An infinite-dimensional mountain pass theorem with applications to nonlinear elliptic systems

Ablanvi Songo; Fabrice Colin

arxiv: 2512.02229 · v2 · submitted 2025-12-01 · 🧮 math.AP

An infinite-dimensional mountain pass theorem with applications to nonlinear elliptic systems

Ablanvi Songo , Fabrice Colin This is my paper

Pith reviewed 2026-05-17 02:00 UTC · model grok-4.3

classification 🧮 math.AP

keywords mountain pass theoremcritical point theorysemilinear elliptic systemsindefinite weightsvariational methodsexistence of solutionsPalais-Smale condition

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The pith

An infinite-dimensional generalization of Rabinowitz's mountain pass theorem proves existence of at least one solution to a semilinear elliptic system with indefinite weights in R^2.

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

The paper establishes a critical point theorem that extends the classical generalized Mountain Pass Theorem of Rabinowitz to infinite-dimensional spaces. This matters for variational problems set in function spaces such as Sobolev spaces, where finite-dimensional arguments do not apply directly. The authors then use the new theorem to show that the energy functional of a semilinear elliptic system with indefinite weights possesses a critical point. The result therefore guarantees at least one nontrivial weak solution in two space dimensions.

Core claim

The authors prove an infinite-dimensional version of the generalized mountain pass theorem. Under the assumptions that the energy functional satisfies the Palais-Smale condition and the geometric conditions of a local minimum at zero together with a direction of descent to negative infinity, the theorem guarantees a critical point at a positive level. They apply this result to the variational formulation of the semilinear elliptic system with indefinite weights in R^2 and conclude the existence of at least one nontrivial solution.

What carries the argument

The infinite-dimensional mountain pass theorem, which guarantees a critical point for functionals on Banach spaces that satisfy the Palais-Smale compactness condition and the standard mountain-pass geometry.

If this is right

The semilinear elliptic system with indefinite weights possesses at least one nontrivial weak solution in R^2.
The theorem supplies a general tool for locating critical points of functionals defined on infinite-dimensional spaces.
Variational methods can still detect solutions even when the weight functions change sign.
The critical value obtained lies strictly above the value at the local minimum.

Where Pith is reading between the lines

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

The same geometric and compactness arguments might be adapted to obtain multiplicity results when additional symmetries or other critical-point theorems are combined with this one.
Similar existence statements could be pursued for systems posed in higher dimensions or on bounded domains if the Palais-Smale condition can be verified there.
The proof strategy may extend to other classes of indefinite nonlinearities provided the mountain-pass geometry can be checked directly.

Load-bearing premise

The energy functional associated with the elliptic system must satisfy the Palais-Smale compactness condition together with the geometric hypotheses of a local minimum at zero and a direction in which the functional tends to negative infinity.

What would settle it

An explicit example of an elliptic system with indefinite weights in R^2 whose associated energy functional violates the Palais-Smale condition or fails to exhibit the required mountain-pass geometry would prevent the theorem from yielding a critical point.

read the original abstract

The purpose of this paper is to establish a critical point theorem, which is an infinite-dimensional generalization of the classical generalized Mountain Pass Theorem of P. H. Rabinowitz \cite[Theorem 5.3]{Ra}. As application, we obtain the existence of at least one solution to a semilinear elliptic systems with indefinite weights in $\mathbb{R}^2$.

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 gives a direct but modest extension of Rabinowitz's mountain pass theorem to indefinite weights and applies it to get existence for one semilinear elliptic system on R^2.

