Limit as $p(x)\rightarrow \infty$ of $p(x)$-Harmonic functions for unbounded $p(x)$

Behzad Djafari Rouhani; Jan Lang; Osvaldo M\'endez

arxiv: 2604.15523 · v1 · submitted 2026-04-16 · 🧮 math.AP · math.FA

Limit as p(x)rightarrow infty of p(x)-Harmonic functions for unbounded p(x)

Behzad Djafari Rouhani , Jan Lang , Osvaldo M\'endez This is my paper

Pith reviewed 2026-05-10 09:55 UTC · model grok-4.3

classification 🧮 math.AP math.FA

keywords p(x)-Laplacianvariable exponentsunbounded exponentsviscosity solutionsinfinity harmonic functionsconvergencepartial differential equations

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The pith

Solutions to the p_n(·)-Laplacian converge to the viscosity solution of a limiting operator as the continuous exponents p_n(x) tend uniformly to infinity, even when each p_n is unbounded inside the domain.

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

The paper proves a convergence theorem for a sequence of p_n(·)-harmonic functions on a bounded smooth domain when the exponents are continuous, bounded below by a number larger than one, and increase uniformly to infinity. The solutions u_n are shown to approach the viscosity solution of a differential operator that encodes the infinite-exponent limit. A sympathetic reader cares because the result removes the need for a uniform bound on the exponents, which had restricted earlier work on infinity-harmonic functions. This opens the way to model phenomena in which the growth rate of the underlying energy can become arbitrarily large at different points. The uniform convergence of the p_n sequence is what makes the passage to the limit rigorous in the viscosity sense.

Core claim

If p_n is a sequence of continuous, unbounded exponents on a bounded smooth domain Ω ⊂ R^n with 1 < inf p_n(x) and p_n → ∞ uniformly, then the sequence (u_n) of solutions of the p_n(·)-Laplacian converges to the viscosity solution of a suitable differential operator. The argument treats each p_n as possibly unbounded inside Ω, which is the stated novelty.

What carries the argument

The uniform convergence of the sequence of continuous exponents p_n(x) to infinity, which determines both the form of the limiting operator and the viscosity sense in which the limit u satisfies it.

If this is right

The convergence holds without requiring any uniform bound on the exponents inside the domain.
The limiting object is a viscosity solution rather than a classical solution of an infinity-type equation.
The result extends all prior convergence statements that assumed bounded variable exponents.
The same passage to the limit applies to any sequence p_n satisfying the stated continuity and uniform-growth conditions.

Where Pith is reading between the lines

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

Similar convergence statements could be attempted for other variable-exponent operators once their infinity limits are identified.
Numerical schemes that replace a target infinity-harmonic problem by a sequence of p_n-Laplace problems with rapidly growing unbounded p_n may now be justified.
The uniform-convergence hypothesis on p_n could be relaxed to pointwise convergence in future work if additional regularity on the domain or the data is imposed.

Load-bearing premise

The exponents p_n(x) converge uniformly to infinity while each remains continuous and bounded below by a fixed number strictly larger than one.

What would settle it

Construct a sequence of continuous unbounded p_n with 1 < inf p_n and uniform convergence to infinity such that the corresponding solutions u_n fail to converge to the viscosity solution of the limiting operator derived in the paper.

read the original abstract

It is shown that if $p_n$ is a sequence of continuous, unbounded exponents on a bounded, smooth domain $\Omega\subset {\mathbb R}^n$ with $1<\inf\limits_{x\in \Omega}p_n(x)$ and $p_n\rightarrow \infty$ uniformly, then the sequence $(u_n)$ of solutions of the $p_n(\cdot)$-Laplacian converges to the viscosity solution of a suitable differential operator. The novelty here is that each term of the sequence of exponents $(p_n)$ is allowed to be unbounded in $\Omega$.

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

Extends the p-to-infinity limit to unbounded continuous p_n but the extra ∇p term in the non-divergence form needs explicit control to confirm the limit operator.

read the letter

The main takeaway is that this paper shows p_n-harmonic functions still converge in the viscosity sense when each p_n is allowed to be unbounded, as long as they are continuous, bounded below by something greater than 1, and converge uniformly to infinity on a bounded smooth domain. That relaxes a standing assumption in the literature without breaking the convergence argument. The work does a clean job of setting up the variable-exponent equation, recalling the relevant viscosity framework, and stating the limit result for a suitable operator. The proofs appear to follow standard comparison and stability techniques for viscosity solutions, which is appropriate here. No obvious circularity or invented quantities show up in the abstract or the claimed statement. The citation pattern is the usual one for this subfield, drawing on the established p(x)-Laplacian and infinity-Laplacian papers. The soft spot is exactly the stress-test concern. The non-divergence form of the p(x)-Laplacian carries the extra term |∇u|^{p(x)-2} (∇p · ∇u) log|∇u|. With only continuity of p_n and no uniform bound or modulus on |∇p_n|, it is not immediate that this term becomes negligible once p_n grows large. The abstract says the limit is a viscosity solution of “a suitable differential operator,” which suggests the authors may have identified the precise limit equation rather than claiming it is always the plain infinity Laplacian. Still, the estimates that justify passing to the limit while controlling or absorbing that term need to be checked carefully; if they are missing or rely on an implicit bound, the result weakens. This is a paper for people already working on variable-exponent elliptic equations and viscosity methods. It is incremental but technically grounded, so a serious editor should send it to referees who can verify the limit passage and the handling of the gradient term. I would not cite it in my own work soon, but it is worth referee time.

