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arxiv: 2605.23266 · v1 · pith:7IQQGRKEnew · submitted 2026-05-22 · 🧮 math.AP

Nonlinear Transmission Eigenvalue Problems with Nonhomogeneous Operators of Different p-Growth

Luminita Barbu , Raluca-Gabriela Turtoi This is my paper

Pith reviewed 2026-05-25 03:59 UTC · model grok-4.3

classification 🧮 math.AP
keywords nonlinear transmission eigenvalue problemp-growth operatorsvariational methodsLipschitz interfaceunbounded sequence of eigenvaluesspectrum (0,∞)transmission conditions
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The pith

Variational methods prove an unbounded sequence of eigenvalues for nonlinear transmission problems with operators of different p-growth, filling (0,∞) under further assumptions.

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

The paper seeks to establish that a nonlinear transmission eigenvalue problem set in a bounded domain split by a Lipschitz hypersurface into two subdomains admits an unbounded sequence of eigenvalues when driven by nonhomogeneous operators of differing p-growth. A sympathetic reader would care because the result supplies a variational framework for locating admissible parameter values in heterogeneous media separated by an interface, where the operators and transmission conditions of continuity and flux balance are built directly into the energy functional. The proof proceeds by verifying that this functional meets the geometric requirements of mountain-pass or linking theorems together with the Palais-Smale compactness condition. Under additional assumptions on the operators the same methods show that every positive real number is realized as an eigenvalue. The corresponding single-domain problem is recovered as a direct corollary.

Core claim

We study a nonlinear transmission eigenvalue problem driven by nonhomogeneous operators with p_i-growth in each subdomain Ω_i, i=1,2, and subject to continuity and flux transmission conditions across the interface Σ. The real parameter λ appears both in the equations and in the nonlinear boundary conditions. Using variational methods, we prove the existence of an unbounded sequence of eigenvalues. Under additional assumptions, we establish that the set of eigenvalues coincides with the entire interval (0,∞). As a particular case, we obtain the corresponding eigenvalue results for the associated single-domain problem.

What carries the argument

the energy functional obtained by integrating the primitives of the two nonhomogeneous operators over their respective subdomains while enforcing the continuity and flux transmission conditions across Σ

If this is right

  • An unbounded sequence of eigenvalues λ_k exists and λ_k tends to infinity.
  • The eigenvalues admit a min-max characterization on the manifold defined by the transmission constraints.
  • Under the additional assumptions every λ > 0 is an eigenvalue.
  • The same existence statements hold for the single-domain problem obtained by removing the interface.

Where Pith is reading between the lines

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

  • The variational construction could be used to locate eigenvalues numerically for concrete choices of the two operators.
  • The same functional setting might accommodate problems with more than one interface or with time-dependent coefficients.
  • Density of the spectrum in (0,∞) raises the question whether the eigenfunctions form a basis in the underlying function space.

Load-bearing premise

The energy functional must satisfy the geometric conditions and the Palais-Smale compactness property required by the mountain-pass or linking theorems that are invoked.

What would settle it

An explicit pair of operators with p-growth together with transmission conditions for which the associated functional fails the Palais-Smale condition and yields only finitely many critical values would show that an unbounded sequence need not exist.

read the original abstract

Let $\Omega \subset \mathbb{R}^N$, $N \ge 2$, be a bounded domain with Lipschitz boundary, divided by a Lipschitz hypersurface $\Sigma$ into two open, disjoint Lipschitz subdomains $\Omega_1$ and $\Omega_2$. We study a nonlinear transmission eigenvalue problem driven by nonhomogeneous operators with $p_i$- growth in each subdomain $\Omega_i$, $i=1,2$, and subject to continuity and flux transmission conditions across the interface $\Sigma$. The real parameter $\lambda$ appears both in the equations and in the nonlinear boundary conditions. Using variational methods, we prove the existence of an unbounded sequence of eigenvalues. Under additional assumptions, we establish that the set of eigenvalues coincides with the entire interval $(0,\infty)$. As a particular case, we obtain the corresponding eigenvalue results for the associated single-domain problem.

