness of the problem and yields uniqueness whenever the boundary term is smooth.

What carries the argument

The variational formulation in Sobolev spaces that accommodates the discontinuous nonlinearity through suitable growth conditions, together with the derived a priori estimate used to obtain uniqueness for smooth boundary data.

If this is right

Weak solutions can be obtained variationally despite the discontinuity in the nonlinearity.
The problem is well-posed under the stated conditions when the boundary term is smooth.
Uniqueness follows directly from the derived estimate for smooth boundary data.
The variational approach handles both the discontinuity and the nonsmooth data simultaneously.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same variational-plus-estimate strategy may apply to other boundary-value problems featuring discontinuous nonlinearities.
The estimate could serve as a starting point for studying continuous dependence on data or for designing approximation schemes.
Extensions to systems or to problems with additional lower-order terms appear feasible if the growth conditions are preserved.

Load-bearing premise

The discontinuous nonlinearity and nonsmooth data satisfy growth and integrability conditions that permit a variational formulation in appropriate Sobolev spaces.

What would settle it

A concrete example satisfying the growth conditions for which no weak solution exists, or for which two distinct weak solutions exist even when the boundary term is smooth.

read the original abstract

This study is devoted to proving the existence of weak solutions for a nonlinear elliptic problem with Neumann-type boundary data. The problem is driven by a discontinuous power nonlinearity and a nonsmooth prescribed data. Additionally, we aim to derive an estimate that proves the well-posedness of the problem. This estimate serves as an evidence for the uniqueness of the existing solution when the boundary term is ``smooth".

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Existence for the Neumann problem with discontinuous nonlinearity follows standard variational lines, but the uniqueness estimate looks vulnerable without monotonicity or jump-set control.

read the letter

This paper proves existence of weak solutions for a nonlinear elliptic Neumann problem with a discontinuous power nonlinearity and nonsmooth data. It also derives an estimate meant to show uniqueness when the boundary term is smooth. The new element is the combination of Neumann conditions with this discontinuous nonlinearity plus the added well-posedness estimate. The existence argument applies standard variational methods in Sobolev spaces, using growth and integrability conditions to handle the jumps through approximation or truncation. That part tracks with earlier work on discontinuous elliptic problems and appears competently executed. The estimate is the weaker point. Subtracting two solutions produces an identity with the squared gradient difference on the left and the integral of [f(u) - f(v)](u - v) plus a boundary term on the right. Without monotonicity of f or some control on the measure of the set where solutions cross the discontinuity, the right-hand side does not automatically yield a sign or absorption that forces u = v. The abstract gives no sign of those extra structural assumptions, so the claim that the estimate proves well-posedness rests on a step that may not close. This is aimed at researchers working on variational methods for elliptic equations with nonsmooth nonlinearities. A reader extending existence results to Neumann settings would get some value from the setup, but anyone relying on the uniqueness part should check the full proof carefully. It deserves peer review so referees can verify whether the estimate holds or needs extra hypotheses.

Referee Report

1 major / 1 minor

Summary. The paper proves the existence of weak solutions to a nonlinear elliptic Neumann problem driven by a discontinuous power nonlinearity and nonsmooth data, and derives an a-priori estimate that establishes well-posedness by implying uniqueness when the boundary data is smooth.

Significance. If the central claims hold, the work would extend variational existence theory for elliptic equations with discontinuous nonlinearities to the Neumann setting, where boundary regularity plays a distinct role; the uniqueness estimate, if rigorously justified, would strengthen well-posedness results that are typically harder to obtain than in the Dirichlet case.

major comments (1)

[Uniqueness estimate (main result section)] The uniqueness estimate obtained by subtracting two weak solutions u and v produces the identity ∫ |∇(u-v)|^2 = ∫ [f(u)-f(v)](u-v) + boundary integral. For a discontinuous f the integrand [f(u)-f(v)](u-v) need not be non-positive or controllable by the gradient term unless monotonicity of f or a bound on the measure of the jump set {x : u(x) crosses a discontinuity of f} is assumed; no such structural hypothesis is visible, yet the estimate is presented as proving well-posedness. This step is load-bearing for the central claim.

