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arxiv: 2506.19793 · v2 · submitted 2025-06-24 · 🧮 math.AP · math.FA

Multiplicity results for mixed local-nonlocal variable exponent problem involving singular and superlinear term

Shammi Malhotra , Ambesh Kumar Pandey , K. Sreenadh This is my paper

Pith reviewed 2026-05-19 07:33 UTC · model grok-4.3

classification 🧮 math.AP math.FA
keywords variable exponentsNehari manifoldsingular nonlinearitysuperlinear termquasilinear elliptic equationslocal-nonlocal operatorsmultiplicity of solutionsvariational methods
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The pith

The mixed local-nonlocal variable exponent elliptic problem with singular and superlinear terms admits two distinct weak solutions.

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

This paper investigates quasilinear elliptic equations that mix local and nonlocal operators under variable exponents. The equations include singular nonlinearities and a subcritical superlinear term modulated by a parameter lambda. Using variational methods, the energy functional is restricted to suitable subsets of the Nehari manifold. Analysis of a splitting property via the topological index and fibering map structure establishes the existence of two distinct solutions. An L infinity bound is also derived for these solutions.

Core claim

We establish the existence of two distinct solutions for the mixed local-nonlocal variable exponent problem involving singular and superlinear terms by restricting the associated energy functional on appropriate subsets of the Nehari manifold and analysing a key splitting property of the associated Nehari manifold using the topological index and the structure of the fibering maps. We also establish the L^∞-bound for the solutions.

What carries the argument

The Nehari manifold of the energy functional, with its splitting into subsets determined by fibering maps and topological index.

Load-bearing premise

The energy functional admits a well-defined Nehari manifold whose fibering maps possess the required geometric structure depending on the range of variable exponents and subcritical growth conditions.

What would settle it

Finding a specific choice of variable exponents or growth conditions where the Nehari manifold does not split and only one solution exists would falsify the multiplicity claim.

read the original abstract

In this paper, we study a class of quasilinear elliptic equations involving both local and nonlocal operators with variable exponents. The problem exhibits singular nonlinearities along with a subcritical superlinear growth term and a parameter $\lambda$. We study the existence of multiple solutions with the help of variational methods by restricting the associated energy functional on appropriate subsets of the Nehari manifold. Using the topological index and the structure of the fibering maps, we analyse a key splitting property of the associated Nehari manifold. This decomposition allows us to establish the existence of two distinct solutions. Additionally, we establish the $L^\infty$-bound for the solutions.

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 paper establishes the existence of two distinct weak solutions to a mixed local-nonlocal quasilinear elliptic equation with variable exponents, a singular nonlinearity, and a superlinear term with parameter λ. The proof proceeds via variational methods by restricting the energy functional to suitable subsets of the Nehari manifold, exploiting a splitting N = N⁺ ∪ N⁻ obtained from the geometry of the fibering maps and the topological index, followed by an L^∞ bound for the obtained solutions.

Significance. If the central claims are verified, the result extends multiplicity theorems for variable-exponent problems to the mixed local-nonlocal setting with singularities, using a combination of Nehari-manifold splitting and topological arguments. The L^∞ regularity step is a useful complement to the existence analysis.

major comments (2)
  1. [§3] §3 (Nehari manifold and fibering maps): The singular term (of the form ∫ |u|^{-α(x)} dx with α(x) > 0) renders J non-differentiable at u = 0. Although N excludes the origin, the paper must explicitly verify that J ∈ C¹(X ∖ {0}), that each fibering map φ_u(t) = J(tu) for t > 0 possesses exactly two critical points with the required sign changes in φ_u'(t), and that these properties hold uniformly under the stated range of variable exponents so that the topological-index argument for the splitting N = N⁺ ∪ N⁻ remains valid.
  2. [§5] §5 (L^∞ bound): The Moser-iteration argument for the L^∞ estimate must incorporate the nonlocal term and the variable exponents; the current sketch does not show how the iteration constants remain controlled when the singular term is present and when the exponent functions are only log-Hölder continuous.
minor comments (2)
  1. [§2] The precise interval for the variable exponent p(x) (and the relation between p(x) and the growth of the superlinear term) should be stated explicitly in the hypotheses rather than only in the abstract.
  2. [§1] Notation for the mixed operator (local p(x)-Laplacian plus nonlocal fractional term) should be introduced once and used consistently; several passages repeat the full expression unnecessarily.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important points that require clarification and expansion in the manuscript. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: §3 (Nehari manifold and fibering maps): The singular term (of the form ∫ |u|^{-α(x)} dx with α(x) > 0) renders J non-differentiable at u = 0. Although N excludes the origin, the paper must explicitly verify that J ∈ C¹(X ∖ {0}), that each fibering map φ_u(t) = J(tu) for t > 0 possesses exactly two critical points with the required sign changes in φ_u'(t), and that these properties hold uniformly under the stated range of variable exponents so that the topological-index argument for the splitting N = N⁺ ∪ N⁻ remains valid.

