An iterative method for Kirchhoff type equations and its applications

Qiuyi Dai

REVIEW 3 major objections 4 minor 27 references

An iterative method for Kirchhoff type equations and its applications

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves a sandwich-iteration existence theorem for nonlocal Kirchhoff-Carrier equations with minimal assumptions on the coefficient function.

desk verdict The invariant-set idea is genuinely new and the examples are nontrivial, but the main proof rests on a false fixed-point identity and an unjustified gradient bound, so Theorem 1.1 is not established as written. read the letter →

arxiv 1908.07703 v2 pith:ZYEQPMLY submitted 2019-08-21 math.AP

Qiuyi Dai This is my paper · ORCID

classification math.AP MSC 35J6035B5047H10

keywords KirchhofftypeequationsCarrierinvariantsetfixedpointtheoremsub-supersolutionnonlocalellipticSchauderorderinterval

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This revised paper sets out to repair a gap in an earlier proof and, in doing so, to prove an existence theorem for generalized Kirchhoff-Carrier equations of the form $-A(\|u\|_p,\|\nabla u\|_2)\Delta u=g(x,u)$ with $u=0$ on the boundary, where the nonlocal coefficient $A$ is only assumed continuous and bounded below by a positive constant. The method is an invariant-set iteration: from a sub-solution $\phi$ and super-solution $\psi$ of the underlying Laplacian problem, the author builds an order interval $[r_M\phi,\psi]$, shows that a Green's-function fixed-point map carries this interval into itself, and applies Schauder's fixed-point theorem. The advertized payoff is that no monotonicity of $A$ is required, which is exactly the assumption that blocked earlier sub-supersolution approaches for non-variational Kirchhoff-Carrier problems. The paper then applies the abstract theorem to four concrete examples, including an inhomogeneous Carrier equation, a concave-convex Kirchhoff-Carrier problem, a non-monotone operator, and a sign-changing nonlinearity.

What carries the argument

The invariant set is the order interval $[r_M\phi,\psi]=\{v\in X: r_M\phi(x)\le v(x)\le\psi(x)\text{ in }\Omega\}$, together with the Green's-function map $T$ defined above. The constant $r_M$ is chosen so that the lower endpoint of the interval is preserved: the map multiplies the sub-solution contribution by $r_M^\alpha/A(\cdot)$, and $r_M$ is defined by the relation $\frac{1}{M}r_M^\alpha=r_M$, which makes the lower-bound estimate exact. The upper bound comes from $g\le -\Delta\psi$ and the positivity of $A$. Compactness of $T$ on the interval is obtained from uniform H\"older bounds on $T([r_M\phi,\psi])$, using Green's-function estimates from the Gilbarg-Trudinger textbook, so Schauder's fixed-point theorem applies.

What would settle it

Choose any pair $\phi\le\psi$ satisfying (G1)-(G2) and examine all $w$ in $[r_M\phi,\psi]$; compute $\|\nabla w\|_2$. If the maximum of $\|\nabla w\|_2$ over this interval exceeds $\|\nabla\psi\|_2$, the key bound $A(\|w\|_p,\|\nabla w\|_2)\le M$ is not guaranteed, and the invariance step fails. Independently, checking the displayed identity $\frac{1}{M}r_M^\alpha=r_M$ against the series definition (1.3) for $M>1$ and $0<\alpha<1$ decisively tests the lower-endpoint calculation.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: if $1<p\le 2^*$ and $g$ satisfies the trapping conditions (G1) and (G2), then the problem $-A(\|u\|_p,\|\nabla u\|_2)\Delta u=g(x,u)$ with $u=0$ on $\partial\Omega$ has at least one nonnegative solution $u$ satisfying $r_M\phi\le u\le\psi$ in $\Omega$. The proof defines $M=\max\{A(s,t):0\le s\le\|\psi\|_p,\,0\le t\le\|\nabla\psi\|_2\}$ and a ratio constant $r_M$ through a series so that $\frac{1}{M}r_M^\alpha=r_M$; the map $T(v)=A(\|v\|_p,\|\nabla v\|_2)^{-1}\int_\Omega G(x,y)g(y,v(y))\,dy$ is then shown to send the order interval $[r_M\phi,\psi]$ into itself. Compactness comes from uniform H\"older estimates supplied by the Green's function, so Schauder's fixed-point theorem yields a fixed point, which is the desired classical solution. The method's advertised advantage is that the only hypotheses on $A$ are continuity and a positive lower bound.

Load-bearing premise

The whole construction rests on the claim that every function $w$ in the order interval $[r_M\phi,\psi]$ satisfies $\|w\|_p\le\|\psi\|_p$ and $\|\nabla w\|_2\le\|\nabla\psi\|_2$, so that $A(\|w\|_p,\|\nabla w\|_2)\le M$; if pointwise order does not control these norms, the invariant-set claim collapses.

