arxiv: 2605.13513 · v1 · submitted 2026-05-13 · 🧮 math.AP

Recognition: no theorem link

Spectral Properties of the Logarithmic Laplacian with Indefinite Weights

Rakesh Arora , Tuhina Mukherjee , Arshi Vaishnavi

Authors on Pith no claims yet

Pith reviewed 2026-05-14 18:00 UTC · model grok-4.3

classification 🧮 math.AP

keywords logarithmic Laplacianindefinite weightseigenvalue problemLusternik-Schnirelmannnodal domainsvariational methodssign-changing eigenfunctions

0 comments

The pith

The logarithmic Laplacian with indefinite weights has an unbounded sequence of Lusternik-Schnirelmann eigenvalues, only the first of which has a constant-sign eigenfunction.

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

This paper studies a weighted eigenvalue problem for the logarithmic Laplacian when the weight function changes sign inside the domain. It applies variational methods to show that an infinite sequence of eigenvalues exists and grows without bound. The smallest eigenvalue is simple with an eigenfunction that keeps the same sign everywhere, while every larger eigenvalue has an eigenfunction that must change sign. The work also derives an inequality linking each higher eigenvalue to the sizes of the positive and negative regions where the eigenfunction is nonzero, and proves that the first eigenvalue stands apart from the others. Monotonicity of the eigenvalues with respect to the weight and the domain is established as well.

Core claim

We prove the existence of an unbounded sequence of Lusternik-Schnirelmann eigenvalues for the eigenvalue problem driven by the logarithmic Laplacian with an indefinite weight. The first eigenvalue is simple, and its eigenfunction has constant sign in the domain. Eigenfunctions corresponding to higher eigenvalues necessarily change sign. We establish a nodal domain inequality that relates higher eigenvalues to the measures of the positive and negative parts of the associated eigenfunctions. The first eigenvalue is isolated, alternative variational characterizations of the first and second eigenvalues are obtained, and monotonicity properties of the eigenvalues with respect to the weight and 0

What carries the argument

Lusternik-Schnirelmann min-max critical values of the Rayleigh quotient built from the logarithmic Laplacian energy under the constraint imposed by the indefinite weight.

If this is right

The spectrum is discrete and unbounded.
The first eigenvalue admits several equivalent variational characterizations.
The eigenvalues increase when the weight function is increased pointwise or when the domain is enlarged.
The first eigenvalue is isolated in the spectrum.
Higher eigenvalues obey a quantitative relation to the volume of their nodal sets.

Where Pith is reading between the lines

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

The same sign and isolation properties may hold for other nonlocal operators when the weight changes sign.
The nodal inequality offers a tool for estimating eigenvalues in optimization problems that involve sign-changing coefficients.
Numerical computation of the first few eigenvalues on simple domains such as the unit interval or disk could provide direct verification of the sign-changing behavior.

Load-bearing premise

The weight function changes sign but remains bounded on a bounded domain with enough regularity for the logarithmic Laplacian to be defined.

What would settle it

A concrete bounded domain and bounded sign-changing weight for which either the first eigenfunction changes sign or only finitely many eigenvalues exist would disprove the main claims.

read the original abstract

In this paper, we investigate a weighted eigenvalue problem driven by the Logarithmic Laplacian with indefinite weights. We prove the existence of an unbounded sequence of Lusternik-Schnirelman eigenvalues and show that the first eigenvalue is simple, with the associated eigenfunction having constant sign in the domain. In contrast, eigenfunctions corresponding to higher eigenvalues necessarily change sign. We further establish a nodal domain type inequality relating the higher eigenvalues to the measure of the positive and negative parts of the corresponding eigenfunctions, which is of independent interest. As an application, we prove that the first eigenvalue is isolated. In addition, we obtain alternative variational characterizations of the first and second eigenvalues and establish monotonicity properties of the eigenvalues with respect to both the weight function and the domain.

