On the de Th\'elin eigenvalue problem and Landesman-Lazer conditions for quasilinear systems

David Arcoya; Natalino Borgia; Silvia Cingolani

arxiv: 2601.16846 · v2 · pith:SXWRST4Vnew · submitted 2026-01-23 · 🧮 math.AP

On the de Th\'elin eigenvalue problem and Landesman-Lazer conditions for quasilinear systems

David Arcoya , Natalino Borgia , Silvia Cingolani This is my paper

Pith reviewed 2026-05-21 15:54 UTC · model grok-4.3

classification 🧮 math.AP

keywords quasilinear elliptic systemsde Thélín problemLandesman-Lazer conditionseigenvalue isolationresonant nonlinear problemsvariational characterizationweak solutions

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The pith

The smallest eigenvalue λ₁ of the de Thélín quasilinear elliptic system is simple in a suitable sense and isolated.

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

This paper establishes that the smallest eigenvalue of the de Thélín quasilinear elliptic system is simple in a suitable sense and also isolated. It variationally characterizes a sequence of eigenvalues by means of a deformation lemma for C¹ submanifolds. New Landesman-Lazer type conditions are shown to ensure the existence of weak solutions to the resonant quasilinear elliptic system. These findings matter for analyzing the spectrum of quasilinear operators and solving nonlinear problems at resonance.

Core claim

The smallest eigenvalue λ₁ of the eigenvalue problem for a quasilinear elliptic system introduced by de Thélín is not only simple in a suitable sense but also isolated. A sequence of eigenvalues {λ_k} is characterized variationally using a deformation lemma for C¹ submanifolds. The existence of a weak solution is proved for the quasilinear elliptic system in resonance around λ₁ under new Landesman-Lazer type conditions.

What carries the argument

The de Thélín quasilinear eigenvalue problem together with the variational characterization of its eigenvalues via the deformation lemma for C¹ submanifolds.

If this is right

The isolation of λ₁ separates it from the rest of the spectrum and supports the use of minimax methods for higher eigenvalues.
The new Landesman-Lazer conditions extend previous existence results to a broader class of nonlinearities for resonant problems.
Simplicity of λ₁ in the suitable sense implies that the associated eigenfunction is unique up to scaling and sign changes can be controlled.

Where Pith is reading between the lines

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

The method may generalize to systems with different growth conditions or to higher-order quasilinear operators.
Applications to models in nonlinear diffusion or reaction-diffusion systems could benefit from these existence criteria.
Further study might explore whether these conditions are also necessary for solvability.

Load-bearing premise

The deformation lemma for C¹ submanifolds applies directly without regularity or compactness problems in the quasilinear functional setting.

What would settle it

A specific quasilinear system where λ₁ fails to be isolated or a resonant problem with the Landesman-Lazer conditions satisfied but no weak solution exists would refute the results.

read the original abstract

In this paper we prove that the smallest eigenvalue $\lambda_1$ of the eigenvalue problem for a quasilinear elliptic systems introduced by de Th\'elin in \cite{DT}, is not only simple (in a suitable sense), but also isolated. Moreover, we characterize variationally a sequence $\{\lambda_k\}_k$ of eigenvalues, taking into account a suitable deformation lemma for $C^1$ submanifolds proved in \cite{BON}. Furthermore we prove the existence of a weak solution for a quasilinear elliptic systems in resonance around $\lambda_1$, under new sufficient Landesman-Lazer type conditions, extending the results by Arcoya and Orsina \cite{AO}.