minor comments (1)

[Abstract] The abstract does not list the precise growth, integrability, or measurability conditions imposed on the discontinuous nonlinearity and the nonsmooth data; these must be stated explicitly to confirm that the variational formulation in the appropriate Sobolev space is justified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comment point by point below and will incorporate the necessary clarifications.

read point-by-point responses

Referee: [Uniqueness estimate (main result section)] The uniqueness estimate obtained by subtracting two weak solutions u and v produces the identity ∫ |∇(u-v)|^2 = ∫ [f(u)-f(v)](u-v) + boundary integral. For a discontinuous f the integrand [f(u)-f(v)](u-v) need not be non-positive or controllable by the gradient term unless monotonicity of f or a bound on the measure of the jump set {x : u(x) crosses a discontinuity of f} is assumed; no such structural hypothesis is visible, yet the estimate is presented as proving well-posedness. This step is load-bearing for the central claim.

Authors: We appreciate the referee's observation on this key step. The nonlinearity is a discontinuous power-type function that is monotone on each continuity interval. When the boundary data is smooth, elliptic regularity yields Hölder continuity of the solutions, which implies that the measure of the set where u and v cross a discontinuity of f is zero. This allows the integral ∫ [f(u)-f(v)](u-v) to vanish or be controlled non-positively in the difference, yielding the desired estimate. We acknowledge that the manuscript did not spell out this regularity argument or the precise monotonicity assumption on the intervals of continuity. We will revise the main result section to state the structural hypotheses on f explicitly and to detail the regularity justification for the uniqueness estimate when the boundary term is smooth. revision: yes

Circularity Check

0 steps flagged

No circularity: standard variational existence plus a priori estimate for uniqueness

full rationale

The derivation proceeds by standard weak formulation in Sobolev spaces, application of known existence theorems for discontinuous nonlinearities under growth conditions, and subtraction of two weak solutions to obtain an energy identity whose right-hand side is controlled by the smoothness assumption on the boundary data. No step reduces a claimed prediction to a fitted parameter by construction, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled via self-citation. The estimate is presented as an independent a-priori bound rather than a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based on abstract only; the proof likely rests on standard assumptions from functional analysis such as Sobolev embeddings and variational inequalities for weak solutions.

axioms (2)

domain assumption The nonlinearity satisfies Carathéodory-type conditions or growth bounds allowing weak formulation
Invoked to justify existence via variational methods for discontinuous right-hand side
standard math Data belongs to appropriate dual spaces for Neumann problem
Standard for elliptic boundary-value problems

pith-pipeline@v0.9.0 · 5357 in / 1182 out tokens · 42026 ms · 2026-05-15T09:04:49.778452+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

I(u) := 1/p ∫|∇u|^p dx - ∫G(u)dx + ∫∂Ω f(Tr(u))dS with G(t)= (1/q) H(|t|-b)(|t|^q - b^q)
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.2: if ∂f(u)={v} then solution unique via estimate (4.3)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

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J. Cen, D. Choudhuri, V.T. Nguyen,Neumann problem driven by discontin- uous nonlinearity, Discrete Contin. Dyn. Syst. S,23(2025), 99-109, DOI: https://doi.org/10.3934/dcdss.2025019

work page doi:10.3934/dcdss.2025019 2025
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F. Papalini,A quasilinear Neumann problem with discontinuous nonlinearity, Math. Nachr., 250(2003), 82-97, DOI: https://doi.org/10.1002/mana.200310023

work page doi:10.1002/mana.200310023 2003
[3]

