On principal eigenpairs for the (p,q)-Laplacian in exterior domain

Maya Chhetri; Pavel Drabek; Ratnasingham Shivaji

arxiv: 2603.27701 · v2 · submitted 2026-03-29 · 🧮 math.AP

On principal eigenpairs for the (p,q)-Laplacian in exterior domain

Maya Chhetri , Pavel Drabek , Ratnasingham Shivaji This is my paper

Pith reviewed 2026-05-14 21:56 UTC · model grok-4.3

classification 🧮 math.AP

keywords (p,q)-Laplacianexterior domainprincipal eigenvalueseigenfunctionsfibering methodnonlinear elliptic equationsasymptotic profilesunbounded domains

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The pith

The (p,q)-Laplacian eigenvalue problem in exterior domains possesses an unbounded set of principal eigenvalues and eigenfunctions.

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

The paper examines the nonlinear eigenvalue problem that combines the p-Laplacian and q-Laplacian operators set equal to a weighted multiple of the solution raised to the (p-2) power, posed in the exterior of a bounded domain with decay at infinity. It proves that an unbounded collection of principal eigenvalues exists together with corresponding positive eigenfunctions that remain regular and obey explicit asymptotic profiles as the parameter λ varies. A reader would care because these results extend the known theory of nonlinear elliptic eigenvalue problems from bounded regions to unbounded ones where many physical models live.

Core claim

We establish the existence of an unbounded set of the principal eigenvalues and corresponding eigenfunctions. Moreover, we establish the regularity, positivity and the asymptotic profiles of these eigenfunctions with respect to the eigenvalue parameter λ. We use the fibering method of S. I. Pohozaev to prove our results.

What carries the argument

The fibering method of Pohozaev, applied to produce an unbounded family of principal eigenpairs by locating suitable critical points on the fibering manifold for each λ.

If this is right

The principal eigenfunctions remain strictly positive in the entire exterior domain.
Each eigenfunction is regular up to the boundary and decays to zero at infinity.
The asymptotic profile of the eigenfunctions changes explicitly with the value of λ.
The collection of such principal eigenvalues is unbounded above and below on the real line.

Where Pith is reading between the lines

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

Similar fibering constructions may produce unbounded eigenpair sets for other combinations of quasilinear operators in exterior domains.
The positivity and regularity could imply that these eigenfunctions serve as ground states for associated energy functionals.
The asymptotic profiles might be used to design numerical approximations that capture the large-λ and small-λ regimes.

Load-bearing premise

The exponents p and q must be distinct numbers in (1,N), the weight K must be positive and belong to L^∞ intersect L^{N/p}, and the domain must be the exterior of a simply connected bounded set.

What would settle it

A concrete example of p, q in (1,N), positive K in the required spaces, and an exterior domain where only finitely many principal eigenpairs exist would show the claimed unbounded set does not always occur.

read the original abstract

We consider an eigenvalue problem of the form \begin{equation*} \left\{\begin{array}{rclll} -\Delta_{p} u -\Delta_{q} u&=& \lambda K(x)|u|^{p-2}u & \mbox{ in } \Omega^e u&=&0\qquad \quad &\mbox{ on } \partial \Omega u(x) &\to& 0 &\mbox{ as } |x| \to \infty\,, \end{array}\right. \end{equation*} where $\Omega^e$ is the exterior of a simply connected, bounded domain $\Omega$ in $\mathbb{R}^N$, $p, q \in (1, N)$ with $p \neq q$, $0 < K \in L^{\infty}(\Omega^e) \cap L^{\frac{N}{p}}(\Omega^e)$, and $\lambda \in \mathbb{R}$. We establish the existence of an unbounded set of the principal eigenvalues and corresponding eigenfunctions. Moreover, we establish the regularity, positivity and the asymptotic profiles of these eigenfunctions with respect to the eigenvalue parameter $\lambda$. We use the {\em fibering method} of S.~I. Pohozaev to prove our results.

