The obstacle problem for singular quasi-linear elliptic equations

Annamaria Barbagallo; Umberto Guarnotta

arxiv: 2605.02562 · v1 · submitted 2026-05-04 · 🧮 math.AP

The obstacle problem for singular quasi-linear elliptic equations

Annamaria Barbagallo , Umberto Guarnotta This is my paper

Pith reviewed 2026-05-08 18:02 UTC · model grok-4.3

classification 🧮 math.AP

keywords obstacle problemp-Laplaciansingular elliptic equationsquasi-linear ellipticexistence of solutionsregularity theorydiscontinuous reaction

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The pith

Existence of solutions is proved for p-Laplacian obstacle problems with singular and possibly discontinuous reactions.

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

The paper establishes existence of weak solutions to an obstacle problem for the p-Laplacian equation when the reaction term is singular and may jump on a Lebesgue-negligible set. It further shows that any such solution is locally C^{1,α} regular outside the set where it touches the obstacle. If the obstacle itself is differentiable, the solution gains C^{1,α} regularity on the entire closed domain. A reader cares because these equations model constrained physical systems with strong nonlinear singularities, where proving both existence and gradient regularity determines whether the model remains well-posed near the constraint.

Core claim

Existence of solutions to an obstacle p-Laplacian problem exhibiting a singular, discontinuous reaction is proved. The reaction term may be discontinuous in a Lebesgue-negligible set. Moreover, solutions are shown to be locally C^{1,α} far away from the contact set. Under a differentiability hypothesis on the obstacle, solutions belong to C^{1,α}(¯Ω).

What carries the argument

The variational formulation of the obstacle problem for the singular p-Laplacian operator, which accommodates the reaction discontinuity on a negligible set through suitable approximation and truncation.

If this is right

Weak solutions exist for the constrained problem even when the reaction is singular and discontinuous on negligible sets.
Solutions are locally C^{1,α} outside the contact set with the obstacle.
Global C^{1,α} regularity on the closed domain holds once the obstacle is differentiable.

Where Pith is reading between the lines

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

The same variational approach could be tested on related quasi-linear operators with different growth exponents.
Numerical schemes for constrained singular equations might incorporate similar truncation techniques to recover gradient regularity away from contact.
The result points toward studying the regularity of the free boundary that separates contact and non-contact regions under the given singularity.

Load-bearing premise

The singular reaction satisfies growth and integrability conditions that allow a variational formulation despite possible blow-up, and the obstacle satisfies a differentiability hypothesis to reach global regularity.

What would settle it

A concrete singular reaction that is discontinuous on a set of positive measure for which the corresponding obstacle problem admits no weak solution.

read the original abstract

Existence of solutions to an obstacle $p$-Laplacian problem exhibiting a singular, discontinuous reaction is proved. The reaction term may be discontinuous in a Lebesgue-negligible set. Moreover, solutions are shown to be locally $C^{1,\alpha}$ far away from the contact set. Under a differentiability hypothesis on the obstacle, solutions belong to $C^{1,\alpha}(\overline{\Omega})$.

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

This paper proves existence for the p-Laplacian obstacle problem with a singular reaction discontinuous on a null set, plus local C^{1,α} regularity off the contact set and global under a differentiability assumption on the obstacle.

read the letter

The main point is that this work establishes existence of solutions for the obstacle problem involving the p-Laplacian with a singular reaction term that may be discontinuous on a Lebesgue null set. Solutions are locally C^{1,α} away from the contact set, and globally so if the obstacle satisfies a differentiability condition. What stands out is the handling of the discontinuous singular reaction in the variational inequality setting. Previous results often required continuity or better regularity on the reaction, so this broadens the applicability to more general data. The authors likely use approximation by smooth problems or truncation to pass to the limit while controlling the singularity. The paper does a good job of combining existence via variational methods with regularity theory for quasilinear elliptic equations. This is a clean way to organize the proof without introducing new technical machinery. The soft spots are minor but worth noting. The abstract does not specify the exact growth conditions or how the singularity is quantified, which makes it difficult to judge the full strength of the result without the details. The global regularity part hinges on differentiability of the obstacle, an assumption that may not be easy to check in applications. If the proofs follow standard lines, there is no major innovation in the methods themselves. This is for people working in nonlinear PDEs, particularly those studying obstacle problems and free boundaries for degenerate operators. A specialist in variational inequalities would find it a useful reference for this class of problems. It deserves serious peer review because the claims are precise and the setting is relevant to current research in the area. I would send it to referees to check the technical steps on the discontinuous case and confirm the regularity estimates hold as stated.