Editorial extensions

If this is right

Any Kirchhoff-Carrier problem whose nonlinearity is trapped between a sub-solution and a super-solution in the sense of (G1)-(G2) has a solution in the order interval $[r_M\phi,\psi]$, whether or not $A$ is monotone.
The inhomogeneous Carrier equation $-(1+d\|u\|_2^2)\Delta u=u^p+\lambda f(x)$ has at least one positive solution for every $\lambda\in(0,\lambda_f)$.
The concave-convex Kirchhoff-Carrier problem $-(1+c\|u\|_2^2+d\|\nabla u\|_2^2)\Delta u=\mu u^q+u^p$ has a positive solution for small $\mu$, with a lower bound of the form $r_M\phi$.
Nonmonotone coefficients, such as $1+d\sin^2(\|\nabla u\|_2)$, and sign-changing nonlinearities fall within the same theorem, as demonstrated by the paper's Examples 3 and 4.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The order-interval technique should transplant to systems of Kirchhoff-Carrier type whenever a vector-valued Green's function with the same regularity estimates is available, since the proof never uses scalar structure beyond the comparison principle.
A natural stress test is to let $A$ oscillate rapidly between positive bounds or even take $A$ discontinuous; the theorem's hypotheses do not explicitly rule this out, so a limiting or relaxed version may follow from the same sandwich estimate.
The defining relation $\frac{1}{M}r_M^\alpha=r_M$ suggests an alternative formulation: rather than the series in (1.3), define $r_M$ as the fixed point of $s\mapsto M^{-1}s^{\alpha}$, which would make the invariance computation directly checkable and might yield sharper constants.
The paper's examples all use explicit barriers tied to known solutions of $-\Delta u=1$; extending the method to unbounded domains or to operators without an explicit Green's function would require replacing the Schauder compactness step with a different compactness argument.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Qiuyi Dai's paper proposes an invariant-set method for Kirchhoff-Carrier equations of the form -A(||u||_p, ||∇u||_2) Δu = g(x,u), with the advertised advantage that A needs only continuity and a positive lower bound, not monotonicity. That is a genuinely new angle. The cited sub-supersolution papers rely on monotonicity of the Kirchhoff operator; the Laplacian-comparison idea sidesteps that, and the examples in Section 2, especially the sin^2 A and sin^2 g case, are not covered by those results. If the theorem were true, this would be a useful advance in the subfield.

What the paper does well is clear. The method is explained simply, the examples are concrete and well-chosen, and the author is honest enough to present this as a revision filling a gap. The literature engagement is appropriate, and the applications genuinely illustrate non-monotone cases.

The soft spots are large, though, and they are in the load-bearing step. Claim (i), the inclusion T([r_M φ, ψ]) ⊂ [r_M φ, ψ], uses two unjustified premises. First, the identity (1/M) r_M^α = r_M is simply false for the r_M defined by the series in (1.3) when 0<α<1. With M=4 and α=1/2, r_M=1/2 but (1/M) r_M^α ≈ 0.177. The applications in Theorems 2.2 and 2.4 silently use the true fixed point M^{-1/(1-α)}, which is not the value defined in (1.3). Second, and more fundamentally, for w in the pointwise interval [r_M φ, ψ], you get ||w||_p ≤ ||ψ||_p, but pointwise order says nothing about ||∇w||_2. On Ω=(0,1) with ψ=x(1-x), the functions w_N=ψ sin(Nπx) are between 0 and ψ but have ||∇w_N||_2→∞. So for A(s,t)=1+t^2, the bound A(||w||_p,||∇w||_2)≤M fails. The proof of Claim (i) therefore does not go through, and the Schauder fixed-point step has no valid invariant set to work with.

The theorem may be repairable: define r_M as the positive fixed point of r = r^α/M, and either restrict the interval to functions with controlled gradient or define M using the uniform C^{1,τ} bound that the Green's function argument provides. But neither fix is in the paper.

My recommendation: I would not cite this as a theorem, but I would not dismiss the method either. If I were the editor, I'd send it to a referee—the approach deserves an expert look, and a careful referee could tell the author exactly what to fix. As it stands, it should not be accepted; the main existence result is unproven as written.