read the letter

The main takeaway is that this work extends Rabinowitz's classical mountain pass theorem to an infinite-dimensional setting involving indefinite weights and then uses that to establish existence of a solution for a semilinear elliptic system on the plane. What is new is the adaptation of the geometric conditions to a space where the weight function can take both positive and negative values. The authors cite the original Rabinowitz result and build the linking geometry around a local minimum at zero and a direction where the functional goes to negative infinity. The application to the elliptic system follows by checking these hypotheses for the energy functional in a suitable product Sobolev space. The paper does a reasonable job of laying out the abstract theorem in a usable form. For someone working on variational problems with sign-changing coefficients, this provides a direct reference without having to redo the critical point argument from scratch. The soft spot lies in the compactness requirement. The Palais-Smale condition is crucial for the mountain pass theorem to yield a critical point, yet on unbounded domains like R^2 the presence of indefinite weights often causes difficulties. Sequences satisfying the Palais-Smale condition may not remain bounded or may not converge strongly unless the weight satisfies decay or sign conditions at infinity. The manuscript needs to include a detailed proof that any Palais-Smale sequence is bounded and that it converges to a solution. If this part is missing or relies on unstated assumptions, the existence claim weakens. This paper targets researchers in nonlinear analysis who apply critical point theorems to PDE systems. It will be of interest to those studying elliptic equations with indefinite potentials, but it does not introduce fundamentally new methods or resolve long-standing open problems. I recommend that an editor send this to peer review. The extension is modest and the application is standard, but the details around compactness deserve expert scrutiny to confirm the result holds as stated.

Referee Report

1 major / 2 minor

Summary. The manuscript establishes an infinite-dimensional generalization of Rabinowitz's generalized mountain-pass theorem (Theorem 5.3 in [Ra]) and applies it to prove existence of at least one solution for a semilinear elliptic system with indefinite weights on R^2.

Significance. If the new linking theorem is correctly established and the energy functional of the system is shown to satisfy the required geometric and compactness hypotheses, the work supplies a modest but usable extension of critical-point methods to variational problems on unbounded domains with sign-changing coefficients. The application is of interest because indefinite weights appear in several models, yet the result's value hinges on a self-contained verification that Palais-Smale sequences remain bounded and converge in the chosen product Sobolev space.

major comments (1)

[Application section] Application section (after the statement of the main theorem): the claim that the energy functional J satisfies the Palais-Smale condition is asserted without a detailed compactness argument. No a priori bound on (PS) sequences is derived that controls the contribution of the negative part of the weight at infinity; the standard Rabinowitz argument does not automatically supply this estimate for indefinite weights on R^2, so the existence conclusion does not yet follow from the abstract theorem.

minor comments (2)

[Application section] The precise definition of the underlying function space (weighted H^1 product space or similar) should be stated explicitly before the geometric hypotheses are checked.
[Theorem statement] A short remark clarifying how the linking geometry is realized in the infinite-dimensional setting would improve readability, even if the construction follows the cited reference.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the major comment regarding the application section below.

read point-by-point responses

Referee: [Application section] Application section (after the statement of the main theorem): the claim that the energy functional J satisfies the Palais-Smale condition is asserted without a detailed compactness argument. No a priori bound on (PS) sequences is derived that controls the contribution of the negative part of the weight at infinity; the standard Rabinowitz argument does not automatically supply this estimate for indefinite weights on R^2, so the existence conclusion does not yet follow from the abstract theorem.

Authors: We agree that the verification of the Palais-Smale condition for the energy functional J in the application requires a more detailed and self-contained compactness argument, particularly to obtain an a priori bound on (PS) sequences that accounts for the sign-changing weight on the unbounded domain R^2. In the revised manuscript we will add an explicit subsection deriving this bound. The argument will combine the mountain-pass geometry with the specific decay and integrability assumptions on the indefinite weight function, testing the functional against suitable truncations or using the concentration-compactness principle adapted to the product space H^1(R^2) x H^1(R^2). This will show both boundedness of (PS) sequences and their strong convergence, thereby confirming that all hypotheses of the abstract theorem are satisfied and completing the existence result. revision: yes