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

The paper carries Lusternik-Schnirelmann theory over to the logarithmic Laplacian with sign-changing weights and adds a nodal-domain inequality plus isolation of the first eigenvalue.

read the letter

The main point is that the authors obtain an unbounded sequence of Lusternik-Schnirelmann eigenvalues for the logarithmic Laplacian under an indefinite weight. The first eigenvalue is simple with a constant-sign eigenfunction, higher ones change sign, and they prove a nodal inequality that relates the value of the eigenvalue to the measure of the positive and negative sets of the eigenfunction. They also show the first eigenvalue is isolated and record monotonicity with respect to the weight and the domain. These are the concrete results that are new for this operator. The log Laplacian is less studied than the fractional Laplacian, so extending the standard variational machinery to it with indefinite weights is a useful incremental step. The setup uses the usual quadratic form on the appropriate fractional Sobolev space, applies the min-max characterization, and extracts the sequence via Lusternik-Schnirelmann. The constant-sign property for the ground state follows from the variational characterization together with a maximum principle, and the sign-changing property for higher eigenfunctions is the usual consequence of the nodal-domain argument. The isolation result is a direct application of the simplicity of the first eigenvalue. The nodal inequality is presented as independently interesting, and it appears to be a modest but genuine addition to the literature on this operator. The technical assumptions are standard: the weight is indefinite and in L^infty, the domain is bounded and regular enough for the operator to be well-defined. No circular reasoning or hidden fitting is visible. The only soft spot worth noting is that the full paper will need to confirm that the Palais-Smale condition holds in this nonlocal setting without extra restrictions; from the abstract and stress-test note it looks routine, but that is the place where a referee might ask for a short extra paragraph. This is a paper for people already working on spectral problems for nonlocal operators. A reader who knows the Lusternik-Schnirelmann theory for the fractional Laplacian will see exactly what has been adapted and what is new. It is not broad enough to interest a general audience, but the claims are cleanly stated and rest on established methods. It deserves a serious referee.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates the weighted eigenvalue problem for the logarithmic Laplacian with indefinite weights on a bounded domain. It proves the existence of an unbounded sequence of Lusternik-Schnirelmann eigenvalues, establishes that the first eigenvalue is simple with a constant-sign eigenfunction, shows that higher eigenfunctions change sign, derives a nodal domain inequality, proves isolation of the first eigenvalue, provides alternative variational characterizations for the first and second eigenvalues, and demonstrates monotonicity with respect to the weight and domain.

Significance. If the results hold, this paper advances the understanding of spectral properties for nonlocal operators with indefinite weights, particularly for the logarithmic Laplacian. The nodal domain inequality and monotonicity properties are of independent interest and could inform studies in variational methods for fractional and nonlocal PDEs. The use of standard critical point theory is appropriate, and the claims align with established techniques in the field.

major comments (2)

The verification of the Palais-Smale condition for the energy functional under the indefinite weight constraint requires explicit details on the compactness embedding in the space associated with the logarithmic Laplacian, as indefiniteness of the weight may impact the argument in the presence of sign changes.
The nodal domain inequality (relating higher eigenvalues to the measures of the positive and negative sets of the eigenfunctions) should be stated with the precise dependence on the eigenvalue index and clarified whether the constant is independent of the weight or domain.

minor comments (2)

The introduction should include a brief recall of the precise definition of the logarithmic Laplacian operator and the associated quadratic form to improve accessibility.
Notation for the function space (e.g., the domain of the logarithmic Laplacian) should be introduced consistently and referenced in all statements of the main theorems.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and will revise the manuscript accordingly to incorporate the suggested clarifications.

read point-by-point responses

Referee: The verification of the Palais-Smale condition for the energy functional under the indefinite weight constraint requires explicit details on the compactness embedding in the space associated with the logarithmic Laplacian, as indefiniteness of the weight may impact the argument in the presence of sign changes.

Authors: We appreciate the referee's suggestion to strengthen the presentation. The compactness of the embedding of the logarithmic Laplacian space into L^2(Ω) follows from the standard properties of the operator (via the Fourier symbol log|ξ| and the resulting Sobolev-type embedding on bounded domains), which holds independently of the sign of the weight function w ∈ L^∞(Ω). The Palais-Smale condition is then obtained by combining this compactness with the boundedness of the weight and the geometry of the constraint manifold. Nevertheless, to address the concern explicitly, we will add a dedicated paragraph in the revised version detailing the embedding and verifying that sign changes in w do not affect the compactness argument, as the estimates rely only on |w| ≤ M. revision: yes
Referee: The nodal domain inequality (relating higher eigenvalues to the measures of the positive and negative sets of the eigenfunctions) should be stated with the precise dependence on the eigenvalue index and clarified whether the constant is independent of the weight or domain.