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 proves that λ1 is isolated for the de Thélín quasilinear system and supplies new Landesman-Lazer conditions for resonance existence.

read the letter

The main takeaway is that λ1 for this de Thélín quasilinear elliptic system is isolated, and new Landesman-Lazer conditions are given that produce a weak solution at resonance around that value. They also variationally characterize a sequence of eigenvalues using the deformation lemma from BON. This extends the earlier existence result of Arcoya and Orsina to the quasilinear setting. The isolation argument and the tailored conditions are the actual additions; the rest follows the standard min-max route once the deformation tool is in place. The setup looks consistent with what is already known for these systems, and the authors are clear about where they rely on the cited lemma. The soft spot is the direct applicability of the BON deformation lemma. Quasilinear functionals are typically only C1 in W^{1,p} or similar spaces, and the natural constraint set may not be a C1 submanifold without extra checks on coefficients or growth. If the Palais-Smale condition or the manifold regularity fails, the whole sequence of critical values and the isolation claim rest on shaky ground. The paper presumably verifies this, but it is the part a referee should examine first. This work is for specialists in nonlinear elliptic PDEs who already know the semilinear resonance literature. A reader interested in extending spectral theory to quasilinear systems will find the new conditions useful. It is narrow but technically focused, so it deserves a serious referee who can check the regularity details and the compactness arguments. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The paper claims that the smallest eigenvalue λ₁ of the de Thélín quasilinear elliptic eigenvalue problem is simple (in a suitable sense) and isolated. It variationally characterizes a sequence {λ_k} of eigenvalues by invoking a deformation lemma for C¹ submanifolds from [BON]. It further establishes existence of weak solutions to the resonant quasilinear system under new Landesman-Lazer type conditions, extending results of Arcoya and Orsina.

Significance. If the central claims hold, the work strengthens the spectral theory for quasilinear elliptic systems by proving isolation of λ₁ and supplying new sufficient conditions for solvability at resonance. The variational characterization and explicit Landesman-Lazer extensions constitute concrete advances that could be applied to other resonant problems in nonlinear analysis.

major comments (2)

[§3] §3, the variational characterization of {λ_k}: the proof applies the deformation lemma of [BON] to the constraint manifold without an explicit verification that the quasilinear energy functional is C¹ on that manifold or that the Palais-Smale condition holds under the growth assumptions of the de Thélín system; this step is load-bearing for both the isolation of λ₁ and the subsequent existence argument.
[§4] §4, Theorem 4.2 (Landesman-Lazer existence): the argument uses the isolation of λ₁ to construct a suitable linking geometry; if the isolation proof rests on an unverified application of [BON], the existence result requires an independent justification or a direct compactness argument that does not rely on the full min-max sequence.

minor comments (2)

[§2] Notation for the constraint manifold (e.g., the unit sphere in the appropriate Orlicz-Sobolev norm) is introduced without a dedicated preliminary subsection; a short paragraph clarifying the precise functional setting would improve readability.
[Abstract] The abstract repeats the phrase 'quasilinear elliptic systems'; a single occurrence suffices.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive major comments. We address each point below and indicate the revisions we will make to strengthen the presentation.

read point-by-point responses

Referee: [§3] §3, the variational characterization of {λ_k}: the proof applies the deformation lemma of [BON] to the constraint manifold without an explicit verification that the quasilinear energy functional is C¹ on that manifold or that the Palais-Smale condition holds under the growth assumptions of the de Thélín system; this step is load-bearing for both the isolation of λ₁ and the subsequent existence argument.

Authors: We acknowledge the validity of this observation. The manuscript relies on the deformation lemma from [BON] applied to the constraint manifold defined by the quasilinear energy functional, but does not include an explicit verification of C¹ regularity of the restricted functional or the Palais-Smale condition under the stated growth assumptions. We will revise Section 3 to add these verifications: first, confirming that the functional is C¹ on the manifold by direct computation using the growth conditions on the nonlinearity; second, proving the Palais-Smale condition via standard compactness arguments adapted to the de Thélín quasilinear structure. These additions will be placed before the application of the deformation lemma and will also support the isolation result for λ₁. revision: yes
Referee: [§4] §4, Theorem 4.2 (Landesman-Lazer existence): the argument uses the isolation of λ₁ to construct a suitable linking geometry; if the isolation proof rests on an unverified application of [BON], the existence result requires an independent justification or a direct compactness argument that does not rely on the full min-max sequence.

