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arxiv: 2606.24827 · v2 · pith:NPTTEOFYnew · submitted 2026-06-23 · 🧮 math.AP

On polyharmonic Kirchhoff double phase problems without AR-conditions

Ashutosh Dixit , Tuhina Mukherjee This is my paper

Pith reviewed 2026-07-01 06:40 UTC · model grok-4.3

classification 🧮 math.AP
keywords polyharmonic Kirchhoff problemsdouble phase operatorMusielak-Orlicz-Sobolev spaceAmbrosetti-Rabinowitz conditionminimax methodsexistence and multiplicityvariational methods
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The pith

Existence and multiplicity of solutions hold for polyharmonic Kirchhoff double phase problems without the Ambrosetti-Rabinowitz condition.

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

The paper proves existence and multiplicity results for a class of polyharmonic Kirchhoff problems driven by a double phase operator. The reaction term is assumed to have only subcritical growth and is not required to meet the Ambrosetti-Rabinowitz condition. Work proceeds in the Musielak-Orlicz-Sobolev space natural to the double phase structure. Suitable modular estimates and compactness arguments set up the variational framework, after which minimax methods yield the solutions. This extends earlier m-polyharmonic Kirchhoff results to the nonhomogeneous double phase setting.

Core claim

Under subcritical growth assumptions on the reaction term that are weaker than the classical Ambrosetti-Rabinowitz condition, the polyharmonic Kirchhoff double phase problem possesses weak solutions, including multiple solutions in some cases, obtained by establishing the variational setting via modular estimates and compactness arguments in the Musielak-Orlicz-Sobolev framework and then applying minimax methods.

What carries the argument

Modular estimates and compactness arguments in the Musielak-Orlicz-Sobolev framework for the double phase operator combined with the nonlocal Kirchhoff term, used to enable minimax methods.

If this is right

  • Existence results apply to a larger family of nonlinearities that violate the Ambrosetti-Rabinowitz condition.
  • Multiplicity theorems extend to higher-order polyharmonic problems with double phase structure.
  • The variational approach functions in nonhomogeneous settings where standard coercivity assumptions fail.
  • Compactness arguments suffice to recover solutions when the usual superlinear growth condition is dropped.

Where Pith is reading between the lines

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

  • The same modular estimates could be tested on related nonlocal problems with variable growth rates.
  • Physical models involving materials with two distinct phases may now be analyzed without forcing the Ambrosetti-Rabinowitz condition on the source term.
  • Numerical verification on explicit examples with slowly growing nonlinearities would check whether the predicted multiplicity actually appears.

Load-bearing premise

The nonlinearity has subcritical growth that can be controlled by modular estimates in the double phase space even without the Ambrosetti-Rabinowitz condition.

What would settle it

A concrete nonlinearity with subcritical growth but no Ambrosetti-Rabinowitz condition for which the corresponding polyharmonic Kirchhoff double phase problem has no weak solution would falsify the existence claim.

read the original abstract

In this paper, we study a class of polyharmonic Kirchhoff problems driven by a double phase operator. The reaction term has subcritical growth but does not satisfy the Ambrosetti--Rabinowitz condition. Motivated by the work of Harrabi-Hamdani-Fiscella \cite{Harrabi-Hamdani-Fiscella-2024} on m-polyharmonic Kirchhoff problems without Ambrosetti--Rabinowitz conditions, we extend their analysis to a nonhomogeneous double phase setting. We study the problem in the natural Musielak--Orlicz--Sobolev framework associated with the double phase structure. The main novelty of the paper lies in combining the nonlocal Kirchhoff term with a higher-order double phase operator under assumptions weaker than the classical Ambrosetti--Rabinowitz condition. By developing suitable modular estimates and compactness arguments, we establish the variational setting and obtain existence and multiplicity results by minimax methods.