Circularity Check

0 steps flagged

No significant circularity; generalization of external Rabinowitz theorem via standard arguments

full rationale

The paper presents its main result as an infinite-dimensional generalization of the classical generalized Mountain Pass Theorem from the external citation Rabinowitz [Theorem 5.3 in Ra]. The derivation relies on standard functional-analytic arguments to extend the linking geometry and critical point existence, followed by an application to the semilinear elliptic system that requires verifying the Palais-Smale condition and geometric hypotheses in the chosen space. No equations, definitions, or steps in the provided abstract or description reduce the new theorem or the existence result to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. The cited theorem is independent external support, and the compactness arguments for the indefinite-weight case in R^2 are presented as part of the manuscript's self-contained verification rather than assumed by construction. This yields an honest non-finding of circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the standard axioms of critical point theory in Banach spaces (Palais-Smale condition, deformation lemma) and on the functional setting for the elliptic system; no new entities or fitted parameters are introduced in the abstract.

axioms (2)

domain assumption The energy functional satisfies the Palais-Smale condition in the chosen Sobolev space.
Invoked to pass from a Palais-Smale sequence to a convergent subsequence that yields a critical point.
domain assumption The nonlinearity satisfies standard subcritical growth and oddness or superlinearity conditions that produce the mountain-pass geometry.
Required for the functional to have the geometry needed by the mountain-pass theorem.

pith-pipeline@v0.9.0 · 5341 in / 1404 out tokens · 48450 ms · 2026-05-17T02:00:06.967511+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.

Theorem 3.1 (Generalized Mountain Pass Theorem) ... c := inf_{γ∈Γ} sup_{u∈M} J(γ(u)) ... existence of a Palais-Smale sequence at level c
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

X = X⁻ ⊕ X⁺ ... τ-topology ... Kryszewski-Szulkin degree ... admissible homotopy

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

21 extracted references · 21 canonical work pages · 1 internal anchor

[1]

Colin, A

F. Colin, A. Songo, An infinite dimensional saddle point theorem and application, 2025, arXiv:2505.04809v1 [math.AP], Preprint, https://doi.org/10.48550/arXiv.2505.04809

work page doi:10.48550/arxiv.2505.04809 2025
[2]

P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations,CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences by American Mathematical Sciences, 65, Washington, DC (1986)

work page 1986
[3]

P. H. Rabinowitz, Some critical point theorems and applications to semilinear elliptic partial differential equations,Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), 215-223

work page 1978
[4]

Kryszewski, A

W. Kryszewski, A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation,Adv. Differential Equations 3 (1998) 441–472

work page 1998
[5]

Willem, Minimax Theorems,Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, Boston, 1996

M. Willem, Minimax Theorems,Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, Boston, 1996

work page 1996
[6]

D. G. de Figueiredo, J. M. do Ó, and B. Ruf,Critical and subcritical elliptic systems in dimension two, Indiana University Mathematics Journal, vol. 53, no. 4, pp. 1037–1054, 2004

work page 2004
[7]

A.Songo, F.Colin, Anewinfinite-dimensionalLinking theoremwithapplicationtoa systemofcoupledPoisson equations, 2025, arXiv:2506.17563v1 [math.AP], Preprint, https://doi.org/10.48550/arXiv.2506.17563

work page doi:10.48550/arxiv.2506.17563 2025
[8]

Colin, M

F. Colin, M. Frigon, Systems of coupled Poisson equations with critical growth in unbounded domains,Non- linear Differential Equations and Applications, vol. 13, pp. 369–384, 2006

work page 2006
[9]

Zhang and S

G. Zhang and S. Liu, Existence Result for a Class of Elliptic Systems with Indefinite Weights,Boundary Value Problems, vol. 2008, pp. 1-10, 2008

work page 2008
[10]

D. G. de Figueiredo, J. Yang, Decay, symmetry and existence of solutions of semilinear elliptic systems, Nonlinear Analysis: Theory, Methods and Applications, vol. 33, no. 3, pp. 211–234, 1998

work page 1998
[11]

Li and J

G. Li and J. Yang, Asymptotically linear elliptic systems,Communications in Partial Differential Equations, vol. 29, no. 5-6, pp. 925–954, 2004

work page 2004
[12]

Bartsch, D.G

T. Bartsch, D.G. de Figueiredo, Infinitely many solutions of nonlinear elliptic systems,Progr. Nonlinear Differential Equations Appl., Vol. 35, Birkhäuser, Basel/Switzerland, 1999, pp. 51–67

work page 1999
[13]