Referee Report

0 major / 3 minor

Summary. The manuscript proves existence of weak solutions to an obstacle problem for the p-Laplacian operator with a singular reaction term that is allowed to be discontinuous on a Lebesgue-null set. It further establishes local C^{1,α} regularity of solutions away from the contact set and global C^{1,α} regularity up to the boundary of Ω under an additional differentiability assumption on the obstacle.

Significance. If the variational existence argument and the subsequent regularity analysis hold, the work extends the theory of obstacle problems for quasilinear elliptic equations to singular and highly irregular reaction terms. This is relevant for models in nonlinear diffusion or elasticity where data may fail to be continuous. The local regularity result away from the coincidence set and the global result under a mild hypothesis on the obstacle are standard but useful additions to the literature on free-boundary regularity.

minor comments (3)

The abstract states that the reaction satisfies growth and integrability conditions allowing the variational formulation, but the introduction should explicitly list the precise structural assumptions (e.g., the precise form of the singularity and the Carathéodory-type conditions) before the existence theorem is stated.
Notation for the contact set and the coincidence set should be introduced once in §1 and used consistently; the current abstract uses both phrases without prior definition.
The differentiability hypothesis on the obstacle is invoked only for the global regularity statement; a brief remark on whether this hypothesis is sharp (or can be weakened to Lipschitz) would clarify the scope of the result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment and recommendation of minor revision. The referee's summary correctly reflects the main results: existence of weak solutions for the obstacle problem with singular discontinuous reaction, local C^{1,α} regularity away from the contact set, and global C^{1,α} regularity under an additional assumption on the obstacle.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes existence of solutions to an obstacle problem for the p-Laplacian with a singular reaction (discontinuous only on a null set) via standard variational methods under growth/integrability assumptions that make the formulation well-posed. Regularity results (local C^{1,α} away from the contact set, global under differentiability of the obstacle) follow from established elliptic regularity techniques once existence is obtained. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations that collapse the central claim to its own inputs appear in the derivation chain. The proof is self-contained against external benchmarks in the literature on quasilinear obstacle problems.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard functional-analytic frameworks for quasilinear elliptic equations and obstacle problems; no free parameters or invented entities are indicated in the abstract.

axioms (2)

standard math The p-Laplacian operator and associated Sobolev spaces admit a variational formulation for the obstacle problem.
Standard background in nonlinear elliptic PDE theory invoked for existence.
domain assumption The singular discontinuous reaction satisfies suitable growth and measurability conditions that permit passage to the limit in approximating problems.
Necessary to handle the singularity and discontinuity on negligible sets, though not detailed in the abstract.

pith-pipeline@v0.9.0 · 5351 in / 1328 out tokens · 79454 ms · 2026-05-08T18:02:53.094165+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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(H_f)(ii) there exists gamma in (0,1) with lim sup s^gamma f(s) < +infty; (H_f)(iii) lim inf f(s)/s^{p-1} > lambda_1; (H_f)(iv) lim sup f(s)/s^{p-1} < lambda_1.