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

0 major / 2 minor

Summary. The manuscript studies a class of polyharmonic Kirchhoff problems driven by a double phase operator. The reaction term has subcritical growth but does not satisfy the Ambrosetti-Rabinowitz condition. Extending the analysis of Harrabi-Hamdani-Fiscella (2024) from the m-polyharmonic Kirchhoff setting, the authors work in the Musielak-Orlicz-Sobolev framework associated with the double phase structure. They develop modular estimates and compactness arguments to establish the variational setting and obtain existence and multiplicity results via minimax methods.

Significance. If the modular estimates and compactness arguments are valid, the work provides a direct extension of variational methods to nonlocal Kirchhoff problems with higher-order double phase operators under assumptions weaker than the classical AR condition. This is a meaningful technical contribution for handling nonhomogeneous operators with variable growth in the Musielak-Orlicz-Sobolev setting.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'suitable modular estimates and compactness arguments' is repeated without indicating the precise growth or phase conditions that make the estimates work; a single sentence clarifying the key structural hypotheses on the double phase integrand would improve immediate readability.
  2. [Introduction] The citation to Harrabi-Hamdani-Fiscella-2024 is central to the motivation; ensure the reference list entry is complete and that any notational differences between the m-polyharmonic and double-phase settings are explicitly contrasted in the introduction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The report correctly identifies the main contribution as extending variational methods for polyharmonic Kirchhoff problems to the double-phase Musielak-Orlicz-Sobolev setting without the Ambrosetti-Rabinowitz condition. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper extends an external result from Harrabi-Hamdani-Fiscella (2024) on m-polyharmonic Kirchhoff problems to the double-phase Musielak-Orlicz-Sobolev setting, using standard modular estimates, compactness arguments, and minimax methods under subcritical growth without the AR condition. The derivation chain relies on established variational techniques applied to a new operator combination; no self-definitional reductions, fitted inputs renamed as predictions, load-bearing self-citations, or ansatz smuggling appear in the stated approach. The central claims remain independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters or invented entities; the work invokes standard background results on function spaces and variational methods.

axioms (1)
  • standard math Standard embedding and compactness properties of Musielak-Orlicz-Sobolev spaces
    Required to set up the variational formulation and apply minimax theorems.

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Works this paper leans on

62 extracted references

  1. [1]

    S. S. Antman,Nonlinear problems of elasticity, 2nd ed., Applied Mathematical Sciences, vol. 107, Springer, New York, 2005

    2005

  2. [2]

    Arora, ´A

    R. Arora, ´A. Crespo-Blanco, and P. Winkert,Logarithmic double phase problems with generalized critical growth, Nonlinear Differential Equations Appl. NoDEA32(2025), no. 5, Article 98

    2025

  3. [3]

    Arora, A

    R. Arora, A. Crespo-Blanco and P. Winkert,On logarithmic double phase problems, J. Differential Equations433 (2025), 113247

    2025

  4. [4]

    Arosio and S

    A. Arosio and S. Panizzi,On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc.348(1996), no. 1, 305–330

    1996

  5. [5]

    Bahrouni, V

    A. Bahrouni, V. D. R˘ adulescu and D. D. Repovˇ s,Double phase transonic flow problems with variable growth: nonlinear patterns and stationary waves, Nonlinearity32(2019), no. 7, 2481–2495

    2019

  6. [6]

    Baroni, M

    P. Baroni, M. Colombo and G. Mingione,Harnack inequalities for double phase functionals, Nonlinear Anal.121 (2015), 206–222

    2015

  7. [7]

    Baroni, M

    P. Baroni, M. Colombo and G. Mingione,Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations57(2018), no. 2, Paper No. 62, 48 pp

    2018

  8. [8]

    Baroni, T

    P. Baroni, T. Kuusi and G. Mingione,Borderline gradient continuity of minima, J. Fixed Point Theory Appl.15 (2014), no. 2, 537–575

    2014

  9. [9]

    Beck and G

    L. Beck and G. Mingione,Lipschitz bounds and nonuniform ellipticity, Comm. Pure Appl. Math.73(2020), no. 5, 944–1034

    2020

  10. [10]