C. J. Batkam, F. Colin, Generalized Fountain theorem and applications to strongly indefinite semilinear problems,J. Math. Anal. Appl. 405 (2013) 438–452

work page 2013
[14]

A. Songo, Existence result for a system of two semilinear coupled Poisson equations with asymptotically linear nonlinearities, 2025, arXiv:2509.22899v1 [math.AP], Preprint, https://doi.org/10.48550/arXiv.2509.22899

work page doi:10.48550/arxiv.2509.22899 2025
[15]

J. M. do Ó, E. Medeiros, U. Severo, A nonhomogeneous elliptic problem involving critical growth in dimension two,Journal of Mathematical Analysis and Applications, Vol. 345, Issue 1, 2008, pp. 286-304

work page 2008
[16]

Adimurthi and S. L. Yadava, Multiplicity results for semilinear elliptic equations in a bounded domain ofR2 involving critical exponents,Annali della Scuola Normale Superiore di Pisa, vol. 17, no. 4, pp. 481–504, 1990

work page 1990
[17]

C. O. Alves, J. M. do Ó, and O. H. Miyagaki, On nonlinear perturbations of a periodic elliptic problem inR2 involving critical growth,Nonlinear Analysis: Theory, Methods and Applications, vol. 56, no. 5, pp. 781–791, 2004

work page 2004
[18]

Liu and Z

S. Liu and Z. Shen, Generalized saddle point theorem and asymptotically linear problems with periodic po- tential,Nonlinear Anal. 86 (2013), 52–57

work page 2013
[19]

J. M. do Ó and M. A. S. Souto, On a class of nonlinear Schrodinger equations inR2 involving critical growth, Journal of Differential Equations, vol. 174, no. 2, pp. 289–311, 2001

work page 2001
[20]

On the Fountain Theorem for Continuous Functionals and Its Application to a Semilinear Elliptic Problem in $\mathbb{R}^2$

A. Songo, F. Colin, On the Fountain Theorem for Continuous Functionals and Its Appli- cation to a Semilinear Elliptic Problem inR 2, 2025, arXiv:2509.16059 [math.AP], Preprint, https://doi.org/10.48550/arXiv.2509.16059

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2509.16059 2025
[21]

Colin, Existence Result for a Class of Nonlinear Elliptic Systems on Punctured Unbounded Domains, Boundary Value Problems, vol

F. Colin, Existence Result for a Class of Nonlinear Elliptic Systems on Punctured Unbounded Domains, Boundary Value Problems, vol. 2010, pp. 1-15, 2010

work page 2010

[1] [1]

Colin, A

F. Colin, A. Songo, An infinite dimensional saddle point theorem and application, 2025, arXiv:2505.04809v1 [math.AP], Preprint, https://doi.org/10.48550/arXiv.2505.04809

work page doi:10.48550/arxiv.2505.04809 2025

[2] [2]

P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations,CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences by American Mathematical Sciences, 65, Washington, DC (1986)

work page 1986

[3] [3]

P. H. Rabinowitz, Some critical point theorems and applications to semilinear elliptic partial differential equations,Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), 215-223

work page 1978

[4] [4]

Kryszewski, A

W. Kryszewski, A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation,Adv. Differential Equations 3 (1998) 441–472

work page 1998

[5] [5]

Willem, Minimax Theorems,Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, Boston, 1996

M. Willem, Minimax Theorems,Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, Boston, 1996

work page 1996

[6] [6]

D. G. de Figueiredo, J. M. do Ó, and B. Ruf,Critical and subcritical elliptic systems in dimension two, Indiana University Mathematics Journal, vol. 53, no. 4, pp. 1037–1054, 2004

work page 2004

[7] [7]

A.Songo, F.Colin, Anewinfinite-dimensionalLinking theoremwithapplicationtoa systemofcoupledPoisson equations, 2025, arXiv:2506.17563v1 [math.AP], Preprint, https://doi.org/10.48550/arXiv.2506.17563

work page doi:10.48550/arxiv.2506.17563 2025

[8] [8]