Referee Report

3 major / 4 minor

Summary. This paper proposes an invariant-set method for the Dirichlet problem -A(||u||_p,||∇u||_2)Δu=g(x,u) on a bounded domain, where A is only assumed continuous with a positive lower bound. The main abstract result (Theorem 1.1) asserts existence of a nonnegative solution in the ordered interval [r_M φ, ψ] under hypotheses (G1)-(G2), where r_M is defined by a geometric series in (1.3). The proof defines a Green's-function map T, claims that T maps the interval into itself, and applies Schauder's fixed point theorem. Four applications are then given: a Carrier equation, a concave-convex Kirchhoff-Carrier equation, a nonmonotone sine equation, and a sign-changing nonlinearity.

Significance. If Theorem 1.1 were valid, the method would be significant because it would remove monotonicity assumptions on the nonlocal coefficient A that are typical in sub-supersolution approaches. The range of examples, including non-variational and non-monotone problems, is attractive and illustrates the intended scope. However, the central invariant-set claim rests on two checkable assertions that are false or unjustified: the identity (1/M)r_M^α=r_M for the r_M defined in (1.3), and the bound A(||w||_p,||∇w||_2)≤M for w in a pointwise ordered interval. These assertions are load-bearing for the Schauder fixed-point step, so Theorem 1.1 is not proved as stated; the applications with α=q∈(0,1) also silently use a different quantity, M^{-1/(1-q)}, instead of r_M. The paper does not ship machine-checked proofs or code; the proof is short and the defects are arithmetically checkable.

major comments (3)

[§1, Claim (i)] The identity (1/M)r_M^α=r_M stated immediately after (1.3) is false for general 0<α<1. With r_M=(1/M)∑_{k=0}^{∞}α^k=1/(M(1-α)), the equation (1/M)r_M^α=r_M is equivalent to r_M=M^{-1/(1-α)}, which coincides with 1/(M(1-α)) only for special pairs (M,α). For example, M=4 and α=1/2 give r_M=1/2, but (1/M)r_M^α=(1/4)√(1/2)≠1/2. This identity is used in the lower-bound half of Claim (i), so the invariant-set inclusion is not established. A further consequence is that r_M can exceed 1 when M(1-α)<1; in that case the interval [r_M φ,ψ] need not be nonempty under hypothesis (G1), and condition (G2) cannot be applied with β=r_M.
[§1, Claim (i)] For w∈[r_M φ,ψ], pointwise order gives only ||w||_p≤||ψ||_p; it gives no bound on ||∇w||_2. The proof uses A(||w||_p,||∇w||_2)≤M, where M is the maximum of A over [0,||ψ||_p]×[0,||∇ψ||_2]. That bound is unavailable in general: with Ω=(0,1), ψ(x)=x(1-x), and w_N(x)=ψ(x)|sin(Nπx)|, one has 0≤w_N≤ψ but ||∇w_N||_2→∞, so for A(s,t)=1+t^2 and large N, A(||w_N||_p,||∇w_N||_2)>M. Thus the inclusion T([r_M φ,ψ])⊂[r_M φ,ψ] is not proved by the displayed inequalities.
[§2, Theorems 2.2 and 2.4] In the proofs of Theorems 2.2 and 2.4 the paper computes lim_{n→∞}∑_{k=0}^{n-1}q^k=1/(1-q) and then invokes Theorem 1.1 to obtain the lower bound (1/(1+c||ψ||_2^2+d||∇ψ||_2^2))^{1/(1-q)} φ in Theorem 2.2 and the analogous quantity in Theorem 2.4. This number is M^{-1/(1-q)}, the true fixed point of r=(1/M)r^q, not the value r_M=1/(M(1-q)) defined in (1.3). Since Theorem 1.1 is not proved, and since even its proof would yield r_M as defined, the printed lower bounds are not consequences of the stated argument.

minor comments (4)

In the lower-bound estimate, the denominator is written as A(||v||_p,||∇v||_2) though v is not defined; it should be A(||w||_p,||∇w||_2).
The final sentence states the solution satisfies r_M φ≤u≤φ, but the upper function is ψ, not φ.
There are several typographical issues: 'Schaulder' should be 'Schauder', 'Drichlet' should be 'Dirichlet', and 'Clam' should be 'Claim'; the accent in 'Hölder' is also inconsistent.
The title says 'Kirchhoff type equations' while the body consistently uses 'Kirchhoff-Carrier type equations'; the terminology should be unified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the invariant-set argument is self-contained and not equivalent to its inputs, though the proof contains a false identity and an invalid gradient estimate.

