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Existence and decay for a Grushin problem in $\mathbb{R}^N$ with singular, convective, critical reaction

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Pith's one-line read The paper proves existence of positive weak solutions for a Grushin problem in $\mathbb{R}^N$ with singular, convective, and critical reaction for small parameters, and decay at infinity when convection is absent.

desk verdict A genuinely new Grushin result with a well-executed decay section, but Lemma 3.20 has a load-bearing scaling error that invalidates the lower-semicontinuity proof as written. read the letter →

arxiv 2506.22177 v1 pith:SFKAGHTG submitted 2025-06-27 math.AP

Laura Baldelli , Paolo Malanchini , Simone Secchi This is my paper · ORCID
classification math.AP MSC 35J7035J2035B3335B0835B45
keywords GrushinoperatorsingularreactionconvectivetermcriticalSobolevexponentset-valuedanalysismountainpasstheoremdecayatinfinityconcentration-compactness
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 aims to prove existence of a positive weak solution, on the entire Euclidean space, for a nonlinear equation driven by the Grushin operator, where the reaction contains a singular term $u^{-\eta}$, a convective term depending on the gradient, and a term at the critical Sobolev growth. The main theorem states that for small positive $\lambda_1$ and any $\lambda_2$ with $0\le \lambda_2\le \lambda_1$, under mild integrability and decay assumptions on the weights, such a solution exists in the Beppo\,--\,Levi space and is bounded and locally H\"older continuous; when the convective term is switched off, the solution decays to zero at infinity. The interest is that the problem lies beyond the reach of variational methods alone, because convection destroys the variational structure, while singularity and critical growth create additional obstructions.

What carries the argument

The proof freezes the convective term at a fixed function $v$ and truncates the singular term below a subsolution $u_{\lambda_1}$, producing a variational energy $J$. Mountain-pass geometry plus concentration-compactness give a critical energy level $\hat c$ below which Palais-Smale sequences recover compactness; for small $\lambda_1,\lambda_2$ the mountain-pass level lies below $\hat c$. The solution map $S(v)$, consisting of low-energy solutions of the truncated frozen problem, is then shown to be compact and lower semicontinuous; its minimal selection $T$ is continuous and compact, so Schauder's fixed point theorem yields a solution of the original problem. A doubling lemma, weak Lebesgue-space estimates, and a local boundedness theorem for X-elliptic operators produce the decay.

What would settle it

Recompute the Grushin gradient of $z^m_n = (u^m_n)_R$: because $\nabla_\gamma g_R = R(\nabla_\gamma g)_R$, the norm is $R^{1-N_\gamma/2}\|\nabla_\gamma g\|_2$, not $R^{-N_\gamma/2}\|\nabla_\gamma g\|_2$. Inserting this factor into condition (3.55) removes the $R$-dependence or reverses its effect, so the claim that a large $R$ satisfies the condition does not follow and the uniform bound on the recursive sequence is not established.

Watch

Extended reading notes

Core claim

Under assumptions (H1)-(H2), there is a threshold $\Lambda>0$ such that for every $\lambda_1\in(0,\Lambda)$ and $\lambda_2\in[0,\lambda_1]$ the problem $-\Delta_\gamma u = \lambda_1 w_1(z) u^{-\eta} + \lambda_2 w_2(z)|\nabla_\gamma u|^{r-1} + u^{2^*_\gamma -1}$ in $\mathbb{R}^N$, $u>0$, has a weak solution $u\in D^0_\gamma(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)\cap C^{0,\tau}(\mathbb{R}^N)$. If $\lambda_2=0$ the solution satisfies $u(z)\to 0$ as $d(z)\to\infty$, and in fact a two-sided power bound $C_0(1+d(z)^{N_\gamma-2})^{-1}\le u(z)\le C_1(1+d(z)^{N_\gamma-2})^{-1}$.

Load-bearing premise

The recursive sequence of rescaled solutions used to prove lower semicontinuity must stay uniformly bounded, and that bound rests on a specific scaling identity for the Grushin gradient of the rescaled functions.

Editorial extensions

If this is right

  • For any admissible weights satisfying (H1)-(H2), sufficiently small $\lambda_1$ dominate the singular and convective effects, so the full problem inherits a solution from the frozen problem.
  • In the purely singular-critical case $\lambda_2=0$, every solution obtained is globally bounded, locally H\"older continuous, and decays like a fundamental solution, with explicit two-sided bounds.
  • The quantitative threshold $\Lambda$ is expressed through explicit conditions (3.37), (3.40), (3.43), and (3.56), so the smallness regime is in principle computable.
  • The same truncation-freezing-set-valued strategy transfers the corresponding p-Laplacian result to a degenerate elliptic setting, where full $C^2$ regularity across the degenerate set is unavailable.

