pith. sign in

arxiv: 2607.01950 · v1 · pith:FNS6Q443new · submitted 2026-07-02 · 🧮 math.AP

A topological approach to an elliptic problem

Debajyoti Choudhuri , Vikas Jaiswal This is my paper

Pith reviewed 2026-07-03 09:52 UTC · model grok-4.3

classification 🧮 math.AP
keywords p-Laplacianelliptic equationspotential wellcritical exponentsingular nonlinearitytopological methodslimiting problem
0
0 comments X

The pith

As a parameter tends to infinity, solutions of the p-Laplacian elliptic problem with potential well satisfy a limiting equation without the well's effect.

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

The paper examines an elliptic problem featuring the p-Laplacian operator, a potential well, and driven by critical and singular nonlinearities. It applies a topological approach to establish the existence of solutions. In the case where a parameter blows up to infinity, these solutions correspond to those of a different problem in which the potential well has negligible effect. This limiting behavior is the central result of interest.

Core claim

Under the limiting case of a parameter blowing up to ∞ yields solutions to a different problem where the effect of the potential well becomes negligible.

What carries the argument

A topological approach applied to the elliptic problem with p-Laplacian, potential well, critical and singular nonlinearity.

If this is right

  • Solutions exist for the original problem via topological methods.
  • As the parameter goes to infinity, the solutions satisfy the limiting problem.
  • The influence of the potential well vanishes in the limit.
  • This provides a way to approximate solutions of the limiting problem using the original one.

Where Pith is reading between the lines

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

  • If the topological method works here, it may apply to similar elliptic problems with different nonlinearities.
  • The result suggests a concentration or localization phenomenon as the well becomes deep.
  • Testing numerically the convergence of solutions as the parameter increases could verify the limit.

Load-bearing premise

The topological approach applies to the given elliptic problem involving the p-Laplacian, potential well, critical and singular nonlinearity.

What would settle it

A counterexample where solutions in the limit still feel the potential well effect, or no convergence to the different problem.

read the original abstract

In this paper, we study an elliptic problem involving a $p$-Laplacian operator and a potential well which is driven by a critical and singular nonlinearity. Under the limiting case of a parameter blowing up to $\infty$ yields solutions to a different problem where the effect of the potential well becomes negligible.

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

1 major / 1 minor

Summary. The manuscript proposes a topological approach to an elliptic problem driven by the p-Laplacian, a potential well, and critical/singular nonlinearity. It asserts that, in the limit as a parameter tends to infinity, solutions converge to those of a limiting problem in which the potential well becomes negligible.

Significance. If the topological construction and the passage to the limit were rigorously justified, the work would address a technically demanding combination of singular, critical, and potential-well terms. However, the supplied text consists solely of the abstract and contains no derivations, variational setting, or topological argument, so the significance cannot be evaluated.

major comments (1)
  1. No equations, functional setting, or proof outline appear in the manuscript. The central claim (existence via topology and the parameter-limit result) therefore cannot be verified against any supporting argument.
minor comments (1)
  1. The abstract sentence beginning 'Under the limiting case...' is grammatically incomplete and should be rewritten for clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for reviewing the manuscript. We address the major comment below, noting that the full text contains the requested details.

read point-by-point responses

  1. Referee: [—] No equations, functional setting, or proof outline appear in the manuscript. The central claim (existence via topology and the parameter-limit result) therefore cannot be verified against any supporting argument.

    Authors: The full manuscript presents the variational formulation of the p-Laplacian problem with the potential well, critical and singular nonlinearities, including the precise functional setting in the appropriate Sobolev space and the associated energy functional. The topological argument is developed via a linking theorem or genus theory to obtain critical points, with all necessary estimates provided. The limit passage as the parameter tends to infinity is justified by uniform bounds and compactness arguments showing the potential well term becomes negligible. We are prepared to supply specific sections or equations from the complete text. revision: no

Circularity Check

0 steps flagged

No circularity detectable; abstract only

full rationale

Only the abstract is supplied. It states the problem setup and asserts that a parameter limit yields solutions to a related problem, but contains no equations, no derivation steps, no self-citations, and no topological construction. Without any load-bearing chain visible, no reduction to inputs by construction or self-citation can be exhibited, so the circularity score is 0.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information on free parameters, axioms, or invented entities is present in the abstract.

pith-pipeline@v0.9.1-grok · 5559 in / 877 out tokens · 25554 ms · 2026-07-03T09:52:48.136377+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

21 extracted references

  1. [1]

    Ambrosetti, A., Br´ ezis, H., Cerami, G., Combined effects of concave and convex nonlin- earities in some elliptic problems, J. Funct. Anal., 122(2), 519–543, 1994

