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REVIEW 2 major objections 4 minor 38 references

Existence of solutions for a nonlocal type problem in fractional Orlicz Sobolev spaces

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Pith's one-line read A fractional nonlocal Dirichlet problem has a nontrivial weak solution

desk verdict Plausible existence theorem for the fractional a-Laplacian in Orlicz spaces, but the proof of J ∈ C^1 rests on a false lemma and needs a standard fix before the main result is rigorous. read the letter →

arxiv 1908.07806 v2 pith:TH6GS7OZ submitted 2019-08-21 math.AP

Elhoussine Azroul , Abdelmoujib Benkirane , Mohammed Srati This is my paper · ORCID
classification math.AP MSC 35R1146E3058E0535J60
keywords fractionalOrlicz-Sobolevspacesa-LaplacianN-functionweaksolutiondirectmethodincalculusofvariationscompactembeddingDelta2condition
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 tries to establish existence of nontrivial weak solutions for the Dirichlet problem driven by the fractional $a$-Laplacian, a nonlocal operator in which the usual power $t^p$ is replaced by a general convex N-function $A$. To do this it first develops the functional setting: the fractional Orlicz-Sobolev space $W^sL_A(\Omega)$ is shown to be a separable reflexive Banach space with a Poincaré inequality and compact embeddings into Lebesgue spaces. It then proves that, when the nonlinearity has subcritical growth with exponent below $p_0$ and is bounded below on a subdomain, the associated energy functional is coercive and weakly lower semicontinuous, so the direct method yields a minimizer. A separate argument shows the minimizer is not zero. The point is that existence theory for fractional $p$-Laplacian problems extends to non-power growth, which broadens the class of nonlinearities and operators for which the problem is known to be solvable.

What carries the argument

The central object is the fractional Orlicz-Sobolev space $W^s_0L_A(\Omega)$, defined as the closure of smooth compactly supported functions under the norm $\|u\|_A + [u]_{s,A}$, where $[u]_{s,A}$ is the infimum of $\lambda>0$ such that $\iint A(|u(x)-u(y)|/(\lambda|x-y|^s))\,dxdy/|x-y|^N \le 1$. The energy functional $I$ carries the argument: Proposition 3.10 converts the double integral of $A$ into powers of the seminorm under condition (2.2), giving coercivity, while convexity of $J(u)=\iint A(h_{x,y}(u))\,dxdy/|x-y|^N$ plus compact embedding into $L^q$ gives weak lower semicontinuity. The direct method in calculus of variations then produces the minimizer, and differentiability of $I$ makes that minimizer a weak solution.

What would settle it

Compute the ratio in Lemma 2.1 for $A(t)=t^p$, $p>2$: then $a(t)=pt^{p-1}$ and $A(a(t))=p^p t^{p(p-1)}$, so $A(a(t))/A(t)=p^p t^{p(p-1)-p}$, which is unbounded as $t\to\infty$. Since the lemma asserts the ratio is bounded, this case directly falsifies the lemma as stated.

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Extended reading notes

Core claim

The central claim is Theorem 4.2: if $A$ is an N-function satisfying the growth condition $1<p_0:=\inf_{t>0} tA'(t)/A(t) \le \sup_{t>0} tA'(t)/A(t)<\infty$ and $f$ is a Carathéodory function with $|f(x,t)|\le \theta_1(1+|t|^{q-1})$ everywhere and $|f(x,t)|\ge \theta_2 |t|^{q-1}$ on a subdomain, with $1<q<p_0$, then the equation $(-\Delta)_a^s u = f(x,u)$ with $u=0$ outside $\Omega$ has a nontrivial weak solution in $W^s_0L_A(\Omega)$. The proof identifies the right function space and shows the energy functional $I(u)=\iint A(|u(x)-u(y)|/|x-y|^s)\,dxdy/|x-y|^N - \int_\Omega F(x,u)\,dx$ attains a negative minimum, which by differentiability of $I$ is a weak solution. The paper also claims the underlying space is complete, separable, reflexive, and compactly embedded in the appropriate Lebesgue spaces, statements it proves in Section 3.

Load-bearing premise

The proof that the energy functional is differentiable relies on the inequality $A(a(t))\le cA(t)$, which fails for the standard power N-function $A(t)=t^p$ with $p>2$; without a valid replacement, the step converting the minimizer into a weak solution does not follow.

