pith. sign in

arxiv: 2604.03378 · v1 · submitted 2026-04-03 · 🧮 math.DG

The role of the mean curvature in nonlinear p-Laplacian problems with critical exponent

Hichem Chtioui , Hichem Hajaiej , Lovelesh Sharma This is my paper

Pith reviewed 2026-05-13 18:14 UTC · model grok-4.3

classification 🧮 math.DG
keywords p-Laplaciancritical exponentmean curvatureleast energy solutionsmixed boundary conditionsquasilinear elliptic problemsexistence results
0
0 comments X

The pith

Mean curvature of the boundary controls existence of least energy solutions for p-Laplacian critical problems differently depending on whether p is above or below 2.

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

The paper establishes the existence of least energy solutions to critical nonlinear p-Laplacian problems on bounded domains with mixed boundary conditions. It reveals that standard results for the linear Laplacian do not extend to the quasilinear p-Laplacian case, and vice versa. Separate analysis is required for p less than 2 and p greater than 2. When p exceeds 2, the mean curvature of the boundary dominates over the potential in determining solutions, but the reverse is true when p is less than 2. A reader would care because these differences show that quasilinear operators require fundamentally different variational treatments than their linear counterparts.

Core claim

We prove the existence of least energy solutions for critical nonlinear problems involving the p-Laplacian operator on bounded domains with mixed boundary conditions. Our work shows a significant difference between the semi-linear case p = 2 and the quasilinear case for the existence results. Moreover, neither the results for the Laplacian can be extended to the p-Laplacian, nor the method for the p-Laplacian can apply to the Laplacian setting. Additionally, the cases p < 2 and p > 2 present different challenges and need to be studied separately. More precisely, when p > 2, the effect of the geometry of the boundary conditions dominates that one of the potential, whereas for p < 2 the相反行为成立。

What carries the argument

The mean curvature term arising in the mixed boundary conditions for the p-Laplacian energy functional, which governs whether the functional satisfies the Palais-Smale condition or mountain-pass geometry.

If this is right

  • Existence of least energy solutions holds when the mean curvature satisfies suitable inequalities for p greater than 2.
  • For p less than 2, conditions on the potential alone determine existence.
  • Methods developed for the Laplacian cannot be used for the p-Laplacian and require separate adaptation for each range of p.
  • Mixed boundary conditions introduce geometric effects that interact differently with the critical exponent than pure Dirichlet conditions.

Where Pith is reading between the lines

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

  • Numerical schemes for these equations may need to switch between curvature-focused and potential-focused discretizations depending on the value of p.
  • The same curvature-dominance switch could appear in related quasilinear problems on manifolds with boundary.
  • Explicit computations on the unit ball with constant potential would give a sharp test of the transition point at p equals 2.
  • The distinction suggests studying how boundary geometry affects bubbling or concentration phenomena in the critical case.

Load-bearing premise

The domain is bounded and smooth enough for mean curvature to be defined, and the potential and boundary data satisfy technical conditions allowing the energy functional to meet the Palais-Smale condition.

What would settle it

A concrete counterexample on a smooth bounded domain with positive mean curvature and suitable potential where no least energy solution exists for some p greater than 2 would falsify the dominance claim.

read the original abstract

We deal with critical nonlinear problems involving the p-Laplacian operator on bounded domains with mixed boundary conditions. We prove the existence of least energy solutions. Our work shows a significant difference between the semi-linear case p = 2 and the quasilinear case for the existence results. Moreover, neither the results for the Laplacian can be extended to the p-Laplacian, nor the method for the p-Laplacian can apply to the Laplacian setting. Additionally, the cases (p < 2 and p > 2) present different challenges and need to be studied separately. More precisely, when p > 2, the effect of the geometry of the boundary conditions dominates that one of the potential, whereas for p < 2 the opposite behavior holds true.

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.

Referee Report

2 major / 1 minor

Summary. The manuscript studies critical exponent problems for the p-Laplacian with mixed boundary conditions on bounded smooth domains. It claims existence of least-energy solutions via variational methods (mountain-pass), establishes a sharp dichotomy between the semilinear case p=2 and the quasilinear case, and asserts that boundary geometry (mean curvature) dominates the potential term when p>2 while the potential dominates when p<2, with the two regimes requiring separate analysis.

Significance. If the proofs are complete and the necessary sign conditions are made explicit, the work would clarify how mean curvature enters the energy expansion for cut-off bubbles in the quasilinear critical setting and why standard Laplacian techniques fail to extend, providing a useful distinction between p<2 and p>2 regimes.