    Benchira, A

    H. Benchira, A. Matallah, M. E. O. El Mokhtar and K. Sabri,The existence result for ap-Kirchhoff-type problem involving critical Sobolev exponent, J. Funct. Spaces2023

  11. [11]

    Benci, P

    V. Benci, P. D’Avenia, D. Fortunato and L. Pisani,Solitons in several space dimensions: Derrick’s problem and infinitely many solutions, Arch. Rational Mech. Anal.154(2000), no. 4, 297–324

    2000

  12. [12]

    Berestycki and P.-L

    H. Berestycki and P.-L. Lions,Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal.82(1983), no. 4, 313–345

    1983

  13. [13]

    Byun and J

    S.-S. Byun and J. Oh,Regularity results for generalized double phase functionals, Anal. PDE13(2020), no. 5, 1269– 1300

    2020

  14. [14]

    S.-S. Byun, J. Ok and K. Song,H¨ older regularity for weak solutions to nonlocal double phase problems, J. Math. Pures Appl. (9)168(2022), 110–142

    2022

  15. [15]

    J. W. Cahn and J. E. Hilliard,Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys.28 (1958), 258–267

    1958

  16. [16]

    Cannone, S

    A. Cannone, S. Cingolani and J. Mederski,Polyharmonic nonlinear scalar field equations, Nonlinear Anal.272(2026), 114176

    2026

  17. [17]

    M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano,Global existence and uniform decay rates for the Kirchhoff–Carrier equation with nonlinear dissipation, Adv. Differential Equations6(2001), 701–730

    2001

  18. [18]

    Chems Eddine and M

    N. Chems Eddine and M. A. Ragusa,Generalized critical Kirchhoff-type potential systems with Neumann boundary conditions, Appl. Anal.101(2022), no. 11, 3958–3988

    2022

  19. [19]

    W. Chen, Z. Fu and Y. Wu,Positive solutions for nonlinear Schr¨ odinger–Kirchhoff equations inR 3, Appl. Math. Lett.104(2020), 106274

    2020

  20. [20]

    W. Chen, Z. Fu and Y. Wu,Positive ground states for nonlinear Schr¨ odinger–Kirchhoff equations with periodic potential or potential well inR 3, Bound. Value Probl.2022(2022), no. 1, 97

    2022

  21. [21]

    Cherfils and Y

    L. Cherfils and Y. Il’yasov,On the stationary solutions of generalized reaction diffusion equations withp&q-Laplacian, Commun. Pure Appl. Anal.4(2005), no. 1, 9–22

    2005

  22. [22]

    Colasuonno and M

    F. Colasuonno and M. Squassina,Eigenvalues for double phase variational integrals, Ann. Mat. Pura Appl. (4)195 (2016), no. 6, 1917–1959

    2016

  23. [23]

    Colombo and G

    M. Colombo and G. Mingione,Bounded minimisers of double phase variational integrals, Arch. Rational Mech. Anal. 218(2015), no. 1, 219–273

    2015

  24. [24]

    Colombo and G

    M. Colombo and G. Mingione,Regularity for double phase variational problems, Arch. Rational Mech. Anal.215 (2015), no. 2, 443–496

    2015

  25. [25]

    Crespo-Blanco, L

    ´A. Crespo-Blanco, L. Gasi´ nski, P. Harjulehto and P. Winkert,A new class of double phase variable exponent problems: existence and uniqueness, J. Differential Equations323(2022), 182–228

    2022

  26. [26]

    Cupini, P

    G. Cupini, P. Marcellini and E. Mascolo,Local boundedness of weak solutions to elliptic equations withp, q-growth, Math. Eng.5(2023), no. 3, 1-28

    2023

  27. [27]

    De Filippis and G

    C. De Filippis and G. Mingione,On the regularity of minima of non-autonomous functionals, J. Geom. Anal.30 (2020), no. 2, 1584–1626

    2020

  28. [28]