Editorial extensions

If this is right

  • For every N-function satisfying (2.2) and every right-hand side growing subcritically below $p_0$, the fractional $a$-Laplacian Dirichlet problem has a nontrivial solution, so the result is not tied to power-type growth.
  • The compact embedding of $W^s_0L_A(\Omega)$ into $L^q(\Omega)$ for $q<p_0^*$ gives a reusable compactness tool for variational problems in fractional Orlicz-Sobolev spaces.
  • The Poincaré-type inequality makes the Gagliardo seminorm an equivalent norm, so coercivity of functionals on this space can be checked using the seminorm alone.
  • In the borderline case $q=p_0$, a weak solution still exists provided the growth constant is below half of the first eigenvalue $\lambda_1$ defined in (4.1).

Reading between the lines

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

  • If the differentiability step that relies on the inequality $A(a(t))\le cA(t)$ is replaced by a valid argument, the same direct-method scheme should handle more general Carathéodory nonlinearities, including sign-changing or oscillatory right-hand sides, as long as the compact embedding remains valid.
  • The structure suggests that an eigenvalue theory for the fractional $a$-Laplacian could be developed around $\lambda_1$ as defined in (4.1), with higher eigenvalues plausibly obtainable by minimax methods on the uniformly convex space $W^s_0L_A(\Omega)$.
  • On bounded domains, one could test whether the global $\Delta_2$ condition can be relaxed to a near-infinity $\Delta_2$ condition; the finite measure of $\Omega$ may still force the compactness and separability properties used in the proof.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper studies the Dirichlet problem driven by the fractional a-Laplacian in fractional Orlicz-Sobolev spaces W^s_0 L_A(Ω). It first establishes basic qualitative properties of these spaces in Section 3: completeness, reflexivity, separability, a Poincaré-type inequality, and continuous and compact embeddings into Lebesgue spaces. In Section 4, using the direct method of the calculus of variations, it claims existence of a nontrivial weak solution when the N-function A satisfies condition (2.2) and the nonlinearity f satisfies (f1)-(f2) with 1 < q < p0; a corollary treats q = p0 under an additional smallness hypothesis. The proof that the energy functional is C^1 relies on Lemma 2.1, and that lemma is the central point of failure in the manuscript as written.

Significance. If completed, the paper would provide a useful extension of fractional Sobolev space theory to Orlicz settings and a corresponding existence theorem for nonlocal problems with nonstandard growth. The Section 3 results on embeddings, the Poincaré-type inequality, and the basic space properties are broadly standard and appear to be correct in outline; the direct-method strategy is appropriate and there are no fitted parameters or circular assumptions. The main weakness is that the differentiability of the energy functional, which is required to identify the minimizer as a weak solution, is not proven because it depends on a false lemma and on an invalid Hölder-type estimate. The existence result may be salvageable with a corrected Orlicz duality argument, but as written the central claim is unsupported.

major comments (2)
  1. [Section 2, Lemma 2.1] Lemma 2.1, which asserts A(a(t)) ≤ c A(t) for all t ≥ 0 under the global Δ2-condition, is false as stated. For A(t) = t^p/p with p > 2, condition (2.2) holds with p0 = p0 = p and a(t) = t^{p-1}, yet A(a(t)) = t^{p(p-1)}/p is not bounded by a constant multiple of A(t) = t^p/p as t → ∞. This counterexample lies inside the paper's own standing hypothesis, so the lemma cannot be used in the form stated.
  2. [Section 4, Lemma 4.5] The proof that J ∈ C^1 uses Lemma 2.1 to conclude that |a(|h_{x,y}(u)|)| belongs to L^A(Ω×Ω,dμ). Since Lemma 2.1 is false, this placement is unjustified. Moreover, even if such a bound held, the subsequent estimate applies Hölder's inequality with both U_{x,y}(u) and h_{x,y}(v) in the same Orlicz space L^A; products of two functions in the same Orlicz space are not generally integrable, so the inequality as written is not valid. The correct argument would need a(|h|) to lie in the complementary Orlicz space L^{\tilde A}, using the standard estimate \tilde A(a(t)) ≤ A(2t) ≤ C A(t), and a genuine duality pairing. Consequently, Lemma 4.5 does not establish that I is C^1, and the minimizer produced in Theorem 4.2 cannot be identified as a weak solution by the argument given.
minor comments (4)
  1. [Section 4, Lemma 4.4] In the proof of Lemma 4.4 the functional being considered is H, but the text writes J'(u_n) - J'(u); the dual norm should also be with respect to W^s_0 L_A(Ω), not W^{s,p}_0(Ω).
  2. [Section 2, equation (2.2)] The notation p0 and p0 in (2.2) and throughout the paper is very easily confused; please use distinguished symbols such as p_- and p_+ or p_0 and p^0 consistently.
  3. [Theorem 3.4] In the completeness proof, the existence of a single λ that works uniformly for the Cauchy sequence is asserted in passing; the authors should justify this explicitly, for example by noting that the sequence is bounded in the Luxemburg seminorm.
  4. [Throughout] There are numerous typographical errors ('Levy', 'Orlicz-Soboliv', 'Soblev', 'integerdivide', and several others); a careful proofreading is needed before resubmission.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the existence proof is a direct-method argument over independently cited fractional Orlicz-Sobolev spaces, with no fitted inputs or self-referential predictions.