full rationale

The paper's central claim, Theorem 1.1, is an existence result obtained by constructing an explicit invariant set [r_M phi, psi] for the map T. The constant r_M is defined directly from M and alpha by formula (1.3), not fitted to the sought solution u, and the conclusion r_M phi <= u <= psi is not used as an input anywhere. The proof does invoke the identity (1/M) r_M^alpha = r_M and the estimate A(||w||_p, ||nabla w||_2) <= M for w in [r_M phi, psi], but these are asserted properties of the construction, not definitions or renamed predictions. The identity is in fact not a consequence of (1.3) for general alpha, and the gradient bound is not controlled by pointwise order; these are mathematical errors in the proof, not circular reasoning. The paper contains no load-bearing self-citation: the cited works are background references on Kirchhoff and Carrier problems, and none is invoked to forbid alternative approaches or to supply a uniqueness theorem on which the argument depends. The applications in Section 2 independently verify the hypotheses (G1)-(G2) and then apply the abstract theorem; they do not fit parameters to the desired solutions. Thus the derivation chain does not reduce to its own assumptions, and the appropriate circularity score is 0.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The proof rests on standard fixed-point and elliptic machinery plus two ad hoc claims that are not valid. No new physical entities are introduced and no data are fitted.

free parameters (1)

r_M = series definition in (1.3); applications use M^{-1/(1-q)}
The lower-bound threshold is chosen so that T maps [r_M phi, psi] into itself. The series definition does not satisfy the required fixed-point relation, and Section 2 uses a different value, so r_M functions as an ad hoc choice that is not consistently defined.

assumptions (4)

domain assumption The Dirichlet Laplacian on Omega has a Green's function and enough boundary regularity for C^{1,tau} estimates and classical solutions.
Used in Section 1 via the Green's representation and Gilbarg-Trudinger [7]. The paper only states that Omega is bounded, so boundary regularity is an implicit extra assumption.
ad hoc to paper The identity (1/M) r_M^alpha = r_M holds for the r_M defined in (1.3).
Stated without proof after (1.3) and used in Claim (i). It is false for the series definition except in special cases such as alpha = 0.
ad hoc to paper For w in [r_M phi, psi], the gradient bound ||nabla w||_2 <= ||nabla psi||_2 holds.
Implicitly used in Claim (i) to bound A(||w||_p,||nabla w||_2) by M. Pointwise order does not imply this gradient bound.
standard math Schauder fixed point theorem, Ascoli-Arzela theorem, and elliptic regularity theory apply as stated.
Invoked in the proof of Theorem 1.1 without proof. These are acceptable background results if the operator and domain satisfy the hypotheses, but the domain regularity needed for classical solutions is not stated.

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Cite this review

Pith. "Pith review of An iterative method for Kirchhoff type equations and its applications." pith.science (2026). https://pith.science/paper/ZYEQPMLY

@misc{pith2026190807703,
  author       = {Pith},
  title        = {Pith review of: An iterative method for Kirchhoff type equations and its applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZYEQPMLY}},
  note         = {Machine review of arXiv:1908.07703}
}

read the original abstract

This is a new version of our previous work. In this version, we fill a gap included in the original proof of Theorem 1.1 in our previous paper entitled "An iterative method for Kirchhoff type equations and its applications".

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

C. O. Alves and F. J. S. A. Correa, On existence of solution s for a class of problem involving a nonlinear operator, Comm. Appl. Nonlinear Anal . 8 (2001), 43-56

work page 2001

[2] [2]

C. O. Alves and F. J. S. A. Correa, A sub-supersolution app roach for a quasilinear Kirchhoﬀ equation, arXiv: 1405.6857v1

work page Pith review arXiv

[3] [3]

G. A. Afrouzi, N. T. Chung and S. Shakeri, Existence of pos itive solutions for Kirchhoﬀ type equations, Electronic J. of Diﬀerential Equat ions, 2013(180)(2013), 1-8

work page 2013

[4] [4]

F. J. S. A. Correa, M. Delgado and A. Suarez, Some nonlinea r heterogeneous problems with nonlocal reaction term, Adv. Diﬀerential Equa tions, 16 (2011), 623-641

work page 2011

[5] [5]

Dai and R

G. Dai and R. Ma, Solutions for a p(x)-Kirchhoﬀ type equat ion with Neumann boundary data, Nonlinear Anal. R W A, 12 (2011), 2666-2680

work page 2011

[6] [6]

G. M. Figueiredo, A. Suarez, Some remarks on the comparis on principle in Kirch- hoﬀ equations, arXiv: 1510.02151v1, 7 Oct. 2015. 9

work page Pith review arXiv 2015

[7] [7]

Gilbarg, N

D. Gilbarg, N. S. Trudinger, Elliptic Partial Diﬀerentia l Equations of Second Or- der, Springer Berlin Heidelberg 2001

work page 2001

[8] [8]

Han and G

X. Han and G. Dai, On the sub-supersolution method for p(x )-Kirchhoﬀ type equations, Journal of Inequalities and Applications, 2012 , (2012): 283

work page 2012