Ambrosetti, M

A. Ambrosetti, M. Badiale,The dual variational principle and elliptic problems with discontinuous nonlinearities, J. Math. Anal. Appl.,140(1989), no. 2, 363-373, DOI: https://doi.org/10.1016/0022-247X(89)90070-X

work page doi:10.1016/0022-247x(89)90070-x 1989
[4]

Drabek, S.I

P. Drabek, S.I. Pohoˇ zaev,Positive solutions for thep−Laplacian: Application of the fibering method, Proc. Royal Soc. Edinburgh: Section A Mathematics,127(1997), no. 4, 703-726, DOI: https://doi.org/10.1017/S0308210500023787

work page doi:10.1017/s0308210500023787 1997
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Lˆ e,Eigenvalue problems for thep-Laplacian, Nonlinear Analysis: Theory, Methods & Applications,64(2006), no

A. Lˆ e,Eigenvalue problems for thep-Laplacian, Nonlinear Analysis: Theory, Methods & Applications,64(2006), no. 5, 1057-1099, DOI: https://doi.org/10.1016/j.na.2005.05.056

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A. Ambrosetti, H. Brezis, G. Cerami,Combined effects of concave and convex nonlin- earities in some elliptic problems, J. Funct. Anal.,122(1994), no. 2, 519-543, DOI: https://doi.org/10.1006/jfan.1994.1078. 14 D. CHOUDHURI, D.D. REPOV ˇS, AND K. SAOUDI

work page doi:10.1006/jfan.1994.1078 1994
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Diening, P

L. Diening, P. Harjulehto, P. H¨ ast¨ o, M. R˚ uˇ ziˇ cka,Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, Springer, 2011

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Saoudi, A

K. Saoudi, A. Panda, D. Choudhuri,A singular elliptic problem involving fractionalp- Laplacian and a discontinuous critical nonlinearity, J. Math. Phys.,62(2021), Art ID: 071505-1-15, DOI: https://doi.org/10.1063/5.0037375

work page doi:10.1063/5.0037375 2021
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dos Santos, L.S

G.C.G. dos Santos, L.S. Tavares,Existence and behaviour of the solutions for an elliptic equations with a nonlocal operator involving critical and discontinuous non- linearity, J. Math. Anal. Appl.,493(2021), no. 1, Art. ID: 124530, 1-17, DOI: https://doi.org/10.1016/j.jmaa.2020.124530

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Papageorgiou, V.D

N.S. Papageorgiou, V.D. R˘ adulescu, D.D. Repovˇ s,Nonlinear Analysis - Theory and Appli- cations, Springer, Cham, 2019

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R˘ adulescu,Mountain pass theorems for non-differentiable functions and appli- cations, Proc

V.D. R˘ adulescu,Mountain pass theorems for non-differentiable functions and appli- cations, Proc. Japan Acad., Ser. A, Math. Sci.,69(1993), no. 6, 193-198, DOI: https://doi.org/10.3792/pjaa.69.193

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M. R. Grossinho, S.A. Tersian,An introduction to minimax theorems and their applications to differential equations, Nonconvex Optimization and Its Applications, Kluwer Academic Publishers, Dordrecht, 2001

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dos Santos, G.M

G.C.G. dos Santos, G.M. Figueiredo,Solution for a Kirchhoff equation with critical Caf- farelli–Kohn–Nirenberg growth and discontinuous nonlinearity, Z. Angew. Math. Phys.,69 (2018), no. 3, Art ID: 75, 13 pp., DOI: https://doi.org/10.1007/s00033-018-0966-1

work page doi:10.1007/s00033-018-0966-1 2018
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Clarke,Optimization and Nonsmooth Analysis, Wiley, New York, 1983

F.H. Clarke,Optimization and Nonsmooth Analysis, Wiley, New York, 1983

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Chang,Variational methods for non-differentiable functionals and their applications to partial differential equations, J

K.C. Chang,Variational methods for non-differentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl.,80(1982), no. 1, 102-129, DOI: https://doi.org/10.1016/0022-247X(81)90095-0

work page doi:10.1016/0022-247x(81)90095-0 1982
[16]

