Existence and multiplicity results for a new $p(x)$-Kirchhoff problem

A. Harrabi; D.D. Repov\v{s}; F. Mtiri; M.K. Hamdani

REVIEW 4 major objections 4 minor 40 references

Existence and multiplicity results for a new $p(x)$-Kirchhoff problem

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

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Pith's one-line read A variable-exponent Kirchhoff problem has solutions for every real λ

desk verdict New a−b Kirchhoff term is worth studying, but both main theorems overclaim: the proofs only cover a restricted λ-range and the compactness and Fountain arguments have load-bearing gaps. read the letter →

arxiv 1908.08369 v1 pith:C6UROTML submitted 2019-08-22 math.AP

M.K. Hamdani , A. Harrabi , F. Mtiri , D.D. Repovš This is my paper · ORCID

classification math.AP MSC 35J5535J6535B65

keywords p(x)-LaplacianKirchhoffproblemvariableexponentnonlocalPalais-SmaleconditionMountainPasstheoremFountainmultiplicity

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

The paper studies a nonlocal $p(x)$-Kirchhoff problem in which the usual Kirchhoff coefficient is replaced by a minus sign: $a - b\int_\Omega \frac{1}{p(x)}|\nabla u|^{p(x)}dx$. It claims that under subcritical growth and an Ambrosetti–Rabinowitz condition on the nonlinearity, this problem has a nontrivial weak solution for every real parameter $\lambda$, and that if the nonlinearity is odd, it has infinitely many solutions with unbounded energy. The reason this is worth caring about is that the minus sign makes the energy functional noncoercive and bounded above, so standard variational arguments require a delicate compactness analysis below the critical energy level $a^2/(2b)$. The proof is carried out in variable-exponent Sobolev spaces via the mountain pass and fountain theorems.

What carries the argument

The load-bearing object is the nonlocal Kirchhoff term $a - b\int_\Omega \frac{1}{p(x)}|\nabla u|^{p(x)}dx$, which enters the energy as a negative quadratic term in the modular $\int_\Omega \frac{1}{p(x)}|\nabla u|^{p(x)}dx$. This creates a sharp energy ceiling $a^2/(2b)$; the Palais–Smale compactness is proved precisely when the energy level satisfies $c < a^2/(2b)$, and the proof splits on whether the Kirchhoff coefficient $a-b\int\frac{1}{p(x)}|\nabla u|^{p(x)}dx$ converges to zero or stays away from it. The $S_+$ property of the $p(x)$-Laplacian, the Poincaré inequality and compact Sobolev embeddings in variable-exponent spaces, and the mountain pass and fountain theorems carry the variational argument.

What would settle it

Take a bounded interval in one dimension with a non-monotone variable exponent $p(x)$, for which $\lambda_{p(\cdot)}=0$, and choose $g$ satisfying (g1)–(g3). Numerically or analytically check whether the energy functional $J$ has a strict local minimum at $u=0$ for some $\lambda>0$; if no such local minimum exists, the mountain pass geometry behind Theorem 1.1 fails for that $\lambda$.

Watch

Extended reading notes

Core claim

The central claim is that the Dirichlet problem $-\left(a-b\int_\Omega \frac{1}{p(x)}|\nabla u|^{p(x)}dx\right)\Delta_{p(x)} u = \lambda |u|^{p(x)-2}u + g(x,u)$ in a bounded smooth domain has a nontrivial weak solution for every $\lambda\in\mathbb{R}$ when $g$ satisfies (g1)–(g3), and infinitely many solutions $\{u_n\}$ with $I(u_n)\to\infty$ when $g$ is also odd (g4). The critical point is the energy functional $J(u)=a\int_\Omega \frac{1}{p(x)}|\nabla u|^{p(x)}dx - \frac{b}{2}\left(\int_\Omega \frac{1}{p(x)}|\nabla u|^{p(x)}dx\right)^2 - \lambda\int_\Omega \frac{1}{p(x)}|u|^{p(x)}dx - \int_\Omega G(x,u)dx$. Because of the negative nonlocal quadratic term, $J$ is bounded above by $a^2/(2b)$; the Palais–Smale condition is proved below that level, and the mountain pass and fountain geometries are established under the exponent range $p^+ < 2p^-$.