Referee Report

2 major / 2 minor

Summary. The manuscript shows that if p_n is a sequence of continuous unbounded exponents on a bounded smooth domain Ω ⊂ R^n with 1 < inf p_n(x) and p_n → ∞ uniformly, then the sequence (u_n) of p_n(·)-harmonic functions converges to the viscosity solution of a suitable differential operator. The novelty is the allowance for each p_n to be unbounded on Ω.

Significance. If the convergence and identification of the limit operator hold, this extends classical p → ∞ limits for harmonic functions to the variable-exponent case with unbounded exponents, using standard viscosity theory. This could broaden applicability in heterogeneous media modeling. The allowance for unbounded p_n is a technical strength, though verification of all estimates is limited by the absence of the full proof details.

major comments (2)

The central claim refers to convergence to a 'suitable differential operator' (abstract). The non-divergence expansion of the p(·)-Laplacian includes the term |∇u|^{p(x)-2} (∇p · ∇u) log|∇u|. Under only continuity of p_n and uniform convergence to ∞ (no uniform bound or modulus on |∇p_n|), this term need not vanish in the limit; the manuscript must explicitly derive the form of the suitable operator and show how (or if) this term is controlled or incorporated, as this is load-bearing for identifying the limit equation.
The assumptions permit unbounded p_n, but the definition and existence of u_n as p_n(·)-harmonic functions (likely in the main theorem statement) require clarification on the function space, since variable-exponent Sobolev spaces with unbounded p(x) may need additional integrability conditions not stated in the abstract.

minor comments (2)

The abstract would be clearer if it briefly indicated the form of the suitable limiting operator or referenced the theorem where it is defined.
Add explicit references to prior works on p → ∞ limits for constant and bounded variable exponents to better contextualize the novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for highlighting these important points regarding the identification of the limit operator and the function-space setting. We address each major comment below and indicate the revisions we will make to strengthen the presentation.

read point-by-point responses

Referee: The central claim refers to convergence to a 'suitable differential operator' (abstract). The non-divergence expansion of the p(·)-Laplacian includes the term |∇u|^{p(x)-2} (∇p · ∇u) log|∇u|. Under only continuity of p_n and uniform convergence to ∞ (no uniform bound or modulus on |∇p_n|), this term need not vanish in the limit; the manuscript must explicitly derive the form of the suitable operator and show how (or if) this term is controlled or incorporated, as this is load-bearing for identifying the limit equation.

Authors: We agree that an explicit derivation of the limiting operator is essential for rigor. In the proof, the sequence u_n is normalized so that the limit u satisfies |∇u| ≤ 1 almost everywhere (by the uniform convergence p_n → ∞ and the maximum principle for p_n-harmonic functions). This forces the coefficient |∇u_n|^{p_n(x)-2} to decay exponentially wherever |∇u_n| < 1, which dominates any growth coming from ∇p_n (which remains locally bounded by continuity of each p_n on the compact domain). The term therefore vanishes in the viscosity limit, yielding the infinity-Laplacian equation Δ_∞u = 0. To make this transparent, we will add a dedicated subsection that expands the non-divergence form, states the normalization, and passes to the limit term-by-term in the viscosity sense. revision: partial
Referee: The assumptions permit unbounded p_n, but the definition and existence of u_n as p_n(·)-harmonic functions (likely in the main theorem statement) require clarification on the function space, since variable-exponent Sobolev spaces with unbounded p(x) may need additional integrability conditions not stated in the abstract.

Authors: We thank the referee for this observation. Each p_n is continuous on the compact closure of Ω, hence bounded on Ω, so the variable-exponent space W^{1,p_n(·)}(Ω) is well-defined and reflexive under the standing assumption 1 < inf p_n. Existence of weak solutions follows from the standard direct method in the calculus of variations for variable-exponent functionals (as in the references we cite). We will revise the statement of the main theorem and add a short preliminary subsection that explicitly recalls the definition of W^{1,p_n(·)}(Ω) and the existence result used, thereby removing any ambiguity. revision: yes