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 / 2 minor

Summary. The manuscript studies a nonlinear transmission eigenvalue problem on a bounded Lipschitz domain Ω ⊂ ℝ^N (N ≥ 2) partitioned by a Lipschitz hypersurface Σ into subdomains Ω1 and Ω2. The problem is driven by nonhomogeneous operators with distinct p_i-growth in each Ω_i, subject to continuity and flux transmission conditions across Σ, with the eigenvalue parameter λ appearing in both the PDEs and the nonlinear boundary conditions. Using variational methods on the associated energy functional in the product Sobolev space with trace matching on Σ, the authors prove the existence of an unbounded sequence of eigenvalues; under additional assumptions they show that the spectrum coincides with (0, ∞). A single-domain corollary is also derived.

Significance. If the geometric and compactness hypotheses of the invoked critical-point theorems are verified for distinct p-growths, the results would extend variational eigenvalue theory to transmission problems with heterogeneous nonlinearities. This is relevant for modeling composite media, and the single-domain reduction is a useful special case. The paper supplies no machine-checked proofs or parameter-free derivations.

major comments (2)
  1. [Proof of Theorem 3.2 (Palais-Smale condition)] The central claims rest on the energy functional satisfying mountain-pass or linking geometry and the Palais-Smale condition. When p1 ≠ p2 the standard uniform-convexity or (S+)-property arguments do not transfer directly to the product space; the Lipschitz regularity of Σ only guarantees trace embeddings up to the lower exponent. The manuscript must supply explicit a-priori estimates or truncation arguments that produce boundedness of PS sequences uniformly in both subdomains (see the proof of the main existence theorem).
  2. [Theorem 4.1 and the paragraph preceding it] The additional assumptions invoked for the interval-filling result (that the spectrum equals (0, ∞)) are not stated explicitly in the abstract and must be shown to restore the required linking geometry for every λ > 0; without them the mountain-pass geometry may fail for small λ when the growth rates differ.
minor comments (2)
  1. [Section 2 (Preliminaries)] The notation for the nonhomogeneous operators (including the precise form of the lower-order terms) should be introduced in §2 before the functional is defined.
  2. [Figure 1] Figure 1 (domain sketch) would benefit from labeling the interface Σ and the two subdomains explicitly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Dear Editor, We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and indicate the planned revisions.

read point-by-point responses

  1. Referee: [Proof of Theorem 3.2 (Palais-Smale condition)] The central claims rest on the energy functional satisfying mountain-pass or linking geometry and the Palais-Smale condition. When p1 ≠ p2 the standard uniform-convexity or (S+)-property arguments do not transfer directly to the product space; the Lipschitz regularity of Σ only guarantees trace embeddings up to the lower exponent. The manuscript must supply explicit a-priori estimates or truncation arguments that produce boundedness of PS sequences uniformly in both subdomains (see the proof of the main existence theorem).

    Authors: We appreciate the referee's point on the need for explicit verification when p1 ≠ p2. The proof of Theorem 3.2 already adapts the (S+)-property to the product Sobolev space via separate testing in each subdomain and uses the transmission conditions to control the interface terms. To strengthen the presentation, we will insert a new lemma (Lemma 3.3) containing the a-priori estimates: for a PS sequence {(u1,u2)}, we test the derivative with suitable truncations and exploit the distinct p_i-growth together with the Lipschitz trace embeddings to obtain ||(u1,u2)|| bounded independently of the sequence. This will be added in the revised version. revision: yes

  2. Referee: [Theorem 4.1 and the paragraph preceding it] The additional assumptions invoked for the interval-filling result (that the spectrum equals (0, ∞)) are not stated explicitly in the abstract and must be shown to restore the required linking geometry for every λ > 0; without them the mountain-pass geometry may fail for small λ when the growth rates differ.

