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arxiv: 1907.09240 · v1 · pith:HO5TGQVRnew · submitted 2019-07-22 · 🧮 math.AP

On the extreme value of the Nehari manifold method for a class of Schr\"{o}dinger equations with indefinite weight functions

Jos\'e Carlos de Albuquerque , Kaye Silva This is my paper

Pith reviewed 2026-05-24 18:10 UTC · model grok-4.3

classification 🧮 math.AP
keywords Schrödinger equationNehari manifoldindefinite weightsp-Laplacianmultiple solutionscritical parameter
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The pith

For Schrödinger equations with indefinite weights, two solutions exist when λ exceeds the critical value λ*.

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

This paper analyzes a p-Laplacian Schrödinger equation in R^N with indefinite weight functions h(x) and f(x). It identifies an extreme value λ* and examines its relation to the Nehari manifold. The central goal is to prove that two distinct solutions exist for all λ larger than λ*. The work extends earlier results on parameter-dependent existence to the indefinite weight setting.

Core claim

The authors show that λ* marks the threshold at which the Nehari manifold acquires the geometry needed to produce two distinct critical points of the energy functional, yielding two solutions of the equation for every λ > λ*.

What carries the argument

The Nehari manifold, the set of functions u where the derivative of the energy functional satisfies <I'(u), u> = 0, used to locate multiple critical points.

If this is right

  • The energy functional admits at least two critical points when λ > λ*.
  • Two distinct nontrivial weak solutions exist in R^N for λ > λ*.
  • The geometry of the Nehari manifold changes precisely at the value λ*.
  • The multiplicity result holds in the indefinite-weight case and complements prior work on definite weights.

Where Pith is reading between the lines

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

  • The same threshold analysis may extend to other quasilinear operators such as the fractional p-Laplacian.
  • For radially symmetric weights, λ* might admit an explicit characterization that allows direct verification of the multiplicity.
  • The approach could apply to equations on bounded domains with similar indefinite nonlinearities.

Load-bearing premise

The weight functions h(x) and f(x) being indefinite together with suitable conditions on the exponents allow the Nehari manifold to possess the geometry needed to produce two distinct critical points for λ > λ*.

What would settle it

An explicit choice of indefinite weights h and f together with exponents where λ* can be computed and for some λ > λ* the equation has only one or zero solutions would falsify the multiplicity claim.

read the original abstract

In this work we are concerned with the following class of equations \[ -\Delta_p u -\lambda h(x)|u|^{p-2}u=f(x)|u|^{\gamma-2}u, \quad \mbox{in } \mathbb{R}^N, \] involving indefinite weight functions. The existence of solution may depend on the parameter $\lambda$. We analyze the extreme value $\lambda^{*}$ and study its relation with the Nehari manifold. Our goal is to establish the existence of two solutions when $\lambda>\lambda^{*}$. This work extends and complements the results obtained by J. Chabrowski and D.G. Costa [Comm. Partial Differential Equations 33 (2008), 1368--1394]

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

0 major / 3 minor

Summary. The manuscript analyzes the parameter-dependent problem −Δ_p u − λ h(x)|u|^{p−2}u = f(x)|u|^{γ−2}u on R^N with indefinite weights h and f. It defines an extreme value λ* via the Nehari manifold, studies its properties, and proves that the manifold geometry yields two distinct critical points (hence two solutions) whenever λ > λ*. The work extends the Chabrowski–Costa framework by focusing on the precise threshold λ* rather than a generic existence interval.

Significance. If the estimates on the fibering maps and the verification that λ* is attained and separates the regimes with one versus two solutions are correct, the paper supplies a sharp characterization of the existence threshold for indefinite-weight p-Laplacian problems. This is a modest but useful refinement of the existing Nehari-manifold literature.

minor comments (3)
  1. §2, definition of λ*: the infimum is taken over a set that depends on the sign-changing weight h; a short remark clarifying whether λ* is attained or only approached would help the reader follow the subsequent geometry arguments.
  2. The statement of Theorem 1.1 (or the main existence theorem) should explicitly list the range p < γ < p* together with the sign conditions on h and f that are used throughout the proofs.
  3. Several citations to Chabrowski–Costa appear without page numbers; adding the relevant page or theorem numbers would make the comparison with the earlier work easier to verify.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines λ* as the threshold value for the parameter λ and analyzes its relation to the Nehari manifold geometry to establish existence of two solutions when λ > λ*. This extends the external results of Chabrowski and Costa (2008) under standard assumptions on the indefinite weights h(x), f(x) and the range p < γ < p*. No load-bearing step reduces by construction to a fitted input, self-definition, or self-citation chain; the central variational argument (mountain-pass or linking structure on the manifold) is independent of the present authors' prior work and relies on externally verifiable conditions on the functional setting.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the paper rests on standard domain assumptions for the functional setting of indefinite-weight elliptic problems and the Nehari manifold construction; no free parameters or invented entities are visible.

axioms (2)
  • domain assumption The weight functions h(x) and f(x) are indefinite and belong to suitable function spaces (e.g., L^∞ or with appropriate decay) so that the energy functional is well-defined on W^{1,p}(R^N).
    Invoked implicitly by the equation setup in the abstract; required for the variational formulation.
  • domain assumption The growth exponents satisfy 1 < p < γ < p* (Sobolev critical) allowing subcritical nonlinearity.
    Standard for the Nehari manifold to be a C^1 manifold and for compactness arguments.

pith-pipeline@v0.9.0 · 5661 in / 1404 out tokens · 29535 ms · 2026-05-24T18:10:38.361706+00:00 · methodology

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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