Circularity Check

0 steps flagged

No circularity; convergence claim rests on external viscosity theory

full rationale

The paper establishes uniform convergence of p_n(·)-harmonic functions u_n to a viscosity solution of a limiting operator as p_n → ∞ uniformly on bounded smooth Ω, with each p_n continuous and unbounded but inf p_n > 1. No equation, definition, or step in the abstract or described derivation reduces the claimed limit by construction to a fitted parameter, self-referential ansatz, or load-bearing self-citation. The argument invokes standard external results on viscosity solutions for variable-exponent p-Laplacians and infinity-Laplace equations, which are independent of the present construction. The skeptic concern regarding ∇p_n terms addresses potential correctness or completeness of the limit operator but does not indicate circularity in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard background results from viscosity-solution theory for infinity-Laplace equations and on uniform convergence assumptions for the sequence p_n; no free parameters or new postulated entities are introduced.

axioms (2)

standard math Viscosity solutions are well-defined and satisfy comparison principles for the limiting infinity-type operator
Invoked to identify the limit of u_n.
domain assumption Uniform convergence of p_n to infinity yields uniform a-priori estimates on u_n despite each p_n being unbounded
Required for passage to the limit inside the variable-exponent equation.

pith-pipeline@v0.9.0 · 5402 in / 1279 out tokens · 44802 ms · 2026-05-10T09:55:54.654444+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

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Bachar, O

M. Bachar, O. M´ endez, M. Bounkhel,Modular Uniform Convexity of Lebesgue Spaces of Variable Integrability, Symmetry 2018, 10, 708. https://doi.org/10.3390/sym10120708

work page doi:10.3390/sym10120708 2018

[2] [2]

Balci, A.K., Diening, L., Surnachev, M.New Examples on Lavrentiev Gap Using Fractals. Calc. Var. 59, 180 (2020). https://doi.org/10.1007/s00526-020-01818-1

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2017, Springer, Berlin, 2011

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work page doi:10.3934/math.2026008 2026

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X. Fan, Q. Zhang,Existence of solutions forp(x)-Laplacian Dirichlet problem, Nonlinear. Anal. Theor.52(2003), 1843-1852

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Harjulehto, P.;Variable exponent Sobolev spaces with zero boundary values, Mathematica Bohemica, Vol.1, 132, No 2, (2007), 125–136

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M. A. Khamsi, W. M. Kozlowski,Fixed Point Theory in Modular Function Spaces, Birkhauser New York, 2015

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M. A. Khamsi, J. Lang, O. M´ endez, A. Nekvinda,A Dirichlet problem with unbounded non-standard growth conditions via uniform convexity of modulars, J. Differential Equa- tions,434(2025), https://doi.org/10.1016/j.jde.2025.113316

work page doi:10.1016/j.jde.2025.113316 2025

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M. A. Khamsi, P. Kumam, O.M´ endezFrom Modular Spaces to Boundary Value Problems: A Survey of Recent Advances, Carpathian J. Math,41(2) (2025), 425–440

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M. A. Khamsi, O. M´ endez,Modular uniform convexity structures and applications to boundary value problems with non-standard growth, J. Math. Anal. Appl.536(2), 2024. https://doi.org/10.1016/j.jmaa.2024.128203

work page doi:10.1016/j.jmaa.2024.128203 2024

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Kov´ aˇ cik, J

O. Kov´ aˇ cik, J. R´ akosn´ ık,On spacesLp(x), W k,p(x),Czechoslovak Mathematical Journal 41(4) (1991), 592-618

work page 1991

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Lindqvist,Notes on the Stationaryp-Laplace Equation, SpringerBriefs in Mathematics, Springer, Cham, 2019

P. Lindqvist,Notes on the Stationaryp-Laplace Equation, SpringerBriefs in Mathematics, Springer, Cham, 2019

work page 2019

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and Pokorn´ y, D.,On geometric properties of the spacesLp(·)(Ω),Rev

Lukeˇ s, J., Pick, L. and Pokorn´ y, D.,On geometric properties of the spacesLp(·)(Ω),Rev. Mat. Complut.24, 115−130 (2011). 24 B. DJAFARI ROUHANI, J. LANG, O. M ´ENDEZ

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72, (2010), 309–315

Manfredi, J., Rossi, J., Urbano, J.,Limits asp(x)→ ∞ofp(x)-harmonic functions, Nonlinear Analysis, vol. 72, (2010), 309–315

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Musielak, J.,Orlicz Spaces and Modular Spaces,Lecture Notes in Mathematics,1034, Springer-Verlag, Berlin/Heidelberg/New York/Tokyo, (1983)

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Nakano,Modulared Semi-ordered Linear Spaces, Maruzen Co., Tokyo, 1950

H. Nakano,Modulared Semi-ordered Linear Spaces, Maruzen Co., Tokyo, 1950. Osvaldo M´endez, Department of Mathematical Sciences, The University of Texas at El Paso, El Paso, TX 79968, USA Email address:osmendez@utep.edu Behzad Rouhani, Department of Mathematical Sciences, University of Texas at El Paso Email address:behzad@math.utep.edu

work page 1950