    Authors: We agree with the referee that a more explicit verification of these properties is essential for rigor, particularly given the variable exponents and the singular term. In the revised manuscript, we will include a dedicated lemma in Section 3 that proves J is C¹ on X excluding the origin by showing that the singular integral term is differentiable away from zero under the integrability conditions ensured by the Sobolev embedding. Furthermore, we will provide a detailed analysis of the fibering map φ_u(t) for fixed u ≠ 0, demonstrating that under the assumptions on the exponents (including log-Hölder continuity and the range 1 < p(x) < q(x) < p^*(x)), the equation φ_u'(t) = 0 has exactly two positive solutions t1(u) < t2(u), with φ_u''(t1(u)) > 0 and φ_u''(t2(u)) < 0, corresponding to the local minimum and maximum. This analysis will be uniform in u on the unit sphere, allowing the topological index to be applied consistently for the splitting into N⁺ and N⁻. We believe this addition will fully address the concern. revision: yes

  2. Referee: §5 (L^∞ bound): The Moser-iteration argument for the L^∞ estimate must incorporate the nonlocal term and the variable exponents; the current sketch does not show how the iteration constants remain controlled when the singular term is present and when the exponent functions are only log-Hölder continuous.

    Authors: We appreciate this observation regarding the L^∞ regularity. The current sketch in Section 5 outlines the Moser iteration but indeed requires expansion to handle all aspects of the problem. In the revision, we will detail the Moser iteration process step by step: first, we obtain a preliminary L^r bound for large r using the weak formulation tested against suitable truncations, where the singular term is handled by noting that for the positive solutions obtained, inf u > 0 on compact sets or by absorbing it into the lower order terms. The nonlocal term will be estimated using the fractional Sobolev inequality adapted to variable exponents. For the variable exponents with log-Hölder continuity, we will show that the constants in the iteration (arising from the modular inequalities and embeddings) are controlled uniformly, following standard techniques in variable exponent spaces (e.g., using the log-Hölder condition to bound the oscillation). We will also verify that the iteration converges to an L^∞ bound independent of λ in the given range. This expanded proof will be included in the revised Section 5. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained variational analysis

full rationale

The paper claims existence of two distinct solutions for the mixed local-nonlocal variable-exponent problem by restricting the energy functional to suitable subsets of the Nehari manifold, analyzing its splitting via topological index and fibering-map geometry. This chain relies on standard assumptions about the functional's C^1 regularity away from the origin, subcritical growth, and the geometric properties of the fibering maps φ_u(t) = J(tu) under the stated exponent ranges. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the splitting N = N+ ∪ N− is derived from the explicit structure of the maps and index theory rather than being presupposed. The argument is therefore independent of its target conclusion and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based on abstract only; the result rests on standard assumptions of variable-exponent Sobolev spaces and the geometric properties of the Nehari manifold for the given growth conditions.

axioms (2)
  • domain assumption Variable exponents p(x), q(x) satisfy conditions ensuring the energy functional is well-defined and C^1 on the appropriate function space.
    Invoked implicitly for the Nehari manifold and fibering maps to be applicable.
  • domain assumption The nonlinear terms satisfy subcritical growth and singularity conditions compatible with the embedding theorems used.
    Required for the splitting property and existence arguments.

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