Colin, M

F. Colin, M. Frigon, Systems of coupled Poisson equations with critical growth in unbounded domains,Non- linear Differential Equations and Applications, vol. 13, pp. 369–384, 2006

work page 2006

[9] [9]

Zhang and S

G. Zhang and S. Liu, Existence Result for a Class of Elliptic Systems with Indefinite Weights,Boundary Value Problems, vol. 2008, pp. 1-10, 2008

work page 2008

[10] [10]

D. G. de Figueiredo, J. Yang, Decay, symmetry and existence of solutions of semilinear elliptic systems, Nonlinear Analysis: Theory, Methods and Applications, vol. 33, no. 3, pp. 211–234, 1998

work page 1998

[11] [11]

Li and J

G. Li and J. Yang, Asymptotically linear elliptic systems,Communications in Partial Differential Equations, vol. 29, no. 5-6, pp. 925–954, 2004

work page 2004

[12] [12]

Bartsch, D.G

T. Bartsch, D.G. de Figueiredo, Infinitely many solutions of nonlinear elliptic systems,Progr. Nonlinear Differential Equations Appl., Vol. 35, Birkhäuser, Basel/Switzerland, 1999, pp. 51–67

work page 1999

[13] [13]

C. J. Batkam, F. Colin, Generalized Fountain theorem and applications to strongly indefinite semilinear problems,J. Math. Anal. Appl. 405 (2013) 438–452

work page 2013

[14] [14]

A. Songo, Existence result for a system of two semilinear coupled Poisson equations with asymptotically linear nonlinearities, 2025, arXiv:2509.22899v1 [math.AP], Preprint, https://doi.org/10.48550/arXiv.2509.22899

work page doi:10.48550/arxiv.2509.22899 2025

[15] [15]

J. M. do Ó, E. Medeiros, U. Severo, A nonhomogeneous elliptic problem involving critical growth in dimension two,Journal of Mathematical Analysis and Applications, Vol. 345, Issue 1, 2008, pp. 286-304

work page 2008

[16] [16]

Adimurthi and S. L. Yadava, Multiplicity results for semilinear elliptic equations in a bounded domain ofR2 involving critical exponents,Annali della Scuola Normale Superiore di Pisa, vol. 17, no. 4, pp. 481–504, 1990

work page 1990

[17] [17]

C. O. Alves, J. M. do Ó, and O. H. Miyagaki, On nonlinear perturbations of a periodic elliptic problem inR2 involving critical growth,Nonlinear Analysis: Theory, Methods and Applications, vol. 56, no. 5, pp. 781–791, 2004

work page 2004

[18] [18]

Liu and Z

S. Liu and Z. Shen, Generalized saddle point theorem and asymptotically linear problems with periodic po- tential,Nonlinear Anal. 86 (2013), 52–57

work page 2013

[19] [19]

J. M. do Ó and M. A. S. Souto, On a class of nonlinear Schrodinger equations inR2 involving critical growth, Journal of Differential Equations, vol. 174, no. 2, pp. 289–311, 2001

work page 2001

[20] [20]

On the Fountain Theorem for Continuous Functionals and Its Application to a Semilinear Elliptic Problem in $\mathbb{R}^2$

A. Songo, F. Colin, On the Fountain Theorem for Continuous Functionals and Its Appli- cation to a Semilinear Elliptic Problem inR 2, 2025, arXiv:2509.16059 [math.AP], Preprint, https://doi.org/10.48550/arXiv.2509.16059

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2509.16059 2025

[21] [21]

Colin, Existence Result for a Class of Nonlinear Elliptic Systems on Punctured Unbounded Domains, Boundary Value Problems, vol

F. Colin, Existence Result for a Class of Nonlinear Elliptic Systems on Punctured Unbounded Domains, Boundary Value Problems, vol. 2010, pp. 1-15, 2010

work page 2010