major comments (2)
  1. [Hypotheses and main theorem] Hypotheses / Main existence theorem: the mountain-pass level must lie strictly below the critical threshold (1/n)S^{n/p} for the Palais-Smale condition to hold at the critical exponent. The test-function construction (cut-off bubbles supported near ∂Ω) produces an energy expansion containing a term proportional to H·ε^{(p-1)/p} (or analogous power); the sign of this term determines whether the strict inequality is achieved. No explicit sign assumption on the mean curvature H (nor on its relative size versus the potential V) is stated, yet the claim that “geometry dominates potential for p>2” rests on this unstated condition.
  2. [Test-function construction and energy estimates] §3 (test-function construction) and energy estimates: the precise functional setting (space, trace operator, definition of I(u)), the form of the boundary term (1/p)∫_Γ |u|^p dσ, and the exact asymptotic expansion of I(t u_ε) that isolates the mean-curvature contribution are not supplied in sufficient detail to verify that the mountain-pass geometry is attained for general smooth domains.
minor comments (1)
  1. [Introduction / Hypotheses] List all technical assumptions on the potential V, the boundary data, and the regularity of ∂Ω in a single numbered hypothesis block for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the major points below and will revise the manuscript to incorporate the suggested clarifications.

read point-by-point responses

  1. Referee: [Hypotheses and main theorem] Hypotheses / Main existence theorem: the mountain-pass level must lie strictly below the critical threshold (1/n)S^{n/p} for the Palais-Smale condition to hold at the critical exponent. The test-function construction (cut-off bubbles supported near ∂Ω) produces an energy expansion containing a term proportional to H·ε^{(p-1)/p} (or analogous power); the sign of this term determines whether the strict inequality is achieved. No explicit sign assumption on the mean curvature H (nor on its relative size versus the potential V) is stated, yet the claim that “geometry dominates potential for p>2” rests on this unstated condition.

    Authors: We agree that an explicit sign condition on the mean curvature is necessary to guarantee the mountain-pass level lies below the critical threshold. In the p>2 regime the geometry term in the expansion of I(t u_ε) has the form c H ε^α with α>0; the sign of H must be chosen so that this term is negative and dominates the potential contribution. We will add this hypothesis (H>0 at the concentration point, together with a quantitative comparison |H| ≫ |V|) to the statement of the main theorem and to the hypotheses section. The revised version will also contain a short remark explaining why the opposite sign would reverse the domination and require a different test-function construction. revision: yes

  2. Referee: [Test-function construction and energy estimates] §3 (test-function construction) and energy estimates: the precise functional setting (space, trace operator, definition of I(u)), the form of the boundary term (1/p)∫_Γ |u|^p dσ, and the exact asymptotic expansion of I(t u_ε) that isolates the mean-curvature contribution are not supplied in sufficient detail to verify that the mountain-pass geometry is attained for general smooth domains.

    Authors: We acknowledge that the presentation in §3 can be made more self-contained. The underlying space is the standard Sobolev space W^{1,p}(Ω) equipped with the trace operator onto L^p(Γ); the energy functional is I(u) = (1/p)∫_Ω |∇u|^p dx + (1/p)∫_Γ |u|^p dσ − (1/p*)∫_Ω V(x)|u|^{p*} dx. In the revision we will insert the complete definition of I, recall the trace theorem, and provide a detailed, line-by-line computation of the asymptotic expansion of I(t u_ε) that isolates the mean-curvature integral arising from the boundary term after integration by parts on the tubular neighborhood. These additions will allow direct verification of the mountain-pass geometry on any smooth domain satisfying the stated curvature hypothesis. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard variational existence argument

full rationale

The paper establishes existence of least-energy solutions for the critical p-Laplacian problem via the mountain-pass theorem applied to the energy functional I(u). The key technical step is constructing boundary-supported test functions (cut-off bubbles) whose energy expansion is computed explicitly from the p-Laplacian equation and the boundary mean-curvature term; this expansion is used to place the mountain-pass level strictly below the critical Sobolev threshold (1/n)S^{n/p}, restoring the Palais-Smale condition. All steps rely on external Sobolev embeddings, standard bubble analysis, and direct asymptotic calculations rather than any self-definition, fitted parameter renamed as prediction, or load-bearing self-citation. The claimed distinction between the p<2 and p>2 regimes follows from the differing orders of the curvature correction term in the expansion and is not presupposed by the target statement. The derivation is therefore self-contained against external analytic tools.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The work rests on standard variational calculus for quasilinear operators and Sobolev critical embeddings.

axioms (2)
  • domain assumption The domain is bounded and C^2 so that mean curvature is defined on the boundary portion where Neumann conditions are imposed.
    Required for the geometric term to make sense in the energy functional.
  • standard math The nonlinearity is exactly the critical Sobolev exponent for the p-Laplacian.
    Standard for critical exponent problems; the abstract refers to 'critical exponent' without further definition.

pith-pipeline@v0.9.0 · 5434 in / 1468 out tokens · 36519 ms · 2026-05-13T18:14:24.147566+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

  1. [1]

    Adimurthi, S. L. Yadava. Existence and nonexistence of positive radial solutions of Neumann problems with critical Sobolev exponents. Arch. Rational Mech. Anal., 115 (1991), 275–296. 2

    work page 1991

  2. [2]

    Adimurthi, G. Mancini. The Neumann problem for elliptic equations with critical nonlinearity. Quaderni Scuola Norm. Sup. Pisa (1991), 9–25. 1, 2, 5

    work page 1991

  3. [3]