full rationale

I walked the paper's derivation chain. The functional framework is the fractional Orlicz-Sobolev space W^sL_A defined in Definition 3.1, with Banach/reflexivity properties proved in Theorem 3.4 by isometry into a product of Orlicz spaces, and embeddings obtained in Theorem 3.11 and Corollary 3.12 from condition (2.2) plus standard fractional Sobolev embedding theorems. The main existence result, Theorem 4.2, is obtained by the direct method: the energy I = J - H is shown to be coercive and weakly lower semicontinuous (Lemmas 4.4 and 4.6), so Theorem 2.4 gives a minimizer, and nontriviality is checked by testing I(t phi) < 0 using condition (f2). At no point does the paper fit a parameter to the target conclusion, rename a hypothesis as a prediction, or invoke a load-bearing result whose only support is a self-citation by the present authors. The only notable defect is Lemma 2.1, quoted from Bonder-Salort [11], which asserts A(a(t)) <= c A(t); this is false for A(t)=t^p/p with p>2, since a(t)=t^{p-1} gives A(a(t)) = p^{p-1} A(t)^p. Lemma 4.5 relies on that assertion to place a(|h|) in L^A and thereby prove J is C^1, so the differentiability argument has a genuine correctness gap. That gap, however, is a mathematical error in a cited auxiliary estimate, not circular reasoning: the existence theorem does not assume the existence of a solution, and the incorrect lemma is not equivalent to the theorem's conclusion. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical or mathematical entities beyond the function spaces, and those spaces already appear in Bonder-Salort [11]. No free parameters are fitted to data; the main quantitative input is the Orlicz index bound (2.2). The most fragile assumption is the cited Lemma 2.1, which is false as written and affects the proof of differentiability of the energy.

assumptions (6)
  • domain assumption A is an N-function satisfying the global index bound 1 < inf t a(t)/A(t) and sup t a(t)/A(t) < infinity, condition (2.2).
    Used throughout to ensure the Delta2 condition, reflexivity, embeddings, and coercivity; it excludes N-functions with very slow or very fast growth.
  • domain assumption Omega is a bounded open subset of R^N with Lipschitz boundary.
    Needed for compact embeddings and for the generalized Poincare inequality in Theorems 3.5, 3.11, and Corollary 3.12.
  • domain assumption f is a Caratheodory function satisfying growth condition (f1) and lower bound (f2) with 1<q<p0.
    These hypotheses make the energy well-defined, coercive, weakly lower semicontinuous, and ensure the minimizer is nontrivial.
  • standard math Embedding theorems for fractional Sobolev spaces W^{s,p}, Theorems 2.2 and 2.3, taken from Di Nezza-Palatucci-Valdinoci and Demengel-Demengel.
    External benchmark used to transfer embeddings of W^s L_A into Lebesgue spaces through the continuous embedding W^s L_A into W^{s,p0}.
  • standard math Struwe's direct method theorem, Theorem 2.4: a coercive and weakly lower semicontinuous functional attains its infimum on a weakly closed subset.
    Standard variational principle used to obtain the minimizer u0 of the energy functional I.
  • ad hoc to paper Lemma 2.1, cited to reference [11]: A(a(t)) <= c A(t) for all t under the global Delta2 condition.
    This lemma is false as stated for A(t)=t^p with p>2, and it is load-bearing for the proof of Lemma 4.5, where it is used to place a(|h|) in the dual Orlicz space.

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Pith. "Pith review of Existence of solutions for a nonlocal type problem in fractional Orlicz Sobolev spaces." pith.science (2026). https://pith.science/paper/TH6GS7OZ

@misc{pith2026190807806,
  author       = {Pith},
  title        = {Pith review of: Existence of solutions for a nonlocal type problem in fractional Orlicz Sobolev spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TH6GS7OZ}},
  note         = {Machine review of arXiv:1908.07806}
}
abstract

In this paper, we investigate the existence of weak solution for a fractional type problems driven by a nonlocal operator of elliptic type in a fractional Orlicz-Sobolev space, with homogeneous Dirichlet boundary conditions. We first extend the fractional Sobolev spaces $W^{s,p}$ to include the general case $W^sL_A$, where $A$ is an N-function and $s\in (0,1)$. We are concerned with some qualitative properties of the space $W^sL_A$ (completeness, reflexivity and separability). Moreover, we prove a continuous and compact embedding theorem of these spaces into Lebesgue spaces.

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