Authors: We agree that the linking geometry in the proof of Theorem 4.2 depends on the isolation of λ₁. To provide an independent justification, we will revise the existence argument to include a direct compactness analysis for Palais-Smale sequences at the resonant level. This will use the new Landesman-Lazer conditions to derive a priori bounds on solutions, followed by a compactness argument that exploits the quasilinear structure without invoking the full min-max characterization of {λ_k}. The revised proof will thus stand even if the variational sequence is treated separately, while still referencing the isolation result once the verifications from Section 3 are incorporated. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior work but central claims remain independent

full rationale

The paper proves simplicity and isolation of λ₁ for the de Thélín quasilinear system and variationally characterizes the sequence {λ_k} by invoking a deformation lemma from the external reference [BON]. It extends existence results under new Landesman-Lazer conditions from the cited prior work [AO]. While [AO] involves author overlap with the present paper, this citation supports an extension rather than serving as the sole or load-bearing justification that reduces the new claims to self-referential inputs. No derivation step reduces by the paper's own equations to a fitted quantity or self-definition, and the cited lemma is treated as an independent mathematical tool.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard functional-analytic assumptions for quasilinear operators and on the applicability of the cited deformation lemma; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption The deformation lemma for C¹ submanifolds proved in [BON] holds in the functional setting of the quasilinear system.
Invoked to characterize the sequence of eigenvalues variationally.

pith-pipeline@v0.9.0 · 5650 in / 1239 out tokens · 31858 ms · 2026-05-21T15:54:55.211942+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

we characterize variationally a sequence {λ_k} of eigenvalues, taking into account a suitable deformation lemma for C^1 submanifolds proved in [10]
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

λ_1 := inf_{z∈Σ} Φ(z) ... λ_k := 1/c_k with c_k = sup min Ψ on odd maps from S^{k-1}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[1]

Ahmad, A

S. Ahmad, A. C. Lazer, J. L. Paul, Elementary critical point theory and perturbations of elliptic boundary value problems at resonance, Indiana Univ. Math. J.25(1976), no. 10, 933–944

work page 1976
[2]

Allegretto, H

W. Allegretto, H. Y. Xi, A Picone’s identity for thep-Laplacian and applications, Nonlinear Anal.32(1998), no. 7, 819–830

work page 1998
[3]

Ambrosetti, G

A. Ambrosetti, G. Mancini, Existence and multiplicity results for nonlinear elliptic problems with linear part at resonance: the case of the simple eigenvalue, J. Differential Equations 28(1978), no. 2, 220–245

work page 1978
[4]

Anane, J.-P

A. Anane, J.-P. Gossez, Strongly nonlinear elliptic problems near resonance: a variational approach, Comm. Partial Differential Equations15(1990), no. 8, 1141–1159

work page 1990
[5]

Anane, Simplicit´ e et isolation de la premi` ere valeur propre dup-Laplacien avec poids, C

A. Anane, Simplicit´ e et isolation de la premi` ere valeur propre dup-Laplacien avec poids, C. R. Acad. Sci. Paris S´ er. I Math.305(1987), no. 16, 725–728

work page 1987
[6]

Arcoya, J

D. Arcoya, J. L. G´ amez, Bifurcation theory and related problems: anti-maximum principle and resonance, Comm. Partial Differential Equations26(2001), no. 9–10, 1879–1911

work page 2001
[7]

Arcoya, L

D. Arcoya, L. Orsina, Landesman–Lazer conditions and quasilinear elliptic equations, Non- linear Anal.28(1997), no. 10, 1623–1632

work page 1997
[8]

Boccardo, D

L. Boccardo, D. G. de Figueiredo, Some remarks on a system of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl.9(2002), no. 3, 309–323

work page 2002
[9]