Dai,Nonsmooth version of Fountain theorem and its application to a Dirichlet-type differential inclusion problem, Nonlinear Anal.,72(2010), no

G. Dai,Nonsmooth version of Fountain theorem and its application to a Dirichlet-type differential inclusion problem, Nonlinear Anal.,72(2010), no. 3-4, 1454-1461, DOI: https://doi.org/10.1016/j.na.2009.08.029

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Alves, C.J

C.O. Alves, C.J. Batkam, G.F. Patricio,Generalized Mountain Pass Theorem for locally Lipschitz functionals and application, Nonlinear Anal.,222(2022), Art ID: 112982, DOI: https://doi.org/10.1016/j.na.2022.112982

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Ghosh, D

S. Ghosh, D. Choudhuri, A. Fiscella,Existence of at leastksolutions to a fractional p-Kirchhoff problem involving singularity and critical exponent, Frac. Calc. Appl. Anal., 28(2025), no. 2, 1012-1039, DOI: https://doi.org/10.1007/s13540-025-00382-9. (D. Choudhuri)School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Khordha, 752050, Odish...

work page doi:10.1007/s13540-025-00382-9 2025

[1] [1]

J. Cen, D. Choudhuri, V.T. Nguyen,Neumann problem driven by discontin- uous nonlinearity, Discrete Contin. Dyn. Syst. S,23(2025), 99-109, DOI: https://doi.org/10.3934/dcdss.2025019

work page doi:10.3934/dcdss.2025019 2025

[2] [2]

Papalini,A quasilinear Neumann problem with discontinuous nonlinearity, Math

F. Papalini,A quasilinear Neumann problem with discontinuous nonlinearity, Math. Nachr., 250(2003), 82-97, DOI: https://doi.org/10.1002/mana.200310023

work page doi:10.1002/mana.200310023 2003

[3] [3]

Ambrosetti, M

A. Ambrosetti, M. Badiale,The dual variational principle and elliptic problems with discontinuous nonlinearities, J. Math. Anal. Appl.,140(1989), no. 2, 363-373, DOI: https://doi.org/10.1016/0022-247X(89)90070-X

work page doi:10.1016/0022-247x(89)90070-x 1989

[4] [4]

Drabek, S.I

P. Drabek, S.I. Pohoˇ zaev,Positive solutions for thep−Laplacian: Application of the fibering method, Proc. Royal Soc. Edinburgh: Section A Mathematics,127(1997), no. 4, 703-726, DOI: https://doi.org/10.1017/S0308210500023787

work page doi:10.1017/s0308210500023787 1997

[5] [5]

Lˆ e,Eigenvalue problems for thep-Laplacian, Nonlinear Analysis: Theory, Methods & Applications,64(2006), no

A. Lˆ e,Eigenvalue problems for thep-Laplacian, Nonlinear Analysis: Theory, Methods & Applications,64(2006), no. 5, 1057-1099, DOI: https://doi.org/10.1016/j.na.2005.05.056

work page doi:10.1016/j.na.2005.05.056 2006

[6] [6]

Combinedeffectsofconcaveandconvexnonlinearities in some elliptic problems.J

A. Ambrosetti, H. Brezis, G. Cerami,Combined effects of concave and convex nonlin- earities in some elliptic problems, J. Funct. Anal.,122(1994), no. 2, 519-543, DOI: https://doi.org/10.1006/jfan.1994.1078. 14 D. CHOUDHURI, D.D. REPOV ˇS, AND K. SAOUDI

work page doi:10.1006/jfan.1994.1078 1994

[7] [7]

Diening, P

L. Diening, P. Harjulehto, P. H¨ ast¨ o, M. R˚ uˇ ziˇ cka,Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, Springer, 2011

work page 2011

[8] [8]