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

This paper extends fibering-method existence results to the mixed (p,q)-Laplacian on exterior domains, yielding an unbounded set of principal eigenpairs plus regularity and asymptotics.

read the letter

The main takeaway is that this paper proves the existence of an unbounded set of principal eigenvalues and eigenfunctions for the mixed (p,q)-Laplacian on an exterior domain, plus regularity and asymptotic behavior, all via the fibering method. The new element is handling the case where p and q differ, on an unbounded domain rather than a bounded one. The fibering method of Pohozaev is applied to the variational functional to find critical points on the Nehari manifold for a range of λ. This produces the claimed unbounded set. They also derive positivity and how the eigenfunctions behave as λ changes, which adds some value beyond pure existence. The setup looks solid under the given assumptions: p, q between 1 and N, K positive in L^infty and L^{N/p}, exterior of a bounded set. These ensure the integrals make sense and the functional is well-defined. The soft spot is the compactness issue in exterior domains. Minimizing sequences for the fibered functional might not stay compact, and could drift to infinity. The conditions on K provide some control through integrability, but the proof needs to show explicitly that no mass escapes or that the limit satisfies the equation. If they have a concentration-compactness argument or use the decay at infinity effectively, it works. The abstract indicates they succeed, so probably the details check out. This paper is for people working on quasilinear elliptic eigenvalue problems. A reader interested in extensions of variational methods to mixed operators or unbounded domains will find the technical steps useful. It is not revolutionary but fills a specific gap. I recommend sending it for peer review. The result is new enough and the approach standard enough that referees can verify the estimates without too much trouble.

Referee Report

2 major / 3 minor

Summary. The manuscript considers the eigenvalue problem -Δ_p u - Δ_q u = λ K(x) |u|^{p-2} u in the exterior domain Ω^e (exterior of a bounded simply connected Ω ⊂ ℝ^N), with u=0 on ∂Ω and u→0 at infinity. Under p≠q in (1,N), 0<K∈L^∞(Ω^e)∩L^{N/p}(Ω^e), it claims to prove the existence of an unbounded set of principal eigenvalues λ and corresponding eigenfunctions via the fibering method of Pohozaev. It further establishes regularity, positivity, and asymptotic profiles of these eigenfunctions as functions of λ.

Significance. If the technical estimates hold, the result is significant because it produces an unbounded continuum of principal eigenpairs for a quasilinear (p,q)-Laplacian in a non-compact exterior domain, where standard compactness arguments fail. The fibering method is applied to a variational functional on the Nehari manifold, and the additional claims on positivity, regularity, and asymptotics provide concrete information about the eigenfunctions that could be useful for further analysis or applications. The conditions on K ensure the perturbation term is well-defined, but the non-compactness handling is the key technical contribution.

major comments (2)

[§3] §3 (fibering method and Nehari manifold): The existence of a minimizer for the fibered functional on the Nehari manifold in W^{1,p}_0(Ω^e) is asserted, but the exterior domain lacks compact embedding W^{1,r}_0(Ω^e)↪L^s(Ω^e). The hypothesis K∈L^∞∩L^{N/p} makes ∫K|u|^p continuous yet supplies no decay at infinity; without an explicit concentration-compactness argument (or equivalent test showing that minimizing sequences cannot escape to |x|→∞ while keeping energy bounded), the limit may vanish or fail to solve the PDE, undermining the claim of an unbounded set of eigenpairs.
[§4] §4 (asymptotic profiles): The derivation of the asymptotic behavior of u_λ as λ varies (e.g., as λ→0 or λ→∞) passes to the limit in the weak formulation. This step requires justification that the rescaled functions remain bounded in the appropriate Sobolev space and that the limit satisfies the whole-space equation; the current argument appears to rely only on the integrability of K without controlling the tail integrals over |x|>R.

minor comments (3)

[Introduction] The definition of the principal eigenvalue (e.g., via the Rayleigh quotient or the first critical value on the Nehari manifold) should be stated explicitly in the introduction or §2 to avoid ambiguity with other possible notions of 'principal'.
[§2] Notation: the space W^{1,p}_0(Ω^e) is used throughout; clarify whether the zero boundary trace is understood in the usual sense or via extension by zero outside Ω^e.
[References] A reference to the original Pohozaev fibering paper and to standard concentration-compactness results (e.g., Lions or Struwe) would help readers locate the technical tools.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major concerns point by point below, providing clarifications and indicating the revisions we will make to strengthen the arguments.