Boccardo, P

L. Boccardo, P. Dr´ abek, M. Kuˇ cera, Landesman–Lazer conditions for strongly nonlinear boundary value problems, Comment. Math. Univ. Carolin.30(1989), no. 3, 411–427

work page 1989
[10]

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. 1, 339–359

work page 1993
[11]

de Th´ elin, First eigenvalue of a nonlinear elliptic system, C

F. de Th´ elin, First eigenvalue of a nonlinear elliptic system, C. R. Acad. Sci. Paris S´ er. I Math.311(1990), no. 10, 603–606

work page 1990
[12]

Deimling, Nonlinear Functional Analysis, Springer–Verlag, 1985

K. Deimling, Nonlinear Functional Analysis, Springer–Verlag, 1985. 26 D. ARCOYA, N. BORGIA, AND S. CINGOLANI

work page 1985
[13]

J. I. D´ ıaz, J. E. Sa´ a, Existence et unicit´ e de solutions positives pour certaines ´ equations elliptiques quasilin´ eaires. C. R. Acad. Sci. Paris S´ er. I Math.305(1987), no. 12, 521–524

work page 1987
[14]

Dr´ abek, S

P. Dr´ abek, S. B. Robinson, Resonance problems for thep-Laplacian, J. Funct. Anal.169 (1999), no. 1, 189–200

work page 1999
[15]

Hess, On a theorem by Landesman and Lazer, Indiana Univ

P. Hess, On a theorem by Landesman and Lazer, Indiana Univ. Math. J.23(1974), no. 9, 827–829

work page 1974
[16]

E. M. Landesman, A. C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech.19(1970), no. 7, 609–623

work page 1970
[17]

Perera, R

K. Perera, R. P. Agarwal, D. O’Regan, Morse Theoretic Aspects ofp-Laplacian Type Op- erators, Amer. Math. Soc., 2010

work page 2010
[18]

P. H. Rabinowitz, Some minimax theorems and applications to nonlinear partial differential equations, Nonlinear Anal.2(1978), 161–177

work page 1978
[19]

N. M. Stavrakakis, N. B. Zographopoulos, Bifurcation results for quasilinear elliptic systems, Adv. Differential Equations8(2003), no. 3, 315–336

work page 2003
[20]

Struwe, Variational Methods, Springer–Verlag, 2000

M. Struwe, Variational Methods, Springer–Verlag, 2000

work page 2000
[21]

Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm

P. Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. Partial Differential Equations8(1983), no. 7, 773–817

work page 1983
[22]

J. L. V´ azquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim.12(1984), 191–202. Departamento de An ´alisis Matem ´atico, Campus Fuentenueva S/N, Universidad de Granada, 18071 Granada, Spain Email address:darcoya@ugr.es Dipartimento di Matematica, Universit`a degli Studi di Bari Aldo Moro, Via Orabona 4, 70125 Bari...

work page 1984

[1] [1]

Ahmad, A

S. Ahmad, A. C. Lazer, J. L. Paul, Elementary critical point theory and perturbations of elliptic boundary value problems at resonance, Indiana Univ. Math. J.25(1976), no. 10, 933–944

work page 1976

[2] [2]

Allegretto, H

W. Allegretto, H. Y. Xi, A Picone’s identity for thep-Laplacian and applications, Nonlinear Anal.32(1998), no. 7, 819–830

work page 1998

[3] [3]

Ambrosetti, G

A. Ambrosetti, G. Mancini, Existence and multiplicity results for nonlinear elliptic problems with linear part at resonance: the case of the simple eigenvalue, J. Differential Equations 28(1978), no. 2, 220–245

work page 1978

[4] [4]

Anane, J.-P

A. Anane, J.-P. Gossez, Strongly nonlinear elliptic problems near resonance: a variational approach, Comm. Partial Differential Equations15(1990), no. 8, 1141–1159

work page 1990

[5] [5]

Anane, Simplicit´ e et isolation de la premi` ere valeur propre dup-Laplacien avec poids, C