Reading between the lines

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

  • The two-sided decay bound suggests the solution behaves like the fundamental solution of $\Delta_\gamma$ at infinity; a testable extension is whether the same bound persists for small $\lambda_2>0$ once gradient estimates for Grushin operators are available.
  • The technical restriction $\ell>4$ in the barrier construction for decay is stated by the authors as likely unnecessary; replacing the barrier near the degenerate set could remove it.
  • Because the fixed-point scheme only needs compactness, lower semicontinuity, and an $(S)_+$ property, the same proof pattern should apply to other subelliptic operators with a Sobolev embedding and a fundamental solution of this type.
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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 / 5 minor

Summary. The paper studies the Grushin problem (P) in the whole space R^N with a singular, convective, and critical reaction. Under assumptions (H1)-(H2) on the weights, Theorem 1.3 claims existence of a positive weak solution u in D_γ^0(R^N) ∩ L^∞(R^N) ∩ C^{0,τ}(R^N) for λ1 small and λ2 ∈ [0,λ1], and decay to zero as d(z)→∞ when λ2=0. The proof combines a variational treatment of a truncated and frozen problem with mountain-pass geometry, concentration-compactness, a set-valued solution map S, a Schauder fixed point argument, and Moser-type estimates for global boundedness and decay. The paper also proves a decay result for the associated subsolution and provides a self-contained local boundedness theorem in an appendix.

Significance. If the proof is completed, the result is significant: it appears to be the first existence theorem for a whole-space Grushin problem simultaneously containing a singular term, a convective term, and a critical nonlinearity. The paper is technically detailed and gives quantitative smallness thresholds for λ1 and λ2, and it adapts a substantial set of tools (concentration-compactness with tight convergence, set-valued fixed point theory, doubling arguments, and weak Lebesgue spaces) to the degenerate Grushin setting. The inclusion of a full appendix for local boundedness is a strength. However, the proof as written contains load-bearing gaps in the lower semicontinuity argument and in the blow-up argument, so the central claims are not yet established.

major comments (2)
  1. [§3.2, Lemma 3.20, Eq. (3.55)] The scaling identity used to verify condition (3.55) is incorrect. Since z_n^0 = (u_n^0)_R = u_R, the Grushin gradient satisfies ∇_γ z_n^0 = R (∇_γ u)_R, not (∇_γ u)_R. Hence ||∇_γ z_n^0||_2 = R^{1-N_γ/2} ||∇_γ u||_2, not R^{-N_γ/2} ||∇_γ u||_2. Substituting the correct value into the left-hand side of (3.55) gives R^2 S^{-2_γ^*/2} (R^{1-N_γ/2} L)^{2_γ^*-2} = S^{-2_γ^*/2} L^{2_γ^*-2}, which is independent of R. With the earlier choice L = 2 S^{N_γ/2} in (3.47), this quantity equals 2^{4/(N_γ-2)} S^{2_γ^*/2} > 1/2. Therefore condition (3.55) cannot be satisfied by choosing R large, contrary to the claim. Since this condition is the only mechanism in Lemma 3.20 that yields the uniform bound on the recursive sequence {z_n^m}, the lower semicontinuity of S is not proved. This gap is load-bearing: continuity of the selection T and the Schauder fixed point argument in Theorem 3.1 depend on it.
  2. [Theorem 4.4, Eqs. (4.11)-(4.13)] The blow-up construction in Theorem 4.4 does not account for the translation by ξ_n. With the definition w = (µ_n^{2/(N_γ-2)} t_x, µ_n^{2(1+γ)/(N_γ-2)} t_y) + ξ_n, the operator acting on t is Δ_{γ,t} = µ_n^{2N_γ/(N_γ-2)} (∂_{w_x}^2 + |w_x - ξ_{x,n}|^{2γ} ∂_{w_y}^2), not the standard Grushin operator µ_n^{2N_γ/(N_γ-2)} (∂_{w_x}^2 + |w_x|^{2γ} ∂_{w_y}^2). Correspondingly, Eq. (4.12) should contain |w_x - ξ_{x,n}|^γ, not |w_x|^γ. As a result, the claimed identity ||∇_{γ,t} \tilde u_n||_2 = ||∇_{γ,w} u_n||_2 is not justified, and the boundedness of (\tilde u_n) in D_γ^0(R^N) does not follow directly from (a3), because the standard Grushin gradient norm controls || |w_x|^{γ} ∇_{w_y} u_n ||_2, not || |w_x - ξ_{x,n}|^{γ} ∇_{w_y} u_n ||_2, when ξ_{x,n} is large. Since the limit equation (4.18) and the contradiction κ < S^{N_γ/4} both rely on this rescaling, the decay argument in Theorem 4.4 is incomplete as written.
minor comments (5)
  1. [Abstract/Introduction] There is a typo in the introduction: 'have also bees treated' should read 'have also been treated'.
  2. [Section 2.3 heading] The heading 'F unctional setting' contains an unintended space; it should be 'Functional setting'.
  3. [Section 3.2, notation for Λ] The symbol Λ is defined in Theorem 3.12 as min{Λ1,Λ2} and then redefined in Section 3.2 as min{Λ1,Λ2,Λ3}. To avoid ambiguity, the second threshold should be denoted differently, for example Λ'.
  4. [Remark 3.6] The sentence 'this choice has forced us to impose ℓ >4 and condition (3.8), which is slightly stronger then (H2)' contains a grammar error: 'then' should be 'than'.
  5. [Theorem A.1 statement] In the inequality in Theorem A.1, the term R^{-N_γ/α} ||u||_{L^α(B_γ(z,2R))} should be multiplied by R^{N_γ/α} after the rescaling to match the displayed formula; please check the scaling in the final line of the proof.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the Grushin result is obtained by adapting previous p-Laplacian frameworks from [12] and [13], but nothing reduces to its own inputs or to fitted constants; self-citations are methodological, not load-bearing.