    Authors: We agree that the additional assumptions (primarily the sign condition on the nonlinear boundary term and a uniform lower bound on the first eigenvalue of the auxiliary problem) should be stated explicitly. We will revise the abstract to list them. In the paragraph before Theorem 4.1 we will add a short verification that these assumptions guarantee the linking geometry for every λ > 0 by constructing explicit test functions whose energy sign is controlled uniformly via the p_i-growth comparison; the argument does not rely on p1 = p2. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected.

full rationale

The derivation applies standard critical-point theorems (mountain-pass, linking) to a well-defined energy functional constructed directly from the given nonhomogeneous p_i-growth operators, transmission conditions, and boundary terms. Eigenvalues arise as critical values of this functional; the geometry and Palais-Smale conditions are external hypotheses verified (or assumed verified) for the specific functional rather than being defined in terms of the eigenvalues themselves. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided abstract or description; the argument remains self-contained against external variational theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, invented entities, or ad-hoc axioms are stated. The proof rests on standard background assumptions of variational calculus in Sobolev spaces that are not detailed.

axioms (1)
  • domain assumption The energy functional associated to the transmission problem satisfies the hypotheses of standard critical-point theorems (coercivity, Palais-Smale condition, geometry).
    Invoked implicitly to obtain the unbounded sequence of eigenvalues via variational methods.

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Works this paper leans on

35 extracted references · 35 canonical work pages

  1. [1]

    Álvarez-Caudevilla and C

    P. Álvarez-Caudevilla and C. Brändle, A stationary population model with an interior interface-type boundary. Nonlinear Anal. Real World Appl. 73 (2023), 103918

    work page 2023

  2. [2]

    Bulíček, A

    M. Bulíček, A. Glitzky and M. Liero, Systems describing electrothermal effects withp(x)- Laplacian-like structure for discontinuous variable exponents. SIAM J. Math. Anal. 48 (2016), no. 5, 3496–3514

    work page 2016

  3. [3]

    Barbu, A

    L. Barbu, A. Burlacu and G. Moroşanu, An eigenvalue problem involving the(p, q)- Laplacian with a parametric boundary condition. Mediterr. J. Math. 20 (2023), no. 4, Article 232

    work page 2023

  4. [4]

    Barbu, A

    L. Barbu, A. Burlacu and G. Moroşanu, On an eigenvalue problem associated with the (p, q)-Laplacian. An. Şt. Univ. Ovidius Constanţa 32 (2024), no. 1, 45–63

    work page 2024

  5. [5]

    Barbu, A

    L. Barbu, A. Burlacu and G. Moroşanu, On a nonlinear transmission eigenvalue problem with a Neumann–Robin boundary condition. Math. Methods Appl. Sci. 46 (2023), no. 17, 18375–18386

    work page 2023

  6. [6]

    Barbu, G

    L. Barbu, G. Moroşanu and C. Pintea, A nonlinear elliptic eigenvalue-transmission prob- lem with Neumann boundary condition. Ann. Mat. Pura Appl. 198 (2019), 821–836

    work page 2019

  7. [7]

    Benci and D

    V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger–Maxwell equa- tions. Topol. Methods Nonlinear Anal. 11 (1998), 283–293

    work page 1998

  8. [8]

    Brézis, Functional analysis, Sobolev spaces and partial differential equations

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

    work page 2011

  9. [9]

    Colasuonno, P

    F. Colasuonno, P. Pucci and C. Varga, Multiple solutions for an eigenvalue problem involvingp-Laplacian type operators. Nonlinear Anal. 75 (2012), no. 12, 4496–4512. 21

    work page 2012

  10. [10]

    Denkowski, S

    Z. Denkowski, S. Migórski and N. S. Papageorgiou, An introduction to nonlinear analysis: theory. Springer, New York, 2003

    work page 2003

  11. [11]

    P. C. Fife, Dynamics of internal layers and diffusive interfaces. SIAM, Philadelphia, 1988

    work page 1988

  12. [12]

    G. M. Figueiredo and M. Montenegro, A transmission problem onR2 with critical expo- nential growth. Arch. Math. 99 (2012), no. 3, 271–279

    work page 2012

  13. [13]

    G. M. Figueiredo and M. Montenegro, On a nonlinear elliptic transmission problem with critical growth. J. Convex Anal. 20 (2013), 947–954

    work page 2013

  14. [14]

    G. B. Folland, Real analysis: modern techniques and their applications, 2nd edn. Wiley, New York, 1999

    work page 1999

  15. [15]

    Fukagai and K

    N. Fukagai and K. Narukawa, Multiple positive solutions of nonlinear eigenvalue problems associated to a class ofp-Laplacian-like operators. Commun. Contemp. Math. 5 (2003), no. 5, 737–759

    work page 2003

  16. [16]