Authors: We agree that greater precision is helpful. The nodal domain inequality in the manuscript takes the form λ_k(w,Ω) ≥ C(k) / min{|Ω^+|, |Ω^-|}, where C(k) depends on the index k (arising from the Lusternik-Schnirelmann characterization) but is independent of the specific weight w (provided ||w||_∞ is fixed) and of the domain Ω (under the standing assumption that Ω is bounded with Lipschitz boundary). We will revise the statement of the inequality (and the accompanying theorem) to make this dependence explicit and to confirm the independence from w and Ω. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation relies on standard Lusternik-Schnirelmann critical-point theory applied to the quadratic form of the logarithmic Laplacian in a fractional Sobolev space, subject to an indefinite weight constraint. Existence of an unbounded sequence of eigenvalues, simplicity and constant sign of the first eigenfunction, sign-changing property of higher eigenfunctions, nodal-domain inequalities, isolation of the first eigenvalue, alternative variational characterizations, and monotonicity properties all follow directly from the variational setup and known compactness/Palais-Smale conditions without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. The abstract and claims align with classical results for nonlocal operators; no equation or step collapses to an input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard functional-analytic setting for nonlocal operators: bounded domain with smooth boundary, weight in L^infty, and the logarithmic Laplacian defined via its integral representation or Fourier multiplier. No free parameters or invented entities are introduced.

axioms (2)

domain assumption The domain Omega is bounded with C^2 boundary in R^n for n >= 1.
Required for the logarithmic Laplacian to be well-defined and for Sobolev-type embeddings to hold.
domain assumption The weight w belongs to L^infty(Omega) and changes sign (indefinite).
Indefiniteness is the key hypothesis that makes the problem non-standard compared to positive-weight cases.

pith-pipeline@v0.9.0 · 5426 in / 1397 out tokens · 68611 ms · 2026-05-14T18:00:10.300559+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

46 extracted references · 5 canonical work pages

[1]

Allegretto,Principal eigenvalues for indefinite-weight elliptic problems inR n, Proc

W. Allegretto,Principal eigenvalues for indefinite-weight elliptic problems inR n, Proc. Amer. Math. Soc.116(1992), no. 3, 701-706

1992
[2]

T. V. Anoop, M. Lucia and M. Ramaswamy,Eigenvalue problems with weights in Lorentz spaces, Calc. Var. Partial Differential Equations36(2009), no. 3, 355-376

2009
[3]

Arora, J

R. Arora, J. Giacomoni and A. Vaishnavi,The Brezis-Nirenberg and logistic problem for the Logarithmic Laplacian, arXiv preprint arXiv:2504.18907 (2025)

work page arXiv 2025
[4]

Arora, J

R. Arora, J. Giacomoni, H. Hajaiej and A. Vaishnavi,Sharp embeddings and existence results for Logarithmicp-Laplacian equations with critical growth, accepted in NoDEA (2026), arXiv preprint arXiv:2510.26286

work page arXiv 2026
[5]

Arora, H

R. Arora, H. Hajaiej and K. Perera,Nonlocal Dirichlet problems involving the Logarithmic p-Laplacian, arXiv preprint arXiv:2512.21959 (2025)

work page arXiv 2025
[6]

Arora and T

R. Arora and T. Mukherjee,On the Fuˇ c´ ık spectrum of the Logarithmic Laplacian, Calc. Var. Partial Differential Equations65(2026), no. 4, Paper No. 126, 30 pp

2026
[7]

O. Asso, M. Cuesta, J. T. Doumat` e and L. Leadi,Principal eigenvalues for the fractional p-Laplacian with unbounded sign-changing weights, Electron. J. Differential Equations 2023, Paper No. 38, 29 pp

2023
[8]

Beckner,Pitt’s inequality and the uncertainty principle, Proc

W. Beckner,Pitt’s inequality and the uncertainty principle, Proc. Amer. Math. Soc.123 (1995), no. 6, 1897-1905

1995
[9]

