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arxiv: 2511.05679 · v3 · pith:735FNFIDnew · submitted 2025-11-07 · 🧮 math.AP

On the asymptotically linear problem for an elliptic equation with an indefinite nonlinearity

M\'onica Clapp , Cristian Morales-Encinos , Alberto Salda\~na , Mayra Soares This is my paper

Pith reviewed 2026-05-21 19:18 UTC · model grok-4.3

classification 🧮 math.AP
keywords semilinear elliptic equationsindefinite nonlinearitypositive solutionsnondegeneracyasymptotically linear problemsblow-up techniquesspectral properties
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The pith

For p > 2 close enough to 2, the elliptic equation with indefinite nonlinearity has a unique nondegenerate positive solution.

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

The paper studies the semilinear elliptic equation with a weight that is positive inside a bounded domain and negative outside, creating an indefinite nonlinearity. For exponents p slightly larger than 2, where the problem becomes asymptotically linear, the authors prove there exists exactly one positive solution and that this solution is nondegenerate. The proof rests on a detailed study of an associated eigenvalue problem, combined with variational methods and blow-up analysis to control the limit as p approaches 2. This matters for models of optical waveguides, where such equations describe light propagation and where uniqueness and nondegeneracy indicate stable behavior. The work also derives symmetry results, sharp decay estimates for eigenfunctions, and analogues of the Faber-Krahn and Hong-Krahn-Szegö inequalities in this indefinite setting.

Core claim

The authors establish that for p > 2 sufficiently close to 2, the problem admits a unique positive solution, which is nondegenerate. This is proven by combining a detailed analysis of the eigenvalue problem involving the weight Q_Ω with variational methods and blow-up techniques in the asymptotically linear regime. They also provide a comprehensive study of the spectral properties of the corresponding linear problem, including the existence and qualitative behavior of eigenfunctions, sharp decay estimates, and symmetry results, along with analogues of the Faber-Krahn and Hong-Krahn-Szegö inequalities.

What carries the argument

The sign-changing weight Q_Ω = χ_Ω - χ_{R^N ∖ Ω} that produces the indefinite nonlinearity, together with the associated linear eigenvalue problem that supplies the spectral gap required for uniqueness and nondegeneracy via blow-up analysis.

If this is right

  • There is exactly one positive solution and it remains isolated in the solution set for p near 2.
  • The solution is nondegenerate, so the linearized operator is invertible at that solution.
  • Spectral properties hold, including sharp decay estimates and symmetry of eigenfunctions.
  • Analogues of the Faber-Krahn and Hong-Krahn-Szegö inequalities are valid for the indefinite weight.

Where Pith is reading between the lines

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

  • The restriction to p near 2 suggests that the number of positive solutions may change as p increases further from the linear case.
  • Nondegeneracy opens the possibility of using continuation methods to track the solution for larger p.
  • The blow-up and spectral techniques developed here may apply to other sign-changing weights or to problems posed in bounded domains.

Load-bearing premise

The analysis requires p to be sufficiently close to 2 so that the asymptotically linear regime allows control via blow-up techniques and the associated eigenvalue problem yields the necessary spectral gap.

What would settle it

The discovery of either a second distinct positive solution or a positive solution for which the linearized operator has a zero eigenvalue, at some value of p only slightly larger than 2, would falsify the claim.

read the original abstract

We study the semilinear elliptic problem \[ -\Delta u = Q_{\Omega} |u|^{p-2}u \quad \text{in } \mathbb{R}^N, \] where \( Q_{\Omega} = \chi_{\Omega} - \chi_{\mathbb{R}^N \setminus \Omega} \) for a bounded smooth domain \( \Omega \subset \mathbb{R}^N \), \( N \ge 3 \), and \( 1 < p < 2^{*} \). This equation arises in the study of optical waveguides and exhibits indefinite nonlinearity due to the sign-changing weight \( Q_{\Omega} \). We prove that, for \( p > 2 \) sufficiently close to \( 2 \), the problem admits a unique positive solution, which is nondegenerate. Our approach combines a detailed analysis of an associated eigenvalue problem involving \( Q_{\Omega} \) with variational methods and blow-up techniques in the asymptotically linear regime. We also provide a comprehensive study of the spectral properties of the corresponding linear problem, including the existence and qualitative behavior of eigenfunctions, sharp decay estimates, and symmetry results. In particular, we establish analogues of the Faber--Krahn and Hong--Krahn--Szeg{\"o} inequalities in this non-standard setting.