    De Filippis and G

    C. De Filippis and G. Mingione,Lipschitz bounds and nonautonomous integrals, Arch. Rational Mech. Anal.242 (2021), 973–1057

    2021

  29. [29]

    De Filippis and G

    C. De Filippis and G. Palatucci,H¨ older regularity for nonlocal double phase equations, J. Differential Equations267 (2019), no. 1, 547–586

    2019

  30. [30]

    Dipierro and H.-C

    S. Dipierro and H.-C. Grunau,Boggio’s formula for fractional polyharmonic Dirichlet problems, Ann. Mat. Pura Appl. 196(2017), 1327–1344

    2017

  31. [31]

    Dixit, T

    A. Dixit, T. Mukherjee and P. Winkert,On polyharmonic Kirchhoff problems with double phase structure and sub- critical nonlinearities, Appl. Math. Optim.93(2026), no. 2, 56

    2026

  32. [32]

    El Ahmadi, A

    M. El Ahmadi, A. Ayoujil and M. Berrajaa,On Kirchhoff–double phase problems without (AR)-condition, Mathemat- ica66(2024), no. 2

    2024

  33. [33]

    El-Houari, L

    H. El-Houari, L. S. Chadli and H. Moussa,Multiple solutions in fractional Orlicz–Sobolev spaces for a class of nonlocal Kirchhoff systems, Filomat38(2024), no. 8, 2857–2875

    2024

  34. [34]

    Fibich, B

    G. Fibich, B. Ilan and G. Papanicolaou,Self-focusing with fourth-order dispersion, SIAM J. Appl. Math.62(2002), no. 4, 1437–1462. ON POLYHARMONIC KIRCHHOFF DOUBLE PHASE PROBLEMS 19

    2002

  35. [35]

    Gasi´ nski and P

    L. Gasi´ nski and P. Winkert,Existence and uniqueness results for double phase problems with convection term, J. Differential Equations268(2020), no. 8, 4183–4193

    2020

  36. [36]

    Gazzola, H.-C

    F. Gazzola, H.-C. Grunau and G. Sweers,Polyharmonic boundary value problems, Lecture Notes in Mathematics, vol. 1991, Springer-Verlag, Berlin, 2010

    1991

  37. [37]

    Harrabi, M

    A. Harrabi, M. K. Hamdani and A. Fiscella,Existence and multiplicity of solutions form-polyharmonic Kirchhoff problems without Ambrosetti–Rabinowitz conditions, Complex Var. Elliptic Equ.69(2024), no. 11, 1853–1869

    2024

  38. [38]

    H¨ ast¨ o and J

    P. H¨ ast¨ o and J. Ok,Maximal regularity for local minimizers of non-autonomous functionals, J. Eur. Math. Soc.24 (2022), no. 4, 1285–1334

    2022

  39. [39]

    Kirchhoff,Vorlesungen ¨ uber mathematische Physik

    G. Kirchhoff,Vorlesungen ¨ uber mathematische Physik. Mechanik, Teubner, Leipzig, 1883

  40. [40]

    J.-L. Lions,On some questions in boundary value problems of mathematical physics, in: Proceedings of International Symposium on Continuum Mechanics and Partial Differential Equations, Rio de Janeiro, 1977, Math. Stud., vol. 30, North-Holland, Amsterdam, 1978, pp. 284–346

    1977

  41. [41]

    Liu and G

    W. Liu and G. Dai,Existence and multiplicity results for double phase problem, J. Differential Equations265(2018), no. 9, 4311–4334

    2018

  42. [42]

    A. E. H. Love,A Treatise on the Mathematical Theory of Elasticity, 4th ed., Cambridge University Press, Cambridge, 1927

    1927

  43. [43]

    Malchiodi, R

    A. Malchiodi, R. Mandel and M. Rizzi,Periodic solutions to a Cahn–Hilliard–Willmore equation in the plane, Arch. Rational Mech. Anal.228(2018), 821–866

    2018

  44. [44]

    Marcellini,Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions, Arch

    P. Marcellini,Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions, Arch. Rational Mech. Anal.105(1989), no. 3, 267–284