Load-bearing premise

The proof of the existence result for positive $\lambda$ requires that the variable-exponent Rayleigh quotient $\lambda_{p(\cdot)}$ is positive and that $\lambda$ stays below a threshold $\lambda^*$, even though the theorem states the result for every real $\lambda$; the paper itself notes that $\lambda_{p(\cdot)}$ can be zero in general.

Editorial extensions

If this is right

For every real $\lambda$, the problem admits at least one nontrivial weak solution under the stated growth and superlinearity assumptions on $g$.
With odd symmetry of $g$, the problem has infinitely many distinct weak solutions whose energy tends to infinity.
The variational solutions are genuine weak solutions in $W_0^{1,p(x)}(\Omega)$, since critical points of $J$ correspond exactly to weak solutions of the problem.
The compactness threshold $a^2/(2b)$ is sharp in the sense that the Palais–Smale condition fails for levels above this value.
The condition $p^+ < 2p^-$ is what lets the positive $p(x)$-energy dominate the negative nonlocal quadratic term near the origin, giving the mountain pass geometry.

Reading between the lines

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

The theorem as stated claims any $\lambda\in\mathbb{R}$, but the proof of the mountain pass geometry for $\lambda>0$ only covers $\lambda\in(0,\lambda^*)$ with $\lambda^*$ depending on the Rayleigh quotient $\lambda_{p(\cdot)}$; a reader should expect that the full claim needs $\lambda_{p(\cdot)}>0$ and a separate argument for $\lambda\ge\lambda^*$.
The threshold $a^2/(2b)$ resembles a resonance value: when the coefficient $a-b\int\frac{1}{p(x)}|\nabla u|^{p(x)}dx$ crosses zero, compactness at the critical level is lost, so one might test numerically whether solutions persist or new branches appear as $\lambda$ passes $\lambda^*$.
The same energy-ceiling mechanism could be adapted to critical growth nonlinearities if the threshold $a^2/(2b)$ is tuned against a sharp Sobolev constant, a direction the paper does not explore.

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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

Colleague,

The paper is about a genuinely new nonlocal term: a − b times the integral of (1/p(x))|∇u|^{p(x)}. That is a real variant on the usual a + b Kirchhoff problem and it changes the energy landscape, making the nonlocal part concave. The authors also correctly recognize that the Palais–Smale condition can only be expected below the threshold a^2/(2b), and they set up the variational machinery in the standard way. The problem itself is worth studying.

That said, the paper as written is not correct. Theorem 1.1 claims existence for any λ∈R, but the proof of the mountain pass geometry (Lemma 4.1) handles λ≤0 and, for λ>0, only 0<λ<λ*, with λ* defined through a quantity that requires λ_{p(.)}>0. The paper itself notes after (1.3) that λ_{p(.)}=0 in general. For λ larger than a λ_{p(.)}, the origin is not a strict local minimum; along a first eigenfunction the leading term is negative for small t. So the all-λ claim is unsupported and, in that range, false under the stated hypotheses. This is not a cosmetic gap.

The compactness proof also has a load-bearing hole. In Lemma 3.1, Subcase 2.1, the authors reach the equation λ|u|^{p-2}u + g(x,u)=0 and immediately conclude u=0. That does not follow; the equation has nontrivial solutions in general. The contradiction that J(u_n)→a^2/(2b) depends on this, so the (PS)_c proof is incomplete.

Theorem 1.2 inherits these problems and adds one of its own. The Fountain theorem is invoked with only (PS)_c proved for c0. And in the proof of condition (ii), the choice of ρ_k makes the lower bound collapse to zero rather than diverge to +∞. So the multiplicity claim is not established either.

