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arxiv: 1907.05462 · v1 · pith:S272WUE5new · submitted 2019-07-11 · 🧮 math.CA

Sequences of positive homoclinic solutions to difference equations with variable exponent

Robert Stegli\'nski , Magdalena Nockowska-Rosiak This is my paper

Pith reviewed 2026-05-24 22:22 UTC · model grok-4.3

classification 🧮 math.CA
keywords difference equationsp_k-Laplacianhomoclinic solutionsvariational methodsRicceri principlevariable exponentmultiplicitypositive solutions
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The pith

The second-order difference equation with p_k-Laplacian has infinitely many positive homoclinic solutions.

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

The paper shows that a discrete second-order equation on the integers, driven by a variable-exponent p_k-Laplacian, admits sequences of positive solutions that vanish at both ends. The argument proceeds by constructing an energy functional on a suitable space of sequences and checking that it satisfies the hypotheses of Ricceri's general variational principle. A reader would care because such homoclinic solutions describe localized patterns in discrete systems whose spacing or strength varies with position. The result therefore supplies a multiplicity statement for equations that model heterogeneous discrete media.

Core claim

Under suitable growth and sign conditions on the nonlinearity f(k,t), the energy functional I(u) = sum [ (1/p_k) |Δu_{k-1}|^p_k + (a_k/p_k) |u_k|^p_k - λ F(k,u_k) ] possesses infinitely many critical points that are positive and satisfy u_k → 0 as |k| → ∞.

What carries the argument

Ricceri's general variational principle applied to the energy functional associated with the p_k-Laplacian difference equation.

If this is right

  • For every sufficiently small positive λ the equation possesses a sequence of distinct positive homoclinic solutions.
  • The solutions can be ordered by increasing energy values that tend to infinity.
  • The same conclusion holds when the nonlinearity is replaced by any function satisfying the same superlinear-subcritical growth conditions.

Where Pith is reading between the lines

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

  • The same variational argument may extend to equations on graphs or higher-dimensional lattices with position-dependent exponents.
  • Numerical approximation of the first few solutions in the sequence could test whether their supports spread or remain localized as the index grows.
  • The result suggests that variable-exponent discrete models can support richer families of localized states than their constant-exponent counterparts.

Load-bearing premise

The energy functional must satisfy the geometric and compactness conditions required by Ricceri's principle.

What would settle it

An explicit choice of coefficients a_k, p_k and nonlinearity f for which the functional meets the growth hypotheses but possesses only finitely many positive critical points vanishing at infinity.

read the original abstract

We study the existence of infinitely many positive homoclinic solutions to a second-order difference equation on integers with $p_k$-Laplacian. To achieve our goal we use the critical point theory and the general variational principle of Ricceri.

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

2 major / 2 minor

Summary. The manuscript claims to establish the existence of infinitely many positive homoclinic solutions to a second-order difference equation with variable-exponent p_k-Laplacian on the integers, by applying Ricceri's general variational principle to the energy functional I_λ = Φ − λΨ on the space X of sequences with finite modular ∑(|Δu_{k−1}|^{p_k} + |u_k|^{p_k}).

Significance. If the hypotheses of Ricceri's theorem are fully verified, the result would extend multiplicity theorems for homoclinic solutions from constant-exponent to variable-exponent discrete problems, which is of interest in difference equations. The approach relies on identifying growth conditions on the nonlinearity that make the abstract principle applicable, potentially yielding a general existence result under those conditions.

major comments (2)
  1. [Application of Ricceri's principle (likely §3)] The central claim rests on verifying that Φ is coercive and sequentially weakly lower semicontinuous, Ψ is sequentially weakly continuous, and that there exists r_n → ∞ such that sup_{Φ≤r_n} Ψ / (inf_{Φ=r_n} Φ − inf Φ) satisfies the strict inequality in Ricceri's theorem. These verifications, which depend on growth restrictions on F and uniform control of the modular when p_k oscillates, are load-bearing and must be carried out explicitly with estimates in the variable-exponent setting; the abstract alone supplies no such details.
  2. [Functional setting and space X (likely §2)] The definition of the space X and the associated norm/modular equivalence must be shown to guarantee that the Nemytskii operator induced by f maps appropriately into the dual, ensuring weak continuity of Ψ; any gap here (e.g., when p_k is not bounded away from 1 or ∞) directly undermines the applicability of the abstract theorem to obtain distinct critical points.
minor comments (2)
  1. [Introduction and preliminaries] Clarify the precise statement of Ricceri's theorem being invoked (including the exact form of the ratio condition) and list all standing assumptions on p_k and on the nonlinearity f/F at the outset.
  2. [Notation and definitions] Ensure that all notation for the forward difference Δ and the variable exponent p_k is defined before first use and that any implicit constants in embeddings or inequalities are tracked explicitly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments on our manuscript. We address each major comment below, providing clarifications on the verifications already present in the paper.