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Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

[1]

ANTONINI, Local and globalC 1,β-regularity for uniformly elliptic quasilinear equations of p-Laplace and Orlicz-Laplace type.Preprint, arXiv:2601.07140

C.A. Antonini,Local and globalC 1,β-regularity for uniformly elliptic quasilinear equations of p-Laplace and Orlicz-Laplace type, preprint (arXiv:2601.07140)

work page arXiv
[2]

Baldelli and U

L. Baldelli and U. Guarnotta,Existence and regularity for ap-Laplacian problem inR N with singular, convective, critical reaction, Adv. Nonlinear Anal.13(2024), Paper no. 20240033

work page 2024
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V. Barbu,Nonlinear semigroups and differential equations in Banach spaces, Noordhoff International Pub- lishing, Leiden, 1976

work page 1976
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Barletta, A

G. Barletta, A. Cianchi, and G. Marino,Boundedness of solutions to Dirichlet, Neumann and Robin problems for elliptic equations in Orlicz spaces, Calc. Var. Partial Differ. Equ., 62 (2023), Paper No. 65, 42 pp

work page 2023
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H. Brezis,Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011

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Candito, U

P. Candito, U. Guarnotta and R. Livrea,Existence of two solutions for singularΦ-Laplacian problems, Adv. Nonlinear Stud., 22 (2022), pp. 659—683

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S. Carl, V. K. Le, and D. Motreanu,Nonsmooth variational problems and their inequalities. Comparison principles and applications, Springer Monographs in Mathematics, Springer, New York, 2007

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K.C. Chang,Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), pp. 102—129

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Cianchi and V.G

A. Cianchi and V.G. Maz’ya,Second-order two-sided estimates in nonlinear elliptic problems, Arch. Ration. Mech. Anal., 229 (2018), pp. 569—599

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DiBenedetto,C 1+α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), pp

E. DiBenedetto,C 1+α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), pp. 827–850

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Frehse,On the regularity of the solution of a second order variational inequality, Boll

J. Frehse,On the regularity of the solution of a second order variational inequality, Boll. Unione Mat. Ital., (4) 6 (1972), pp. 312—315

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Gigli and S

N. Gigli and S. Mosconi,The abstract Lewy–Stampacchia inequality and applications, J. Math. Pures Appl., (9) 104 (2015), pp. 258—275

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Gilbarg and N.S

D. Gilbarg and N.S. Trudinger,Elliptic partial differential equations of second order, Classics Math., Springer-Verlag, Berlin, 2001

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Guarnotta, R

U. Guarnotta, R. Livrea and S.A. Marano,Some recent results on singularp-Laplacian equations, Demonstr. Math., 55 (2022), pp. 416-–428

work page 2022
[15]

Guarnotta, R

U. Guarnotta, R. Livrea and S.A. Marano,Some recent results on singularp-Laplacian systems, Discrete Contin. Dyn. Syst. Ser. S, 16 (2023), pp. 1435–1451

work page 2023
[16]

Guarnotta and S.A

U. Guarnotta and S.A. Marano,Strong solutions to singular discontinuousp-Laplacian problems, Commun. Contemp. Math. (2025), https://doi.org/10.1142/S0219199725500671

work page doi:10.1142/s0219199725500671 2025
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Guarnotta and S.A

U. Guarnotta and S.A. Marano,On singularp-Laplacian problems with discontinuous convection terms, preprint (arXiv:2603.13811)

work page arXiv
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Hai,On a class of singularp-Laplacian boundary value problems, J

D.D. Hai,On a class of singularp-Laplacian boundary value problems, J. Math. Anal. Appl., 383 (2011), pp. 619-–626

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Kinderlehrer,The coincidence set of solutions of certain variational inequalities, Arch

D. Kinderlehrer,The coincidence set of solutions of certain variational inequalities, Arch. Ration. Mech. Anal., 40 (1970/71), pp. 231-–250

work page 1970
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Kinderlehrer and G

D. Kinderlehrer and G. Stampacchia,An introduction to variational inequalities and their applications, Pure Appl. Math., 88, Academic Press Inc., New York-London, 1980

work page 1980
[21]

Kyritsi and N.S

S.Th. Kyritsi and N.S. Papageorgiou,An obstacle problem for nonlinear hemivariational inequalities at resonance, J. Math. Anal. Appl., 276 (2002), pp. 292-–313. THE OBSTACLE PROBLEM FOR SINGULAR QUASI-LINEAR ELLIPTIC EQUATIONS 17

work page 2002
[22]