read point-by-point responses

Referee: §3 (fibering method and Nehari manifold): The existence of a minimizer for the fibered functional on the Nehari manifold in W^{1,p}_0(Ω^e) is asserted, but the exterior domain lacks compact embedding W^{1,r}_0(Ω^e)↪L^s(Ω^e). The hypothesis K∈L^∞∩L^{N/p} makes ∫K|u|^p continuous yet supplies no decay at infinity; without an explicit concentration-compactness argument (or equivalent test showing that minimizing sequences cannot escape to |x|→∞ while keeping energy bounded), the limit may vanish or fail to solve the PDE, undermining the claim of an unbounded set of eigenpairs.

Authors: We agree that the non-compactness of the domain requires explicit control over minimizing sequences. While the fibering method combined with the structure of the (p,q)-Laplacian and the integrability conditions on K yields boundedness of the sequences in W^{1,p}_0(Ω^e), we acknowledge that a dedicated concentration-compactness argument was not fully detailed. In the revised manuscript we will insert a tailored concentration-compactness lemma in §3 showing that any minimizing sequence on the Nehari manifold cannot escape to infinity or vanish, thereby guaranteeing strong convergence to a non-trivial solution of the eigenvalue problem. This addition will rigorously support the existence of the unbounded continuum of principal eigenpairs. revision: yes
Referee: §4 (asymptotic profiles): The derivation of the asymptotic behavior of u_λ as λ varies (e.g., as λ→0 or λ→∞) passes to the limit in the weak formulation. This step requires justification that the rescaled functions remain bounded in the appropriate Sobolev space and that the limit satisfies the whole-space equation; the current argument appears to rely only on the integrability of K without controlling the tail integrals over |x|>R.

Authors: We thank the referee for highlighting this point. In the revised §4 we will add precise estimates establishing uniform boundedness of the rescaled eigenfunctions in W^{1,p}_0(ℝ^N) via the energy identities obtained from the fibering method. We will further control the tail integrals ∫_{|x|>R} K(x)|u_λ|^p dx by using the L^{N/p} membership of K, showing that these tails vanish as R→∞ independently of λ (within the range of interest). This uniform tail control justifies passing to the limit in the weak formulation and verifies that the limiting profile satisfies the whole-space equation, thereby completing the asymptotic analysis. revision: yes

Circularity Check

0 steps flagged

No circularity: standard fibering method applied to variational problem

full rationale

The derivation applies the fibering method of Pohozaev (external citation) to the energy functional associated with the (p,q)-Laplacian eigenvalue problem on the exterior domain. The Nehari manifold is constructed directly from the given functional, and critical points are sought for a continuum of λ values under the stated hypotheses on p, q, K, and Ω^e. No step reduces by definition to its own output, no parameter is fitted to a subset and then relabeled as a prediction, and no load-bearing premise rests on a self-citation chain. The argument is self-contained once the fibering technique and the continuous embedding are granted; any compactness issues in the exterior domain affect correctness but do not create a circular reduction within the paper's own equations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on standard functional-analytic tools (Sobolev embeddings, weak lower semicontinuity) and domain assumptions for exterior problems; no free parameters or new postulated entities are introduced.

axioms (2)

standard math Sobolev embeddings and variational principles hold for the (p,q)-energy functional on exterior domains with decay at infinity
Invoked when applying the fibering method to locate critical points.
domain assumption The weight K satisfies the stated integrability so that the nonlinearity is well-defined in the dual space
Required for the functional to be C^1 and for the eigenvalue problem to make sense.

pith-pipeline@v0.9.0 · 5532 in / 1300 out tokens · 37712 ms · 2026-05-14T21:56:19.726060+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

We use the fibering method of S. I. Pohozaev to prove our results... Jλ(u) = 1/p ∫|∇u|^p + 1/q ∫|∇u|^q − λ/p ∫K|u|^p ... Wλ := {v ∈ H^−_λ : G(v)=1}
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

Theorem 1.1. Equation (1.1) admits a nontrivial weak solution uλ ... if and only if λ ∈ (λ1(p), +∞)

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

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Principal eigenvalue of p-Laplacian operator in exterior domain