The citations are fine, and the self-citations are not load-bearing. This is not a case of a clever proof with a small gap; the statements simply outrun the arguments. I would send it to a serious referee if it crosses a desk, but I would expect a reject with a clear request to fix the λ-range and the compactness argument, or to scale the claims down to what the proof actually shows.

Referee Report

4 major / 4 minor

Summary. The manuscript studies the nonlocal p(x)-Kirchhoff Dirichlet problem (1.1), whose Kirchhoff coefficient is a - b∫Ω (1/p(x))|∇u|^{p(x)} dx. The energy functional J is defined in (2.2). Theorem 1.1 claims that under the exponent condition (1.4) and hypotheses (g1)-(g3), for every real λ the problem has a nontrivial weak solution; Theorem 1.2 claims that adding the oddness hypothesis (g4) yields infinitely many solutions {u_n} with I(u_n)→∞. The proof strategy is: prove a Palais-Smale condition below the level a²/(2b) (Lemma 3.1), prove a local-minimum lemma (Lemma 4.1) and a mountain-pass geometry lemma (Lemma 4.2), invoke the Mountain Pass theorem for Theorem 1.1, and invoke the Fountain theorem for Theorem 1.2, with a compactness lemma for the finite-dimensional subspaces (Lemma 5.1).

Significance. If the claims were correct, the paper would add a new nonlocal Kirchhoff-type problem to the variable-exponent literature, and the explicit compactness threshold a²/(2b) is an interesting feature. The paper is appropriately ambitious in treating all real λ and in aiming at infinitely many high-energy solutions. However, several load-bearing steps are either unsupported or incorrect, and the proof as written does not establish either theorem. The paper contains no machine-checked proofs, no numerical verification, and no falsifiable quantitative predictions; its value rests entirely on the validity of the variational arguments.

major comments (4)

[§4, Lemma 4.1] Lemma 4.1 proves the local minimum only for λ≤0 and for 0<λ<λ*, with λ* defined in (4.2); for positive λ the proof divides by λ_{p(.)} and requires λ<a λ_{p(.)}. The range λ≥λ* is never treated, yet Theorem 1.1 asserts existence for every λ∈R. Moreover, Section 1 after (1.3) notes that λ_{p(.)} is zero in general, so the estimates in Lemma 4.1 are not available under the stated assumptions. In the constant-exponent case, taking a first eigenfunction φ of the p-Laplacian shows that for λ>a λ_p one has J(tφ)<0 for all sufficiently small t, so the claimed local minimum fails without an additional assumption.
[§3, Lemma 3.1, Step 2, Subcase 2] The proof asserts that the pointwise identity λ|u|^{p(x)-2}u+g(x,u)=0 a.e. forces u=0. This is not correct as stated. For example, take λ=0 and let g(x,s)=0 for |s|≤1 and g(x,s)=|s|^{q(x)-2}s for |s|>1, with q satisfying (1.4); this g satisfies (g1)-(g3) and (g4). Any nonzero W^{1,p(x)}_0 function with values in [0,1/2] a.e. then satisfies g(x,u)=0 a.e. and hence φ'(u)=0, so u need not vanish. The conclusion φ(u_n)→0 and J(u_n)→a²/(2b) is therefore not justified by the arguments given; additional reasoning would be needed to show that the energy limit contradicts c<a²/(2b).
[§4, proof of Theorem 1.1] The mountain pass level c is never shown to satisfy c<a²/(2b). Lemma 3.1 gives the Palais-Smale condition only at levels below a²/(2b), but the proof of Theorem 1.1 simply states that the Mountain Pass theorem applies after Lemmas 3.1-4.2. The minimax level is not estimated. This is not automatic: for λ<0, the term -λ∫(1/p)|u|^p is positive and can push the energy above a²/(2b) along the relevant paths. Thus the application of Lemma 3.1 to the mountain pass sequence is not justified.
[§5, proof of Theorem 1.2] The application of the Fountain theorem is invalid in two independent ways. First, Lemma 3.1 only proves the (PS)_c condition for c<a²/(2b), whereas Theorem A requires the condition for every positive level and the conclusion produces critical points with I(u_n)→∞. Second, condition (ii) of Theorem A is not established: the choice of ρ_k in the proof only shows J(tφ)≥0 for one particular t and φ∈Z_k with ||φ||=1; it does not show that b_k=inf_{u∈Z_k, ||u||=ρ_k} J(u) tends to +∞. In fact, along every fixed nonzero direction u one has J(tu)→−∞ as t→∞ because of the term -(b/2)(∫(1/p)|∇u|^p)^2, so the desired b_k→+∞ is incompatible with the functional's behavior as it is described in the paper.