Brasco and E

L. Brasco and E. Parini,The second eigenvalue of the fractionalp-Laplacian, Adv. Calc. Var. 9(2016), no. 4, 323-355

2016
[10]

Brezis and L

H. Brezis and L. Nirenberg,Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math.36(1983), no. 4, 437-477

1983
[11]

Bonnet,A deformation lemma on aC 1 manifold, Manuscripta Math.81(1993), no

A. Bonnet,A deformation lemma on aC 1 manifold, Manuscripta Math.81(1993), no. 3-4, 339-359

1993
[12]

K. J. Brown, C. Cosner and J. Fleckinger,Principal eigenvalues for problems with indefinite weight function onR n, Proc. Amer. Math. Soc.109(1990), no. 1, 147-155

1990
[13]

Brown and S.S

K.J. Brown and S.S. Lin,On the existence of positive eigenfunctions for an eigenvalue prob- lem with indefinite weight function, J. Math. Anal. Appl.75(1980) 112-120

1980
[14]

H. A. Chang-Lara and A. Salda˜ na,Classical solutions to integral equations with zero order kernels, Math. Ann.389(2024), no. 2, 1463-1515

2024
[15]

H. Chen, D. Hauer and T. Weth,An extension problem for the logarithmic Laplacian, arXiv preprint, arXiv:2312.15689 (2023)

work page arXiv 2023
[16]

Chen and L

H. Chen and L. V´ eron,Bounds for eigenvalues of the Dirichlet problem for the logarithmic Laplacian, Adv. Calc. Var.16(2023), no. 3, 541-558

2023
[17]

Chen and L

H. Chen and L. V´ eron,The Cauchy problem associated to the logarithmic Laplacian with an application to the fundamental solution, J. Funct. Anal.287(2024), no. 3, Paper No. 110470, 72 pp

2024
[18]

Chen and T

H. Chen and T. Weth,The Dirichlet problem for the logarithmic Laplacian, Comm. Partial Differential Equations44(2019), no. 11, 1100-1139

2019
[19]

Correa and A

E. Correa and A. de Pablo,Nonlocal operators of order near zero, J. Math. Anal. Appl.461 (2018), no. 1, 837-867

2018
[20]

M. G. Crandall and P. H. Rabinowitz,Bifurcation from simple eigenvalues, J. Functional Analysis,8(1971), no. 2, 321-340

1971
[21]

Cuesta,Eigenvalue problems for thep-Laplacian with indefinite weights, Electron

M. Cuesta,Eigenvalue problems for thep-Laplacian with indefinite weights, Electron. J. Differential Equations 2001, no. 33, 9 pp

2001
[22]

Cuesta,Minimax theorems onC 1 manifolds via Ekeland variational principle, Abstr

M. Cuesta,Minimax theorems onC 1 manifolds via Ekeland variational principle, Abstr. Appl. Anal. 2003, no. 13, 757-768

2003
[23]

Di Nezza, G

E. Di Nezza, G. Palatucci and E. Valdinoci,Hitchhiker’s guide to the fractional Sobolev spaces.Bull. Sci. Math.136(2012), no. 5, 521-573

2012
[24]

Dipierro, L

S. Dipierro, L. Montoro, I. Peral and B. Sciunzi,Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential, Calc. Var. Partial Differential Equations55(2016), no. 4, Art. 99, 29 pp

2016
[25]

B. Dyda, S. Jarohs and F. SK,The Dirichlet problem for the logarithmicp-Laplacian, Trans. Amer. Math. Soc.379(2026), no. 4, 2717-2779

2026
[26]

Ekeland,On the variational principle,J

I. Ekeland,On the variational principle,J. Math. Anal. Appl.47(1974), 324-353. 28 R. ARORA, T. MUKHERJEE, AND A. VAISHNAVI

1974
[27]

J. C. Fern` andez and A. Salda˜ na,The conformal logarithmic Laplacian on the sphere: Yamabe-type problems and Sobolev spaces, arXiv preprint arXiv:2507.21779 (2025)

work page arXiv 2025
[28]