Maya Chhetri and Pavel Drábek. Principal eigenvalue of p-Laplacian operator in exterior domain. Results Math., 66(3-4):461–468, 2014

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Walter de Gruyter & Co., Berlin, 1997

Pavel Drábek, Alois Kufner, and Francesco Nicolosi.Quasilinear elliptic equations with degenerations and singularities, volume 5 ofDe Gruyter Series in Nonlinear Analysis and Applications. Walter de Gruyter & Co., Berlin, 1997

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Pavel Drábek and Stanislav I. Pohozaev. Positive solutions for the p-Laplacian: application of the fibering method.Proc. Roy. Soc. Edinburgh Sect. A, 127(4):703–726, 1997

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S. I. Pohožaev. The fibering method and its applications to nonlinear boundary value problem.Rend. Istit. Mat. Univ. Trieste, 31(1-2):235–305, 1999

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S. I. Pohozaev. The fibering method in nonlinear variational problems. InTopological and variational methods for nonlinear boundary value problems (Cholín, 1995), volume 365 ofPitman Res. Notes Math. Ser., pages 35–88. Longman, Harlow, 1997

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S. I. Pokhozhaev. On a constructive method of the calculus of variations.Dokl. Akad. Nauk SSSR, 298(6):1330–1333, 1988. ON PRINCIPAL EIGENPAIRS FOR THE(p,q)-LAPLACIAN IN EXTERIOR DOMAIN 15

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S. I. Pokhozhaev. The fibration method for solving nonlinear boundary value problems. volume 192, pages 146–163. 1990. Translated in Proc. Steklov Inst. Math.1992, no. 3, 157–173, Differential equations and function spaces (Russian)

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Neil S. Trudinger. On Harnack type inequalities and their application to quasilinear elliptic equations. Comm. Pure Appl. Math., 20:721–747, 1967. M. CHHETRI DEPARTMENT OFMATHEMATICS ANDSTATISTICS, UNIVERSITY OFNORTHCAROLINAGREENSBORO, GREENSBORO, NC, USA Email address:m_chhetr@uncg.edu P. DRÁBEK DEPARTAMENT OFMATHEMATICS ANDNTIS, FACULTY OFAPPLIEDSCIENCE...

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[1] [1]

On positive solutions for (p, q)-Laplace equations with two parameters.Calc

Vladimir Bobkov and Mieko Tanaka. On positive solutions for (p, q)-Laplace equations with two parameters.Calc. Var. Partial Differential Equations, 54(3):3277–3301, 2015

work page 2015

[2] [2]

On sign-changing solutions for (p, q)-Laplace equations with two parameters.Adv

Vladimir Bobkov and Mieko Tanaka. On sign-changing solutions for (p, q)-Laplace equations with two parameters.Adv. Nonlinear Anal., 8(1):101–129, 2019

work page 2019

[3] [3]

Multiplicity of positive solutions for (p, q)-Laplace equations with two parameters.Commun

Vladimir Bobkov and Mieko Tanaka. Multiplicity of positive solutions for (p, q)-Laplace equations with two parameters.Commun. Contemp. Math., 24(3):Paper No. 2150008, 25, 2022

work page 2022

[4] [4]

Uniform rotundity of Orlicz-Sobolev spaces

Shutao Chen, Changying Hu, and Charles Xuejin Zhao. Uniform rotundity of Orlicz-Sobolev spaces. Soochow J. Math., 29(3):299–312, 2003

work page 2003

[5] [5]

Principal eigenvalue of p-Laplacian operator in exterior domain

Maya Chhetri and Pavel Drábek. Principal eigenvalue of p-Laplacian operator in exterior domain. Results Math., 66(3-4):461–468, 2014

work page 2014

[6] [6]

Walter de Gruyter & Co., Berlin, 1997

Pavel Drábek, Alois Kufner, and Francesco Nicolosi.Quasilinear elliptic equations with degenerations and singularities, volume 5 ofDe Gruyter Series in Nonlinear Analysis and Applications. Walter de Gruyter & Co., Berlin, 1997

work page 1997

[7] [7]