minor comments (4)

[§3, Lemma 3.2] Parts (i) and (ii) are written as equalities, but the proofs establish convergence to zero; they should state that the integrals tend to 0 as n→∞.
[§4, Lemma 4.2] The displayed chain 'θ>2p^->p^+>p^-' is inconsistent with hypothesis (g3), because (g3) gives θ<2(p^-)^2/p^+<2p^-; the conclusion J(tψ)→−∞ still follows from the negative t^{2p^-} term, but the inequality chain should be corrected.
[§3, Remark 3.1] The inequality J(u)≤a∫(1/p)|∇u|^p -(b/2)(∫(1/p)|∇u|^p)^2 requires -λ∫(1/p)|u|^p -∫G(x,u)≤0, which is not a consequence of (g1)-(g3) for arbitrary λ∈R; the remark is therefore not correct as stated.
[§5, proof of Theorem 1.2] In the paragraph after the definition of ρ_k, the text says 'ρ_k ∈ Z_k with ||φ||=1'; this should be 't=ρ_k with φ∈Z_k and ||φ||=1'. More importantly, the estimate at that radius only gives a lower bound of zero for a single element, not the infimum over a sphere.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the variational derivation is self-contained and does not reduce to its own inputs.

full rationale

The paper's derivation chain is a standard variational argument: it defines the energy functional J in (2.2), proves a Palais-Smale compactness condition (Lemma 3.1) at levels c < a^2/(2b), establishes mountain-pass geometry (Lemmas 4.1 and 4.2), and then invokes the classical Mountain Pass and Fountain theorems. No parameter is fitted to a subset of data and later called a prediction; no existence claim is defined in terms of the object it is supposed to derive. The cited results used in the proof are standard theorems on variable-exponent Sobolev spaces and the Mountain Pass theorem, not self-cited 'uniqueness' theorems. The self-citations [22, 23] appear only in the literature overview and are not load-bearing for Theorems 1.1 and 1.2. The mismatch between the advertised claim 'for any λ ∈ R' and the proof's explicit restriction to λ ≤ 0 and 0 < λ < λ* in Lemma 4.1 is a correctness gap or unstated-assumption issue, not a circular dependency: the proof does not assume the conclusion to prove it, nor does it rename a fitted quantity as a prediction. Therefore no circular step satisfies the evidentiary standard required for a nonzero score.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The paper relies on standard variable exponent Sobolev space theory and on the Mountain Pass and Fountain theorems. It additionally uses, without stating, a positive Rayleigh quotient lambda_{p(.)}, a nonnegativity condition on G, and an exponent restriction p+ < 2(p^-)^2. No data-driven parameters or invented entities appear.

free parameters (1)

lambda* = given by (4.2), depends on rho, epsilon, lambda_{p(x)}
A threshold introduced in Lemma 4.1 to prove the local minimum for lambda in (0, lambda*). The theorem claims all lambda, so this parameter marks the gap between proof and claim.