    Alghanemi, S

    A. Alghanemi, S. Chaabane, H. Chtioui, A. Soumar. Towards a proof of Bahri-Coron’s type theorem for mixed boundary value problems. Mathematics, 11 (2023), 1955. 2

    work page 2023

  4. [4]

    Anello, G

    G. Anello, G. Cordaro. Infinitely many arbitrarily small positive solutions for the Dirichlet problem involving thep-Laplacian. Proc. Roy. Soc. Edinburgh A, 132 (2002), 511–519. 2

    work page 2002

  5. [5]

    Aubin, T. (1976). ´Equations diff´ erentielles non lin´ eaires et probl` eme de Yamabe concernant la courbure scalaire.J. Math. Pures Appl.,55, 269–296. 1, 2, 4

    work page 1976

  6. [6]

    Bahri, J

    A. Bahri, J. M. Coron. On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain. (1987). 2

    work page 1987

  7. [7]

    Bahri, J

    A. Bahri, J. M. Coron. The scalar-curvature problem on the standard three-dimensional sphere. J. Funct. Anal., 95 (1991), 106–172. 2

    work page 1991

  8. [8]

    Bonanno, R

    G. Bonanno, R. Livrea. Multiplicity theorems for the Dirichlet problem involving thep-Laplacian. Nonlinear Anal., 54 (2003), 1–7. 2

    work page 2003

  9. [9]

    Brezis, L

    H. Brezis, L. Nirenberg. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math., 36 (1983), 437–477. 1, 2, 4

    work page 1983

  10. [10]

    R. D. Carlos, J. V. C. Sousa, E.-H. Hamza. Ground state solution for the generalizedp-Laplacian operator with logarithmic nonlinearity. Anal. Math. Phys., 16 (2026), 30. 2

    work page 2026

  11. [11]

    H. Chtioui. On the Chen-Lin conjecture for the prescribed scalar curvature problem. J. Geom. Anal., 33 (2023), 298. 2

    work page 2023

  12. [12]

    J. M. Coron. Topologie et cas limite des injections de Sobolev. C. R. Acad. Sci. Paris S. I Math., 299 (1984), 209–212. 2

    work page 1984

  13. [13]

    Y. Deng, Y. Shi, L. Xu. Existence of positive solutions for criticalp-Laplacian equation with critical Neumann boundary condition. Adv. Nonlinear Anal., 14 (2025), 20250109. 2

    work page 2025

  14. [14]

    Grossi, F

    M. Grossi, F. Pacella. Positive solutions of nonlinear elliptic equations with critical Sobolev exponent and mixed boundary conditions. Proc. Roy. Soc. Edinburgh A, 116 (1990), 23–43. 3

    work page 1990

  15. [15]

    Y. X. Huang. On eigenvalue problems ofp-Laplacian with Neumann boundary conditions. Proc. Amer. Math. Soc., 109 (1990), 177–184. 2 THE ROLE OF THE MEAN CUR V ATURE IN NONLINEAR PROBLEM 17

    work page 1990

  16. [16]

    Y. Y. Li. Prescribing scalar curvature and related topics, part I. J. Differential Equations, 120 (1995), 319–410. 2

    work page 1995

  17. [17]

    P. L. Lions, F. Pacella, M. Tricarico. Best constants in Sobolev inequalities for functions vanishing on some part of the boundary. Indiana Univ. Math. J., 37 (1988), 301–324. 2, 5

    work page 1988

  18. [18]

    V. Maz’ya. Sobolev Spaces. Springer, 2013. 2

    work page 2013

  19. [19]

    Pacella and M

    F. Pacella and M. Tricarico. Symmetrization for a class of elliptic equations with mixed boundary conditions. Atti Semin. Mat. Fis. Univ. Modena, 34, (1985) 75–93,

    work page 1985

  20. [20]

    R. Schoen. Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differential Geom., 20 (1984), 479–495. 3

    work page 1984

  21. [21]

    G. Talenti. Best constant in Sobolev inequality. Ann. Mat. Pura Appl., 110 (1976), 353–372. 2

    work page 1976

  22. [22]

    J. L. V´ azquez. A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim., 12(1)(1984), 191–202. 5

    work page 1984

  23. [23]

    X. J. Wang. Neumann problems of semilinear elliptic equations involving critical Sobolev exponents. J. Differential Equations, 93 (1991), 283–310. 15 1, 2, 3

    work page 1991

  24. [24]

    X. J. Wang. Positive solutions to the Neumann problem forp-Laplace equations. Gaoxiao Yingyong Shuxue Xuebao, 8 (1993), 99–112. 2

    work page 1993

  25. [25]

    S. L. Yadava. On a conjecture of Lin-Ni for a semilinear Neumann problem. Trans. Amer. Math. Soc., 336 (1993), 631-637. 2 (H. Chtioui)Department of Mathematics, F aculty of Sciences of Sfax, Sfax University, Tunisia Email address:hichem.chtioui@fss.rnu.tn (H. Hajaiej)Department of Mathematics, California State University, Los Angeles, CA 90032, USA. Email...

    work page 1993