Lieberman,Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal.12 (1988), pp

G.M. Lieberman,Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal.12 (1988), pp. 1203–1219

work page 1988
[23]

Lindqvist,Regularity for the gradient of the solution to a nonlinear obstacle problem with degenerate ellipticity, Nonlinear Anal., 12 (1988), pp

P. Lindqvist,Regularity for the gradient of the solution to a nonlinear obstacle problem with degenerate ellipticity, Nonlinear Anal., 12 (1988), pp. 1245–1255

work page 1988
[24]

Lindqvist,Notes on the p -Laplace equation, Rep

P. Lindqvist,Notes on the p -Laplace equation, Rep. Univ. Jyv¨ askyl¨ a Dep. Math. Stat., 102, University of Jyv¨ askyl¨ a, Jyv¨ askyl¨ a, 2006

work page 2006
[25]

Livrea and S.A

R. Livrea and S.A. Marano,Non-smooth critical point theory, Handbook of nonconvex analysis and appli- cations, 353–407, International Press, Somerville, 2010

work page 2010
[26]

Renato Caccioppoli

D. Motreanu and V. R˘ adulescu,Variational and non-variational methods in nonlinear analysis and boundary value problems, Nonconvex Optim. Appl., 67, Kluwer Academic Publishers, Dordrecht, 2003. (A. Barbagallo)Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Universit `a di Napoli Federico II, Via Cinthia, 80126 Napoli, Italy Email address:...

work page 2003

[1] [1]

ANTONINI, Local and globalC 1,β-regularity for uniformly elliptic quasilinear equations of p-Laplace and Orlicz-Laplace type.Preprint, arXiv:2601.07140

C.A. Antonini,Local and globalC 1,β-regularity for uniformly elliptic quasilinear equations of p-Laplace and Orlicz-Laplace type, preprint (arXiv:2601.07140)

work page arXiv

[2] [2]

Baldelli and U

L. Baldelli and U. Guarnotta,Existence and regularity for ap-Laplacian problem inR N with singular, convective, critical reaction, Adv. Nonlinear Anal.13(2024), Paper no. 20240033

work page 2024

[3] [3]

Barbu,Nonlinear semigroups and differential equations in Banach spaces, Noordhoff International Pub- lishing, Leiden, 1976

V. Barbu,Nonlinear semigroups and differential equations in Banach spaces, Noordhoff International Pub- lishing, Leiden, 1976

work page 1976

[4] [4]

Barletta, A

G. Barletta, A. Cianchi, and G. Marino,Boundedness of solutions to Dirichlet, Neumann and Robin problems for elliptic equations in Orlicz spaces, Calc. Var. Partial Differ. Equ., 62 (2023), Paper No. 65, 42 pp

work page 2023

[5] [5]

Brezis,Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011

H. Brezis,Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011

work page 2011

[6] [6]

Candito, U

P. Candito, U. Guarnotta and R. Livrea,Existence of two solutions for singularΦ-Laplacian problems, Adv. Nonlinear Stud., 22 (2022), pp. 659—683

work page 2022

[7] [7]

S. Carl, V. K. Le, and D. Motreanu,Nonsmooth variational problems and their inequalities. Comparison principles and applications, Springer Monographs in Mathematics, Springer, New York, 2007

work page 2007

[8] [8]

Chang,Variational methods for nondifferentiable functionals and their applications to partial differential equations, J

K.C. Chang,Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), pp. 102—129

work page 1981

[9] [9]

Cianchi and V.G

A. Cianchi and V.G. Maz’ya,Second-order two-sided estimates in nonlinear elliptic problems, Arch. Ration. Mech. Anal., 229 (2018), pp. 569—599

work page 2018

[10] [10]

DiBenedetto,C 1+α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), pp

E. DiBenedetto,C 1+α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), pp. 827–850

work page 1983

[11] [11]

Frehse,On the regularity of the solution of a second order variational inequality, Boll

J. Frehse,On the regularity of the solution of a second order variational inequality, Boll. Unione Mat. Ital., (4) 6 (1972), pp. 312—315

work page 1972

[12] [12]