assumptions (5)

standard math Standard properties of variable exponent Lebesgue and Sobolev spaces: reflexivity, compact Sobolev embeddings for q < p*, Poincare inequality, and S+ property of the p(x)-Laplacian.
Invoked throughout Section 2 and used in Lemma 3.1; citations [14,18,21].
standard math Mountain Pass theorem and Fountain theorem (Willem [38]) are applicable.
Used in Sections 4 and 5 to convert geometric and compactness conditions into critical points.
ad hoc to paper The principal Rayleigh quotient lambda_{p(.)} is positive and lambda < a lambda_{p(.)} for positive lambda.
Lemma 4.1 uses division by lambda_{p(x)} and requires a - lambda/lambda_{p(x)} > 0. The paper only notes lambda_{p(.)} > 0 under special conditions, not assumed in Theorems 1.1-1.2.
ad hoc to paper The nonlinearity satisfies G(x,u) >= 0 and lambda >= 0 so that J(u) <= a^2/(2b).
Remark 3.1 and Lemma 3.1 use the upper bound a^2/(2b) for J, which requires -lambda integral |u|^p/p - integral G <= 0. The stated (g1)-(g3) allow G < 0 and lambda < 0.
ad hoc to paper Exponent condition p+ < 2(p^-)^2 holds.
Proof of Theorem 1.2 condition (ii) needs a/p+ - b/(2p-^2) > 0, equivalent to p+ < 2(p^-)^2, stronger than the stated p+ < 2p^-.

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Pith. "Pith review of Existence and multiplicity results for a new $p(x)$-Kirchhoff problem." pith.science (2026). https://pith.science/paper/C6UROTML

@misc{pith2026190808369,
  author       = {Pith},
  title        = {Pith review of: Existence and multiplicity results for a new $p(x)$-Kirchhoff problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C6UROTML}},
  note         = {Machine review of arXiv:1908.08369}
}

abstract

We study the existence and multiplicity results for the following nonlocal $p(x)$-Kirchhoff problem: \begin{equation} \label{10} \begin{cases} -\left(a-b\int_\Omega\frac{1}{p(x)}| \nabla u| ^{p(x)}dx\right)div(|\nabla u| ^{p(x)-2}\nabla u)=\lambda |u| ^{p(x)-2}u+g(x,u) \mbox{ in } \Omega, \\ u=0,\mbox{ on } \partial\Omega, \end{cases} \end{equation} where $a\geq b > 0$ are constants, $\Omega\subset \mathbb{R}^N$ is a bounded smooth domain, $p\in C(\overline{\Omega})$ with $N>p(x)>1$, $\lambda$ is a real parameter and $g$ is a continuous function. The analysis developed in this paper proposes an approach based on the idea of considering a new nonlocal term which presents interesting difficulties.

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Pith tools

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

[1] [1]

Aboulaich, D

R. Aboulaich, D. Meskine, and A. Souissi. New di ﬀusion models in image processing. Comput. Math. Appl., 56(4):874–882, 2008

work page 2008

[2] [2]

Acerbi and G

E. Acerbi and G. Mingione. Gradient estimates for the p(x)-Laplacean system. J. Reine Angew. Math., 584:117–148, 2005

work page 2005

[3] [3]

Adamowicz and P

T. Adamowicz and P . H¨ ast¨ o. Harnack’s inequality and the strong p(·)-Laplacian. J. Diﬀerential Equations, 250(3):1631–1649, 2011

work page 2011

[4] [4]

C. O. Alves and J. L. P . Barreiro. Multiple solutions for a class of quasilinear problems involving variable exponent s. Asymptot. Anal. , 96(2):161–184, 2016

work page 2016

[5] [5]

Antontsev, M

S. Antontsev, M. Chipot, and Y . Xie. Uniqueness results f or equations of the p(x)-Laplacian type. Adv. Math. Sci. Appl. , 17(1):287–304, 2007

work page 2007

[6] [6]

S. N. Antontsev and J. F. Rodrigues. On stationary thermo -rheological viscous ﬂows. Ann. Univ. Ferrara Sez. VII Sci. Mat. , 52(1):19–36, 2006

work page 2006

[7] [7]

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