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 / 3 minor

Summary. The manuscript studies the semilinear elliptic problem −Δu = Q_Ω |u|^{p−2}u in R^N (N≥3), where Q_Ω = χ_Ω − χ_{R^N∖Ω} is a sign-changing weight for a bounded smooth domain Ω. It claims that for p>2 sufficiently close to 2 the problem admits a unique positive solution that is nondegenerate. The approach combines a detailed spectral analysis of the associated linear eigenvalue problem with weight Q_Ω (including analogues of the Faber–Krahn and Hong–Krahn–Szegő inequalities, eigenfunction properties, decay estimates and symmetry results) with variational methods and blow-up techniques in the asymptotically linear regime.

Significance. If the central claims hold, the work advances the theory of elliptic equations with indefinite nonlinearities in the asymptotically linear setting. The spectral results for the weighted linear problem with sign-changing weight are of independent interest and supply the spectral gap needed for nondegeneracy. The combination of variational and blow-up arguments provides a concrete route to uniqueness and nondegeneracy near p=2, with potential relevance to optical-waveguide models.

minor comments (3)
  1. [Abstract and Theorem 1.1] The abstract and main theorem statement should make explicit the dependence of the 'sufficiently close to 2' threshold on N and on the geometry of Ω (e.g., via the principal eigenvalue of the weighted problem).
  2. [Introduction] In the discussion of the physical motivation, a short paragraph or reference to the specific optical-waveguide model that leads to this indefinite nonlinearity would help readers outside the immediate community.
  3. [Section 2] Notation for the principal eigenvalue λ_1(Q_Ω) and its eigenfunction should be introduced once and used consistently; currently the same symbol appears with slightly varying normalizations in different sections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and constructive report, which recognizes the contributions of our work on the asymptotically linear indefinite elliptic problem. We appreciate the recommendation for minor revision and the acknowledgment that the spectral results are of independent interest. Since no specific major comments or requested changes were detailed in the report, we interpret the minor revision as pertaining to possible typographical or presentational improvements, which we will address in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external spectral analysis and standard blow-up techniques

full rationale

The paper establishes spectral properties of the weighted linear problem (analogues of Faber-Krahn and Hong-Krahn-Szegő inequalities) using variational methods on the indefinite weight Q_Ω, then applies these to control the asymptotically linear regime for p close to 2 via blow-up analysis and eigenvalue gaps. These steps draw on standard external tools (variational methods, blow-up techniques, and known inequalities adapted to the sign-changing setting) without reducing any central claim to a fitted parameter, self-definition, or self-citation chain. The uniqueness and nondegeneracy for p sufficiently close to 2 follow from the spectral gap of the principal eigenvalue, which is derived independently of the nonlinear solution itself. No load-bearing step collapses to an input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain being bounded and smooth and on p lying in a narrow interval above 2; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Ω is a bounded smooth domain in R^N, N ≥ 3
    Required to define the sign-changing weight Q_Ω and to guarantee regularity and embedding properties.
  • ad hoc to paper 1 < p < 2* and p sufficiently close to 2
    The regime in which the blow-up analysis and spectral gap arguments are claimed to work.