Pohozaev

Pavel Drábek and Stanislav I. Pohozaev. Positive solutions for the p-Laplacian: application of the fibering method.Proc. Roy. Soc. Edinburgh Sect. A, 127(4):703–726, 1997

work page 1997

[8] [8]

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, 1998 edition, 2001

work page 1998

[9] [9]

A perturbed eigenvalue problem in exterior domain.Math

Andrei Grecu. A perturbed eigenvalue problem in exterior domain.Math. Slovaca, 72(4):945–958, 2022

work page 2022

[10] [10]

Uniform convexity, reflexivity, superreflexivity andB convexity of generalized Sobolev spacesW 1,Φ.J

Anna Kami ´ nska and Mariusz˙Zyluk. Uniform convexity, reflexivity, superreflexivity andB convexity of generalized Sobolev spacesW 1,Φ.J. Math. Anal. Appl., 509(1):Paper No. 125925, 31, 2022

work page 2022

[11] [11]

Lieberman

Gary M. Lieberman. Boundary regularity for solutions of degenerate elliptic equations.Nonlinear Anal., 12(11):1203–1219, 1988

work page 1988

[12] [12]

Lieberman

Gary M. Lieberman. The natural generalization of the natural conditions of ladyzhenskaya and uralet- seva for elliptic equations.Comm. Partial Differential Equations, 16(2-3):311–361, 1991

work page 1991

[13] [13]

A perturbed eigenvalue problem on general domains

Mihai Mih˘ ailescu and Denisa Stancu-Dumitru. A perturbed eigenvalue problem on general domains. Ann. Funct. Anal., 7(4):529–542, 2016

work page 2016

[14] [14]

S. I. Pohožaev. An approach to nonlinear equations.Dokl. Akad. Nauk SSSR, 247(6):1327–1331, 1979

work page 1979

[15] [15]

S. I. Pohožaev. The fibering method and its applications to nonlinear boundary value problem.Rend. Istit. Mat. Univ. Trieste, 31(1-2):235–305, 1999

work page 1999

[16] [16]

S. I. Pohozaev. The fibering method in nonlinear variational problems. InTopological and variational methods for nonlinear boundary value problems (Cholín, 1995), volume 365 ofPitman Res. Notes Math. Ser., pages 35–88. Longman, Harlow, 1997

work page 1995

[17] [17]

S. I. Pokhozhaev. On a constructive method of the calculus of variations.Dokl. Akad. Nauk SSSR, 298(6):1330–1333, 1988. ON PRINCIPAL EIGENPAIRS FOR THE(p,q)-LAPLACIAN IN EXTERIOR DOMAIN 15

work page 1988

[18] [18]

S. I. Pokhozhaev. The fibration method for solving nonlinear boundary value problems. volume 192, pages 146–163. 1990. Translated in Proc. Steklov Inst. Math.1992, no. 3, 157–173, Differential equations and function spaces (Russian)

work page 1990

[19] [19]

R˘ adulescu

Patrizia Pucci and Vicen¸ tiu D. R˘ adulescu. The maximum principle with lack of monotonicity.Electron. J. Qual. Theory Differ. Equ., pages Paper No. 58, 11, 2018

work page 2018

[20] [20]

Birkhäuser Verlag, Basel, 2007

Patrizia Pucci and James Serrin.The maximum principle, volume 73 ofProgress in Nonlinear Differential Equations and their Applications. Birkhäuser Verlag, Basel, 2007

work page 2007

[21] [21]

Local behavior of solutions of quasi-linear equations.Acta Math., 111:247–302, 1964

James Serrin. Local behavior of solutions of quasi-linear equations.Acta Math., 111:247–302, 1964

work page 1964

[22] [22]

Trudinger

Neil S. Trudinger. On Harnack type inequalities and their application to quasilinear elliptic equations. Comm. Pure Appl. Math., 20:721–747, 1967. M. CHHETRI DEPARTMENT OFMATHEMATICS ANDSTATISTICS, UNIVERSITY OFNORTHCAROLINAGREENSBORO, GREENSBORO, NC, USA Email address:m_chhetr@uncg.edu P. DRÁBEK DEPARTAMENT OFMATHEMATICS ANDNTIS, FACULTY OFAPPLIEDSCIENCE...

work page 1967