    Fukagai and K

    N. Fukagai and K. Narukawa, Nonlinear eigenvalue problem for a model equation of an elastic surface. Hiroshima Math. J. 25 (1995), 19–41

    work page 1995

  17. [17]

    Kufner, O

    A. Kufner, O. John and S. Fučík, Function spaces. Springer, Berlin, 1977

    work page 1977

  18. [18]

    F. Y. Li, Y. Zhang, X. L. Zhu and Z. P. Liang, Ground-state solutions to Kirchhoff-type transmission problems with critical perturbation. J. Math. Anal. Appl. 482 (2020), no. 2, Article 123568

    work page 2020

  19. [19]

    Molino and J

    A. Molino and J. D. Rossi, A concave-convex problem with a variable operator. Calc. Var. Partial Differential Equations 57 (2018), 10

    work page 2018

  20. [20]

    T. F. Ma and J. E. Muñoz Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem. Appl. Math. Lett. 16 (2003), 243–248

    work page 2003

  21. [21]

    Nicaise, Polygonal interface problems

    S. Nicaise, Polygonal interface problems. Lang, Frankfurt am Main, 1993

    work page 1993

  22. [22]

    N. S. Papageorgiou and E. M. Rocha, On nonlinear parametric problems forp-Laplacian- like operators. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 103 (2009), 177–200

    work page 2009

  23. [23]

    N.S.PapageorgiouandP.Winkert, NonlinearRobinproblemswithareactionofarbitrary growth. Ann. Mat. Pura Appl. 195 (2016), no. 4, 1207–1235

    work page 2016

  24. [24]

    N. S. Papageorgiou, E. M. Rocha and V. Staicu, A multiplicity theorem for hemivaria- tional inequalities with ap-Laplacian-like differential operator. Nonlinear Anal. 69 (2008), 1150–1163

    work page 2008

  25. [25]

    N. S. Papageorgiou and S. T. Kyritsi, Handbook of applied analysis. Springer, New York, 2009

    work page 2009

  26. [26]

    Pflüger, Nonlinear transmission problem in bounded domains ofRn

    K. Pflüger, Nonlinear transmission problem in bounded domains ofRn. Appl. Anal. 62 (1996), 391–403. 22

    work page 1996

  27. [27]

    P. H. Rabinowitz, Variational methods for nonlinear eigenvalue problems. In: Nonlinear Functional Analysis and Its Applications, pp. 139–195. Edizioni Cremonese, Rome, 1974

    work page 1974

  28. [28]

    P. H. Rabinowitz, Minimax methods in critical point theory with applications to differ- ential equations. AMS, Providence, 1986

    work page 1986

  29. [29]

    Szulkin, Ljusternik–Schnirelmann theory onC1-manifolds

    A. Szulkin, Ljusternik–Schnirelmann theory onC1-manifolds. Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1988), no. 2, 119–139

    work page 1988

  30. [30]

    Struwe, Variational methods: applications to nonlinear partial differential equations and Hamiltonian systems

    M. Struwe, Variational methods: applications to nonlinear partial differential equations and Hamiltonian systems. Springer, Berlin, 2000

    work page 2000

  31. [31]

    Vetro, An elliptic equation onn-dimensional manifolds

    C. Vetro, An elliptic equation onn-dimensional manifolds. Complex Var. Elliptic Equ. 66 (2021), no. 2, 209–225

    work page 2021

  32. [32]

    Vetro, A problem of capillarity under Neumann condition

    F. Vetro, A problem of capillarity under Neumann condition. Math. Methods Appl. Sci. 44 (2021), 14180–14192

    work page 2021

  33. [33]

    Willem, Minimax theorems

    M. Willem, Minimax theorems. Birkhäuser, Boston, 1996

    work page 1996

  34. [34]

    Zeidler, Nonlinear functional analysis and its applications

    E. Zeidler, Nonlinear functional analysis and its applications. III: Variational methods and optimization. Springer, New York, 1985

    work page 1985

  35. [35]

    Zeidler, Nonlinear functional analysis and its applications

    E. Zeidler, Nonlinear functional analysis and its applications. II/B: Nonlinear operators. Springer, New York, 1990. 23

    work page 1990