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Works this paper leans on

27 extracted references · 27 canonical work pages

  1. [1]

    A concentration phenomenon for semilinear elliptic equations.Arch

    Nils Ackermann and Andrzej Szulkin. A concentration phenomenon for semilinear elliptic equations.Arch. Ration. Mech. Anal., 207(3):1075–1089, 2013

    work page 2013

  2. [2]

    A Picone’s identity for thep-Laplacian and applications.Nonlinear Anal., 32(7):819–830, 1998

    Walter Allegretto and Yin Xi Huang. A Picone’s identity for thep-Laplacian and applications.Nonlinear Anal., 32(7):819–830, 1998

    work page 1998

  3. [3]

    Symmetry and monotonicity of solutions to some variational problems in cylinders and annuli

    Friedemann Brock. Symmetry and monotonicity of solutions to some variational problems in cylinders and annuli. Electron. J. Differential Equations, pages No. 108, 20, 2003

    work page 2003

  4. [4]

    John E Brothers and William P. Ziemer. Minimal rearrangements of Sobolev functions.J. Reine Angew. Math., 384:153–179, 1988

    work page 1988

  5. [5]

    Critical equations with a sharp change of sign in the nonlinearity

    Mónica Clapp, Jorge Faya, and Alberto Saldaña. Critical equations with a sharp change of sign in the nonlinearity. arXiv preprint arXiv:2502.19563, 2025

    work page arXiv 2025

  6. [6]

    Positive and nodal limiting profiles for a semi- linear elliptic equation with a shrinking region of attraction.Nonlinear Anal., 251:Paper No

    Mónica Clapp, Víctor Hernández-Santamaría, and Alberto Saldaña. Positive and nodal limiting profiles for a semi- linear elliptic equation with a shrinking region of attraction.Nonlinear Anal., 251:Paper No. 113680, 15, 2025

    work page 2025

  7. [7]

    Multiple solutions to a semilinear elliptic equation with a sharp change of sign in the nonlinearity.J

    Mónica Clapp, Angela Pistoia, and Alberto Saldaña. Multiple solutions to a semilinear elliptic equation with a sharp change of sign in the nonlinearity.J. Math. Pures Appl. (9), 205:Paper No. 103783, 35, 2026

    work page 2026

  8. [8]

    A concentration phenomenon for a semilinear Schrödinger equation with periodic self-focusing core.arXiv preprint arXiv:2507.14541, 2025

    Mónica Clapp, Alberto Saldaña, and Andrzej Szulkin. A concentration phenomenon for a semilinear Schrödinger equation with periodic self-focusing core.arXiv preprint arXiv:2507.14541, 2025

    work page arXiv 2025

  9. [9]

    On a Schrödinger system with shrinking regions of attraction

    Mónica Clapp, Alberto Saldaña, and Andrzej Szulkin. On a Schrödinger system with shrinking regions of attraction. Z. Angew. Math. Phys., 76(1):Paper No. 37, 23, 2025

    work page 2025

  10. [10]

    Richard Courant and David Hilbert.Methods of mathematical physics. Vol. I. Interscience Publishers, Inc., New York, 1953

    work page 1953

  11. [11]

    Qualitative properties of positive solutions of semilinear elliptic equations in symmetric domains via the maximum principle.Ann

    Lucio Damascelli, Massimo Grossi, and Filomena Pacella. Qualitative properties of positive solutions of semilinear elliptic equations in symmetric domains via the maximum principle.Ann. Inst. H. Poincaré C Anal. Non Linéaire, 16(5):631–652, 1999

    work page 1999

  12. [12]

    Edward N. Dancer. Real analyticity and non-degeneracy.Math. Ann., 325(2):369–392, 2003

    work page 2003

  13. [13]

    UniquenessandnondegeneracyforDirichletfractionalproblems in bounded domains via asymptotic methods.Nonlinear Anal., 236:Paper No

    AbdelrazekDieb, IsabellaIanni, andAlbertoSaldaña. UniquenessandnondegeneracyforDirichletfractionalproblems in bounded domains via asymptotic methods.Nonlinear Anal., 236:Paper No. 113354, 21, 2023

    work page 2023

  14. [14]

    Limiting profile of solutions for Schrödinger equations with shrinking self- focusing core.Calc

    Xiang-Dong Fang and Zhi-Qiang Wang. Limiting profile of solutions for Schrödinger equations with shrinking self- focusing core.Calc. Var. Partial Differential Equations, 59(4):Paper No. 129, 18, 2020

    work page 2020

  15. [15]