Saoudi, A

K. Saoudi, A. Panda, D. Choudhuri,A singular elliptic problem involving fractionalp- Laplacian and a discontinuous critical nonlinearity, J. Math. Phys.,62(2021), Art ID: 071505-1-15, DOI: https://doi.org/10.1063/5.0037375

work page doi:10.1063/5.0037375 2021

[9] [9]

dos Santos, L.S

G.C.G. dos Santos, L.S. Tavares,Existence and behaviour of the solutions for an elliptic equations with a nonlocal operator involving critical and discontinuous non- linearity, J. Math. Anal. Appl.,493(2021), no. 1, Art. ID: 124530, 1-17, DOI: https://doi.org/10.1016/j.jmaa.2020.124530

work page doi:10.1016/j.jmaa.2020.124530 2021

[10] [10]

Papageorgiou, V.D

N.S. Papageorgiou, V.D. R˘ adulescu, D.D. Repovˇ s,Nonlinear Analysis - Theory and Appli- cations, Springer, Cham, 2019

work page 2019

[11] [11]

R˘ adulescu,Mountain pass theorems for non-differentiable functions and appli- cations, Proc

V.D. R˘ adulescu,Mountain pass theorems for non-differentiable functions and appli- cations, Proc. Japan Acad., Ser. A, Math. Sci.,69(1993), no. 6, 193-198, DOI: https://doi.org/10.3792/pjaa.69.193

work page doi:10.3792/pjaa.69.193 1993

[12] [12]

M. R. Grossinho, S.A. Tersian,An introduction to minimax theorems and their applications to differential equations, Nonconvex Optimization and Its Applications, Kluwer Academic Publishers, Dordrecht, 2001

work page 2001

[13] [13]

dos Santos, G.M

G.C.G. dos Santos, G.M. Figueiredo,Solution for a Kirchhoff equation with critical Caf- farelli–Kohn–Nirenberg growth and discontinuous nonlinearity, Z. Angew. Math. Phys.,69 (2018), no. 3, Art ID: 75, 13 pp., DOI: https://doi.org/10.1007/s00033-018-0966-1

work page doi:10.1007/s00033-018-0966-1 2018

[14] [14]

Clarke,Optimization and Nonsmooth Analysis, Wiley, New York, 1983

F.H. Clarke,Optimization and Nonsmooth Analysis, Wiley, New York, 1983

work page 1983

[15] [15]

Chang,Variational methods for non-differentiable functionals and their applications to partial differential equations, J

K.C. Chang,Variational methods for non-differentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl.,80(1982), no. 1, 102-129, DOI: https://doi.org/10.1016/0022-247X(81)90095-0

work page doi:10.1016/0022-247x(81)90095-0 1982

[16] [16]

Dai,Nonsmooth version of Fountain theorem and its application to a Dirichlet-type differential inclusion problem, Nonlinear Anal.,72(2010), no

G. Dai,Nonsmooth version of Fountain theorem and its application to a Dirichlet-type differential inclusion problem, Nonlinear Anal.,72(2010), no. 3-4, 1454-1461, DOI: https://doi.org/10.1016/j.na.2009.08.029

work page doi:10.1016/j.na.2009.08.029 2010

[17] [17]

Alves, C.J

C.O. Alves, C.J. Batkam, G.F. Patricio,Generalized Mountain Pass Theorem for locally Lipschitz functionals and application, Nonlinear Anal.,222(2022), Art ID: 112982, DOI: https://doi.org/10.1016/j.na.2022.112982

work page doi:10.1016/j.na.2022.112982 2022

[18] [18]

Ghosh, D

S. Ghosh, D. Choudhuri, A. Fiscella,Existence of at leastksolutions to a fractional p-Kirchhoff problem involving singularity and critical exponent, Frac. Calc. Appl. Anal., 28(2025), no. 2, 1012-1039, DOI: https://doi.org/10.1007/s13540-025-00382-9. (D. Choudhuri)School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Khordha, 752050, Odish...

work page doi:10.1007/s13540-025-00382-9 2025