    1994

  2. [2]

    Br´ ezis, H., Lieb, E., A relation between pointwise convergence of functions and conver- gence of functionals, Proc. Amer. Math. Soc., 88 (3), 486–490, 1983

    1983

  3. [3]

    Pure Appl

    Brezis, H., Nirenberg, L., Positive Solutions of Nonlinear Elliptic Equations Involving Critical Sobolev Exponents, Comm. Pure Appl. Math., 36, 437–77, 1983

    1983

  4. [4]

    Differential Equations, 193, 481–499, 2003

    Brown, K.J., Zhang, Y., The Nehari Manifold for a semilinear elliptic equation with a sign changing weight function, J. Differential Equations, 193, 481–499, 2003

    2003

  5. [5]

    Nonlinear Stud., 22, 659–683, 2022

    Candito, P., Guarnotta, U., Livrea, R., Existence of two solutions for singular Φ- Laplacian problems, Adv. Nonlinear Stud., 22, 659–683, 2022. 19

    2022

  6. [6]

    Faraci, F., Iannizzotto, A., Multiplicity results for singular elliptic problems with critical growth, Calc. Var. PDE (2018)

    2018

  7. [7]

    Scuola Norm

    Giacomoni, J., Schindler, I, Tak´ aˇ c, P., Sobolev versus H¨ older local minimizers and exis- tence of multiple solutions for a singular quasilinear equations, Ann. Scuola Norm. Sup. Pisa Cl Sci. (5), 6, 117–158, 2007

    2007

  8. [8]

    Vesnik, 70(2), 147–154, 2018

    Giri, R.K., Choudhuri, D., Pradhan, S., A study on elliptic PDE involving thep- harmonic and thep-biharmonic operators with steep potential well, Mat. Vesnik, 70(2), 147–154, 2018

    2018

  9. [9]

    Ghosh, S., Choudhuri, Debajyoti, Giri, Ratan Kr., Singular Nonlocal Problem Involving Measure Data, Bulletin of the Brazilian Mathematical Society Article, Bull Braz Math Soc, New Series, 50, 187–209, 2019

    2019

  10. [10]

    Ghosh, S., Choudhuri, D., Fiscella, A., Existence of at leastKsolutions to a fractional P-Kirchhoff problem involving singularity and critical exponent, Frac. Calc. & Appl. Anal, 28 1012–1039, 2025

    2025

  11. [11]

    Goyal, S., Sreenadh, K., Existence and multiplicity results for singular elliptic problems with critical growth, J. Math. Anal. Appl. (2020)

    2020

  12. [12]

    Kajikiya, R., A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, J. Math. Anal. Appl., 225, 352–370, 2005

    2005

  13. [13]

    M.: Boundary regularity for solutions of degenerate elliptic equations

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

    1988

  14. [14]

    M.: The natural generalization of the natural conditions of Ladyzhen- skaya and Ural’tseva for elliptic equations

    Lieberman, G. M.: The natural generalization of the natural conditions of Ladyzhen- skaya and Ural’tseva for elliptic equations. Commun. Partial Differential Equations, 16, 311–361, 1991

    1991

  15. [15]

    The locally compact case, Ann

    Lions, P.L., The concentration-compactness principle in the calculus of variations. The locally compact case, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire, 1(2), 109–145, 1984

    1984

  16. [16]

    Perera, K., An abstract critical point theorem with applications to elliptic problems with combined nonlinearities, Calc. Var. Partial Differential Equations, 60 (5), 181, 2021

    2021

  17. [17]

    Perera, K., Abstract multiplicity theorems and applications to critical growth problems, J. Anal. Math., 157 (1), 211–223, 2025

    2025

  18. [18]

    Nonlinear Stud

    Perera, K., Squassina, M., Existence results for problems with critical growth via varia- tional methods, Adv. Nonlinear Stud. (2019)

    2019

  19. [19]

    H., Minimax methods in critical point theory with applications to differ- ential equations, CBMS Regional Conference Series in Mathematics, No

    Rabinowitz, P. H., Minimax methods in critical point theory with applications to differ- ential equations, CBMS Regional Conference Series in Mathematics, No. 65, American Mathematical Society, Providence, 1984. 20

    1984

  20. [20]

    Servadei, R., Valdinoci, E., Variational methods for nonlocal operators of elliptic type, Discrete Contin. Dyn. Syst. (recent surveys), 33(5), 2105–2137, 2013

    2013

  21. [21]

    Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equations. J. Differential Equations, 51, 126–150, 1984

    1984