    Asymptotic location and shape of the optimal favorable region in a Neumann spectral problem.J

    Lorenzo Ferreri, Dario Mazzoleni, Benedetta Pellacci, and Gianmaria Verzini. Asymptotic location and shape of the optimal favorable region in a Neumann spectral problem.J. Math. Pures Appl. (9), 205:Paper No. 103815, 2026

    work page 2026

  16. [16]

    Monotonicity properties of variational integrals,Ap weights and unique continu- ation.Indiana Univ

    Nicola Garofalo and Fang-Hua Lin. Monotonicity properties of variational integrals,Ap weights and unique continu- ation.Indiana Univ. Math. J., 35(2):245–268, 1986

    work page 1986

  17. [17]

    Symmetry of positive solutions of nonlinear elliptic equations inR n

    Basilis Gidas, Wei Ming Ni, and Louis Nirenberg. Symmetry of positive solutions of nonlinear elliptic equations inR n. InMathematical analysis and applications, Part A, volume 7a ofAdv. Math. Suppl. Stud., pages 369–402. Academic Press, New York-London, 1981

    work page 1981

  18. [18]

    Trudinger.Elliptic partial differential equations of second order

    David Gilbarg and Neil S. Trudinger.Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition

    work page 2001

  19. [19]

    Lieb and Michael Loss.Analysis, volume 14

    Elliott H. Lieb and Michael Loss.Analysis, volume 14. American Mathematical Society, Providence, RI, 2001

    work page 2001

  20. [20]

    Uniqueness of least energy solutions to a semilinear elliptic equation inR2.Manuscripta Math., 84(1):13–19, 1994

    Chang Shou Lin. Uniqueness of least energy solutions to a semilinear elliptic equation inR2.Manuscripta Math., 84(1):13–19, 1994

    work page 1994

  21. [21]

    On the least-energy solutions of the pure Neumann Lane-Emden equation

    Alberto Saldaña and Hugo Tavares. On the least-energy solutions of the pure Neumann Lane-Emden equation. NoDEA Nonlinear Differential Equations Appl., 29(3):Paper No. 30, 36, 2022. 23

    work page 2022

  22. [22]

    Asymptotic axial symmetry of solutions of parabolic equations in bounded radial domains.J

    Alberto Saldaña and Tobias Weth. Asymptotic axial symmetry of solutions of parabolic equations in bounded radial domains.J. Evol. Equ., 12(3):697–712, 2012

    work page 2012

  23. [23]

    Self-trapping of an electromagnetic field and bifurcation from the essential spectrum.Arch

    Charles A Stuart. Self-trapping of an electromagnetic field and bifurcation from the essential spectrum.Arch. Rational Mech. Anal., 113:65–96, 1990

    work page 1990

  24. [24]

    Guidance properties of nonlinear planar waveguides.Arch

    Charles A Stuart. Guidance properties of nonlinear planar waveguides.Arch. Rational Mech. Anal., 125:145–200, 1993

    work page 1993

  25. [25]

    Eigenvalue problems with indefinite weight.Studia Math., 135(2):191–201, 1999

    Andrzej Szulkin and Michel Willem. Eigenvalue problems with indefinite weight.Studia Math., 135(2):191–201, 1999

    work page 1999

  26. [26]

    Symmetry of solutions to variational problems for nonlinear elliptic equations via reflection methods

    Tobias Weth. Symmetry of solutions to variational problems for nonlinear elliptic equations via reflection methods. Jahresber. Dtsch. Math.-Ver., 112(3):119–158, 2010

    work page 2010

  27. [27]

    Birkhäuser Boston, Inc., Boston, MA, 1996

    Michel Willem.Minimax theorems, volume 24 ofProgress in Nonlinear Differential Equations and their Applications. Birkhäuser Boston, Inc., Boston, MA, 1996. Mónica Clapp Instituto de Matemáticas Universidad Nacional Autónoma de México Campus Juriquilla Boulevard Juriquilla 3001 76230 Querétaro, Qro., Mexico monica.clapp@im.unam.mx Cristian Morales-Encinosa...

    work page 1996