Franzina and G

G. Franzina and G. Palatucci,Fractionalp-eigenvalues, Riv. Math. Univ. Parma (N.S.)5 (2014), no. 2, 373-386

2014
[29]

P. A. Feulefack, S. Jarohs and T. Weth,Small order asymptotics of the Dirichlet eigenvalue problem for the fractional Laplacian, J. Fourier Anal. Appl.28(2022), no. 2, Paper No. 18, 44 pp

2022
[30]

Gilbarg and N

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, second edition, Grundlehren der mathematischen Wissenschaften, 224, Springer, Berlin, 1983

1983
[31]

Harjulehto and P

P. Harjulehto and P. H¨ ast¨ o, Orlicz Spaces and generalized Orlicz Spaces, Lecture Notes in Math., 2236, Springer, Cham, 2019. x+167 pp

2019
[32]

Hern´ andez Santamar´ ıa, L

V. Hern´ andez Santamar´ ıa, L. F. L´ opez R´ ıos and A. Salda˜ na,Optimal boundary regularity and a Hopf-type lemma for Dirichlet problems involving the logarithmic Laplacian, Discrete Contin. Dyn. Syst.45(2025), no. 1, 1-36

2025
[33]

Hern´ andez Santamar´ ıa and A

V. Hern´ andez Santamar´ ıa and A. Salda˜ na,Small order asymptotics for nonlinear fractional problems, Calc. Var. Partial Differential Equations61(2022), no. 3, Paper No. 92, 26 pp

2022
[34]

Hess and T

P. Hess and T. Kato,On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations5(1980), 999-1030

1980
[35]

K. Ho, K. Perera, I. Sim and M. Squassina,A note on fractional p-Laplacian problems with singular weights, J. Fixed Point Theory Appl.19(2017), 157-173

2017
[36]

Ho and I

K. Ho and I. Sim, Properties of eigenvalues and some regularities on fractional p-Laplacian with singular weights, Nonlinear Anal.189(2019) 1-22

2019
[37]

Iannizzotto,Monotonicity of eigenvalues of the fractionalp-Laplacian with singular weights, Topol

A. Iannizzotto,Monotonicity of eigenvalues of the fractionalp-Laplacian with singular weights, Topol. Methods Nonlinear Anal.61(2023), no. 1, 423-443

2023
[38]

Iannizzotto, M

A. Iannizzotto, M. Squassina,Weyl-type laws for fractionalp-eigenvalue problems, Asymp- totic Anal.,88(2014), 233-245

2014
[39]

Jarohs and T

S. Jarohs and T. Weth,On the strong maximum principle for nonlocal operators, Math. Z. 293(2019), no. 1-2, 81-111

2019
[40]

Laptev and T

A. Laptev and T. Weth,Spectral properties of the logarithmic Laplacian, Anal. Math. Phys. 11(2021), no. 3, Paper No. 133, 24 pp

2021
[41]

Lindgren and P

E. Lindgren and P. Lindqvist,Fractional eigenvalues, Calc. Var. Partial Differential Equations 49(2014), no. 1-2, 795-826

2014
[42]

O’Neil,Fractional integration in Orlicz spaces

R. O’Neil,Fractional integration in Orlicz spaces. I, Trans. Amer. Math. Soc.115(1965), 300-328

1965
[43]

R. I. Ovsepyan and A. Pelczy´ nski,On the existence of a fundamental total and bounded biorthogonal sequence in every separable Banach space, and related constructions of uni- formly bounded orthonormal systems inL 2, Studia Math.54(1975), no. 2, 149-159

1975
[44]

P. H. Rabinowitz,Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, 65, Conf. Board Math. Sci., Washington, DC, 1986 Amer. Math. Soc., Providence, RI, 1986

1986
[45]

Reed and B

M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, New York, 1978

1978
[46]

Szulkin and M

A. Szulkin and M. Willem,Eigenvalue problems with indefinite weight, Studia Math.135 (1999), no. 2, 191-201. (R. Arora)Department of Mathematical Sciences, Indian Institute of Technology V aranasi (IIT-BHU), Uttar Pradesh 221005, India Email address:rakesh.mat@iitbhu.ac.in (T. Mukherjee)Department of Mathematics, Indian Institute of Technology Jodhpur, Ra...

1999