full rationale

Theorem 1.3 is proved through a truncated and frozen auxiliary problem, a mountain-pass argument with a concentration-compactness threshold, and a set-valued fixed point procedure. The paper explicitly says the first part is inspired by [12], where the analogous p-Laplacian problem was treated, and that the decay part follows [13]. Baldelli is a coauthor of both [12] and [13], so there is repeated self-citation. This is not circular, however: the cited works supply structural ideas, a recursion bound (Lemma 2.9), and a decay strategy, not the conclusion of the Grushin theorem. The threshold Lambda is fixed by explicit inequalities (3.37), (3.43), and (3.56) in terms of known constants S, w1, w2, and the energy threshold (3.23) is an estimate, not a fitted parameter renamed as a prediction. The final solution is obtained by Schauder's fixed point theorem after proving compactness and lower semicontinuity of the set-valued map S. The lower-semicontinuity proof in Lemma 3.20 contains what appears to be an incorrect scaling identity for the Grushin gradient under anisotropic rescaling; the paper claims ||nabla_gamma z_n^0||_2 = ||(nabla_gamma u)_R||_2 = R^{-N_gamma/2}||nabla_gamma u||_2, while the correct identity introduces a factor R. This is a serious correctness risk in the submitted proof, but it is an internal mathematical error, not a circular reduction: it does not make the theorem's output equivalent to its inputs. No fitted constant is presented as a prediction, no defined object is defined in terms of the target quantity, and the load-bearing self-citations are not used to forbid alternatives or to import a uniqueness theorem on which the whole argument depends. The honest circularity finding is therefore no significant circularity, with the minor score reflecting the paper's reliance on the authors' own prior framework.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The proof rests on standard Grushin analysis tools and explicit assumptions on the weights. No free parameters are fitted and no new entities are postulated. The main structural inputs are the weight assumptions H1-H2 and known embedding, concentration-compactness, Harnack, and fixed point theorems.

assumptions (7)
  • domain assumption H1: w1, w2 are nonnegative, belong to L1(RN) ∩ L∞(RN), and there exists a ball B(z0, rho) with integral of w1 at least omega > 0.
    Assumed to guarantee the sub-solution construction and integrability of the singular weight terms.
  • domain assumption H2: w1(z) <= c1 d(z)^{-delta-2gamma} with delta > Ngamma + eta(Ngamma-2).
    This decay condition is needed to prove w1 u^{-eta} in L1(RN) ∩ L∞(RN) and the decay of solutions at infinity.
  • standard math The Grushin Sobolev embedding Dgamma0(RN) embeds into L^{2*gamma}(RN), with best constant S, and requires Ngamma > 2.
    Used throughout for energy estimates, concentration compactness, and critical level thresholds.
  • standard math Concentration-compactness principles for the Grushin operator, including the escape to infinity lemma.
    Invoked to rule out concentration at points and at infinity below the critical energy level.
  • standard math Non-homogeneous Harnack inequality for X-elliptic operators, as in Gutiérrez and Lanconelli [34, Theorem 5.5].
    Used to establish local Hölder continuity of solutions and the subsolution u_lambda1.
  • standard math Mountain Pass Theorem, Schauder fixed point theorem, Minty-Browder theorem, and weak comparison principles for the Grushin operator.
    Core tools for existence, unfreezing the convection term, uniqueness of linear subproblems, and barrier arguments.
  • standard math Fundamental solution for the Grushin operator, Gamma(z) = C d(z)^{2-Ngamma}.
    Used in the exterior comparison and lower decay estimates.

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Pith. "Pith review of Existence and decay for a Grushin problem in $\mathbb{R}^N$ with singular, convective, critical reaction." pith.science (2026). https://pith.science/paper/SFKAGHTG

@misc{pith2026250622177,
  author       = {Pith},
  title        = {Pith review of: Existence and decay for a Grushin problem in $\mathbbR^N$ with singular, convective, critical reaction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SFKAGHTG}},
  note         = {Machine review of arXiv:2506.22177}
}
read the original abstract

We establish an existence result for a problem set in the whole Euclidean space involving the Grushin operator and featuring a critical term perturbed by a singular, convective reaction. Our approach combines variational methods, truncation techniques, and concentration-compactness arguments, together with set-valued analysis and fixed point theory. Additionally, we prove the decay at infinity of solutions in the absence of the convective term. The result is new even in the case where more than one feature between singularity, convectivity and criticality is taken into account.

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    2021

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Reviewed August 6, 2026 · model on record in the stance chip above.