    1989

  45. [45]

    Marcellini,Regularity and existence of solutions of elliptic equations withp, q-growth conditions, J

    P. Marcellini,Regularity and existence of solutions of elliptic equations withp, q-growth conditions, J. Differential Equations90(1991), no. 1, 1–30

    1991

  46. [46]

    Marcellini,Local Lipschitz continuity forp, q-PDEs with explicitu-dependence, Nonlinear Anal.226(2023), 113066, 26 pp

    P. Marcellini,Local Lipschitz continuity forp, q-PDEs with explicitu-dependence, Nonlinear Anal.226(2023), 113066, 26 pp

    2023

  47. [47]

    Mederski,Nonradial solutions of nonlinear scalar field equations, Nonlinearity33(2020), no

    J. Mederski,Nonradial solutions of nonlinear scalar field equations, Nonlinearity33(2020), no. 12, 6349–6380

    2020

  48. [48]

    Mederski,General class of optimal Sobolev inequalities and nonlinear scalar field equations, J

    J. Mederski,General class of optimal Sobolev inequalities and nonlinear scalar field equations, J. Differential Equations 281(2021), 411–441

    2021

  49. [49]

    Mederski and J

    J. Mederski and J. Siemianowski,Biharmonic nonlinear scalar field equations, Int. Math. Res. Not. IMRN2023 (2023), no. 23, 19963–19995

    2023

  50. [50]

    V. V. Meleshko,Selected topics in the history of the two-dimensional biharmonic problem, Appl. Mech. Rev.56 (2003), 33–85

    2003

  51. [51]

    R. D. Mindlin,Micro-structure in linear elasticity, Arch. Rational Mech. Anal.16(1964), no. 1, 51–78

    1964

  52. [52]

    Moujane and M

    N. Moujane and M. El Ouaarabi,On a class of Schr¨ odinger–Kirchhoff-double phase problems with convection term and variable exponents, Commun. Nonlinear Sci. Numer. Simul.141(2025), 108453

    2025

  53. [53]

    Musielak,Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, Vol

    J. Musielak,Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, Vol. 1034, Springer, Berlin, 1983

    1983

  54. [54]

    Pucci and J

    P. Pucci and J. Serrin,A general variational identity, Indiana Univ. Math. J.35(1986), no. 3, 681–703

    1986

  55. [55]

    P. H. Rabinowitz,Minimax methods in critical point theory with applications to differential equations, CBMS Reg. Conf. Ser. Math.65, American Mathematical Society, Providence, RI, 1986

    1986

  56. [56]

    Rizzi,Clifford tori and the singularly perturbed Cahn–Hilliard equation, J

    M. Rizzi,Clifford tori and the singularly perturbed Cahn–Hilliard equation, J. Differential Equations262(2017), no. 10, 5306–5362

    2017

  57. [57]

    A. P. S. Selvadurai,Partial differential equations in mechanics. 2. The biharmonic equation, Poisson’s equation, Springer-Verlag, Berlin, 2000

    2000

  58. [58]

    B. S. Wang, G. L. Hou and B. Ge,Existence of solutions for double-phase problems by topological degree, J. Fixed Point Theory Appl.23(2021), 1–10

    2021

  59. [59]

    V. V. Zhikov,Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat.50(1986), no. 4, 675–710

    1986

  60. [60]

    V. V. Zhikov,On Lavrentiev’s phenomenon, Russ. J. Math. Phys.3(1995), no. 2, 249–269

    1995

  61. [61]

    V. V. Zhikov,On variational problems and nonlinear elliptic equations with nonstandard growth conditions, J. Math. Sci.173(2011), no. 5, 463–570

    2011

  62. [62]

    J. Zuo, W. Zhang and V. D. R˘ adulescu,Multiplicity and concentration properties for(p, q)-Kirchhoff non-autonomous problems with Choquard nonlinearity, Bull. Sci. Math.191(2024), 103398

    2024