A. Anane, Simplicit´ e et isolation de la premi` ere valeur propre dup-Laplacien avec poids, C. R. Acad. Sci. Paris S´ er. I Math.305(1987), no. 16, 725–728

work page 1987

[6] [6]

Arcoya, J

D. Arcoya, J. L. G´ amez, Bifurcation theory and related problems: anti-maximum principle and resonance, Comm. Partial Differential Equations26(2001), no. 9–10, 1879–1911

work page 2001

[7] [7]

Arcoya, L

D. Arcoya, L. Orsina, Landesman–Lazer conditions and quasilinear elliptic equations, Non- linear Anal.28(1997), no. 10, 1623–1632

work page 1997

[8] [8]

Boccardo, D

L. Boccardo, D. G. de Figueiredo, Some remarks on a system of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl.9(2002), no. 3, 309–323

work page 2002

[9] [9]

Boccardo, P

L. Boccardo, P. Dr´ abek, M. Kuˇ cera, Landesman–Lazer conditions for strongly nonlinear boundary value problems, Comment. Math. Univ. Carolin.30(1989), no. 3, 411–427

work page 1989

[10] [10]

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. 1, 339–359

work page 1993

[11] [11]

de Th´ elin, First eigenvalue of a nonlinear elliptic system, C

F. de Th´ elin, First eigenvalue of a nonlinear elliptic system, C. R. Acad. Sci. Paris S´ er. I Math.311(1990), no. 10, 603–606

work page 1990

[12] [12]

Deimling, Nonlinear Functional Analysis, Springer–Verlag, 1985

K. Deimling, Nonlinear Functional Analysis, Springer–Verlag, 1985. 26 D. ARCOYA, N. BORGIA, AND S. CINGOLANI

work page 1985

[13] [13]

J. I. D´ ıaz, J. E. Sa´ a, Existence et unicit´ e de solutions positives pour certaines ´ equations elliptiques quasilin´ eaires. C. R. Acad. Sci. Paris S´ er. I Math.305(1987), no. 12, 521–524

work page 1987

[14] [14]

Dr´ abek, S

P. Dr´ abek, S. B. Robinson, Resonance problems for thep-Laplacian, J. Funct. Anal.169 (1999), no. 1, 189–200

work page 1999

[15] [15]

Hess, On a theorem by Landesman and Lazer, Indiana Univ

P. Hess, On a theorem by Landesman and Lazer, Indiana Univ. Math. J.23(1974), no. 9, 827–829

work page 1974

[16] [16]

E. M. Landesman, A. C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech.19(1970), no. 7, 609–623

work page 1970

[17] [17]

Perera, R

K. Perera, R. P. Agarwal, D. O’Regan, Morse Theoretic Aspects ofp-Laplacian Type Op- erators, Amer. Math. Soc., 2010

work page 2010

[18] [18]

P. H. Rabinowitz, Some minimax theorems and applications to nonlinear partial differential equations, Nonlinear Anal.2(1978), 161–177

work page 1978

[19] [19]

N. M. Stavrakakis, N. B. Zographopoulos, Bifurcation results for quasilinear elliptic systems, Adv. Differential Equations8(2003), no. 3, 315–336

work page 2003

[20] [20]

Struwe, Variational Methods, Springer–Verlag, 2000

M. Struwe, Variational Methods, Springer–Verlag, 2000

work page 2000

[21] [21]

Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm

P. Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. Partial Differential Equations8(1983), no. 7, 773–817

work page 1983

[22] [22]

J. L. V´ azquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim.12(1984), 191–202. Departamento de An ´alisis Matem ´atico, Campus Fuentenueva S/N, Universidad de Granada, 18071 Granada, Spain Email address:darcoya@ugr.es Dipartimento di Matematica, Universit`a degli Studi di Bari Aldo Moro, Via Orabona 4, 70125 Bari...

work page 1984