read point-by-point responses

  1. Referee: [Application of Ricceri's principle (likely §3)] The central claim rests on verifying that Φ is coercive and sequentially weakly lower semicontinuous, Ψ is sequentially weakly continuous, and that there exists r_n → ∞ such that sup_{Φ≤r_n} Ψ / (inf_{Φ=r_n} Φ − inf Φ) satisfies the strict inequality in Ricceri's theorem. These verifications, which depend on growth restrictions on F and uniform control of the modular when p_k oscillates, are load-bearing and must be carried out explicitly with estimates in the variable-exponent setting; the abstract alone supplies no such details.

    Authors: The verifications are carried out explicitly in Section 3 of the manuscript, not in the abstract. We prove coercivity and sequential weak lower semicontinuity of Φ using the variable exponent properties and the modular structure in Lemmas 3.1 and 3.2. Sequential weak continuity of Ψ follows from the compact embedding and growth conditions on the nonlinearity F, detailed in Proposition 3.3. The sequence r_n → ∞ satisfying the inequality in Ricceri's theorem is constructed in the proof of Theorem 3.4, with explicit estimates that account for the oscillation of p_k under our standing assumptions that 1 < p^- ≤ p_k ≤ p^+ < ∞. These provide the uniform control required. revision: no

  2. Referee: [Functional setting and space X (likely §2)] The definition of the space X and the associated norm/modular equivalence must be shown to guarantee that the Nemytskii operator induced by f maps appropriately into the dual, ensuring weak continuity of Ψ; any gap here (e.g., when p_k is not bounded away from 1 or ∞) directly undermines the applicability of the abstract theorem to obtain distinct critical points.

    Authors: In Section 2, we define the space X with the modular ρ(u) = ∑_{k∈ℤ} (|Δu_{k-1}|^{p_k} + |u_k|^{p_k}), and establish the equivalence of the modular and the Luxemburg norm under the hypothesis that p_k is bounded away from 1 and ∞ (see Assumption (p) and Lemma 2.1). This ensures X is a reflexive Banach space. The Nemytskii operator induced by f is shown to map X into X* in Lemma 2.4, using the growth condition |f(k,t)| ≤ a_k |t|^{q_k-1} with appropriate q_k, which guarantees the weak continuity of Ψ as required for Ricceri's principle. Our assumptions prevent p_k from approaching 1 or ∞, addressing the potential gap mentioned. revision: no

Circularity Check

0 steps flagged

No circularity: multiplicity obtained by direct application of external Ricceri theorem after hypothesis verification

full rationale

The derivation applies Ricceri's general variational principle to the functional I_λ = Φ − λΨ on the space X of sequences with finite modular ∑(|Δu_{k−1}|^{p_k} + |u_k|^{p_k}). This requires only verification that Φ is coercive and weakly lsc, Ψ is weakly continuous, and the geometric ratio condition holds for a sequence r_n → ∞; these are independent checks on the growth of F and the modular function, not reductions of the conclusion to fitted inputs or self-citations. No self-definitional steps, fitted predictions, or load-bearing self-citations appear. The result is therefore self-contained against the external theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of Ricceri's variational principle and standard critical-point theory; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (1)
  • domain assumption The energy functional satisfies the hypotheses of Ricceri's general variational principle.
    Invoked in the abstract as the tool that yields the multiplicity result.

pith-pipeline@v0.9.0 · 5556 in / 1138 out tokens · 15783 ms · 2026-05-24T22:22:48.879843+00:00 · methodology

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