Gigli and S

N. Gigli and S. Mosconi,The abstract Lewy–Stampacchia inequality and applications, J. Math. Pures Appl., (9) 104 (2015), pp. 258—275

work page 2015

[13] [13]

Gilbarg and N.S

D. Gilbarg and N.S. Trudinger,Elliptic partial differential equations of second order, Classics Math., Springer-Verlag, Berlin, 2001

work page 2001

[14] [14]

Guarnotta, R

U. Guarnotta, R. Livrea and S.A. Marano,Some recent results on singularp-Laplacian equations, Demonstr. Math., 55 (2022), pp. 416-–428

work page 2022

[15] [15]

Guarnotta, R

U. Guarnotta, R. Livrea and S.A. Marano,Some recent results on singularp-Laplacian systems, Discrete Contin. Dyn. Syst. Ser. S, 16 (2023), pp. 1435–1451

work page 2023

[16] [16]

Guarnotta and S.A

U. Guarnotta and S.A. Marano,Strong solutions to singular discontinuousp-Laplacian problems, Commun. Contemp. Math. (2025), https://doi.org/10.1142/S0219199725500671

work page doi:10.1142/s0219199725500671 2025

[17] [17]

Guarnotta and S.A

U. Guarnotta and S.A. Marano,On singularp-Laplacian problems with discontinuous convection terms, preprint (arXiv:2603.13811)

work page arXiv

[18] [18]

Hai,On a class of singularp-Laplacian boundary value problems, J

D.D. Hai,On a class of singularp-Laplacian boundary value problems, J. Math. Anal. Appl., 383 (2011), pp. 619-–626

work page 2011

[19] [19]

Kinderlehrer,The coincidence set of solutions of certain variational inequalities, Arch

D. Kinderlehrer,The coincidence set of solutions of certain variational inequalities, Arch. Ration. Mech. Anal., 40 (1970/71), pp. 231-–250

work page 1970

[20] [20]

Kinderlehrer and G

D. Kinderlehrer and G. Stampacchia,An introduction to variational inequalities and their applications, Pure Appl. Math., 88, Academic Press Inc., New York-London, 1980

work page 1980

[21] [21]

Kyritsi and N.S

S.Th. Kyritsi and N.S. Papageorgiou,An obstacle problem for nonlinear hemivariational inequalities at resonance, J. Math. Anal. Appl., 276 (2002), pp. 292-–313. THE OBSTACLE PROBLEM FOR SINGULAR QUASI-LINEAR ELLIPTIC EQUATIONS 17

work page 2002

[22] [22]

Lieberman,Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal.12 (1988), pp

G.M. Lieberman,Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal.12 (1988), pp. 1203–1219

work page 1988

[23] [23]

Lindqvist,Regularity for the gradient of the solution to a nonlinear obstacle problem with degenerate ellipticity, Nonlinear Anal., 12 (1988), pp

P. Lindqvist,Regularity for the gradient of the solution to a nonlinear obstacle problem with degenerate ellipticity, Nonlinear Anal., 12 (1988), pp. 1245–1255

work page 1988

[24] [24]

Lindqvist,Notes on the p -Laplace equation, Rep

P. Lindqvist,Notes on the p -Laplace equation, Rep. Univ. Jyv¨ askyl¨ a Dep. Math. Stat., 102, University of Jyv¨ askyl¨ a, Jyv¨ askyl¨ a, 2006

work page 2006

[25] [25]

Livrea and S.A

R. Livrea and S.A. Marano,Non-smooth critical point theory, Handbook of nonconvex analysis and appli- cations, 353–407, International Press, Somerville, 2010

work page 2010

[26] [26]

Renato Caccioppoli

D. Motreanu and V. R˘ adulescu,Variational and non-variational methods in nonlinear analysis and boundary value problems, Nonconvex Optim. Appl., 67, Kluwer Academic Publishers, Dordrecht, 2003. (A. Barbagallo)Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Universit `a di Napoli Federico II, Via Cinthia, 80126 Napoli, Italy Email address:...

work page 2003