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arxiv: 2510.13299 · v2 · pith:SNTQ5WL4new · submitted 2025-10-15 · 🧮 math.AP

Bifurcation and multiplicity results for critical Grushin-Choquard problems

Suman Kanungo , Pawan Kumar Mishra , Giovanni Molica Bisci This is my paper

Pith reviewed 2026-05-21 20:48 UTC · model grok-4.3

classification 🧮 math.AP
keywords Grushin operatorChoquard problemcritical exponentbifurcationmultiplicity resultsnonlocal equationDirichlet problem
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The pith

The critical Grushin-Choquard problem exhibits bifurcation from every eigenvalue of the Grushin operator, with solution multiplicity at least doubled in left neighborhoods.

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

The paper shows that solutions to the nonlocal critical problem with the Grushin operator branch off from the eigenvalues of the linear problem. For parameters λ just below any such eigenvalue, there are at least twice as many nontrivial solutions as the multiplicity of that eigenvalue. This matters because it generalizes known results for the standard Laplacian to operators with degeneracy and to nonlocal Choquard-type nonlinearities. The setting requires the domain to intersect the degeneracy line so that the homogeneous dimension controls the critical growth.

Core claim

We prove the existence of bifurcation from any eigenvalue λ* of −Δ_γ under Dirichlet boundary conditions. Furthermore, we show that in a suitable left neighborhood of λ*, the number of nontrivial solutions to the problem is at least twice the multiplicity of λ*.

What carries the argument

The Grushin operator Δ_γ combined with the critical Choquard nonlocal term, analyzed via variational methods to detect bifurcation points at the spectrum of -Δ_γ.

If this is right

  • Solutions exist in intervals to the left of every eigenvalue.
  • The number of solutions is bounded below by twice the geometric or algebraic multiplicity.
  • Bifurcation occurs for the critical exponent case in the homogeneous dimension N_γ.
  • Nontrivial solutions appear for λ in (λ* - δ, λ*) for some δ>0 depending on the eigenvalue.

Where Pith is reading between the lines

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

  • This could extend to other degenerate elliptic operators with similar homogeneous structures.
  • Similar multiplicity results might hold for right neighborhoods or for non-critical cases.
  • Applications may arise in models of nonlocal interactions in degenerate media.

Load-bearing premise

The bounded domain must intersect the degeneracy set {x=0} nontrivially so that the homogeneous dimension and critical exponent are meaningful for the variational setting.

What would settle it

A numerical computation or explicit construction for a specific eigenvalue and domain showing either no bifurcation or fewer than twice the multiplicity solutions in the left neighborhood would disprove the claim.

read the original abstract

We consider the following nonlocal Br\'ezis-Nirenberg type critical Choquard problem involving the Grushin operator \begin{equation*} \left\{ \begin{aligned} -\Delta_\gamma & u =\lambda u + \left(\displaystyle\int_\Omega \frac{|u(w)|^{2^*_{\gamma,\mu}}}{d(z-w)^\mu}dw\right) |u|^{2^*_{\gamma,\mu}-2}u \quad &&\text{in} \ \Omega, u &= 0 \quad &&\text{on} \, \partial \Omega, \end{aligned} \right. \end{equation*} where $\Omega$ is an open bounded domain in $\mathbb{R}^N$, $N \geq 3$, with $\Omega \cap \{ x=0\} \neq \emptyset$, and $\lambda >0$ is a parameter. Here, $\Delta_\gamma$ represents the Grushin operator, defined as \[ \Delta_\gamma u(z) = \Delta_x u(z) +(1+\gamma)^2 |x|^{2\gamma} \Delta_y u(z), \quad \gamma \geq 0, \] where $z=(x,y)\in \Omega \subset \mathbb{R}^m\times \mathbb{R}^n$, $m+n=N \geq 3$ and $2^*_{\gamma,\mu}= \frac{2N_\gamma-\mu}{N_\gamma-2}$ is the Sobolev critical exponent in the Hardy-Littlewood context with $N_\gamma= m+(1+\gamma)n$ is the homogeneous dimension associated to the Grushin operator and $0<\mu<N_\gamma$. The homogeneous norm related to the Grushin operator is denoted by $d(\cdot)$. In this article, we prove the existence of bifurcation from any eigenvalue $\lambda^*$ of $-\Delta_\gamma$ under Dirichlet boundary conditions. Furthermore, we show that in a suitable left neighborhood of $\lambda^*$, the number of nontrivial solutions to the problem is at least twice the multiplicity of $\lambda^*$.

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.

Referee Report

1 major / 1 minor

Summary. The manuscript considers a critical Choquard problem driven by the Grushin operator −Δ_γ on a bounded domain Ω intersecting the degeneracy set {x=0}. It claims to prove bifurcation from every eigenvalue λ* of −Δ_γ (Dirichlet) and, in a left neighborhood of λ*, the existence of at least 2·mult(λ*) nontrivial solutions, obtained variationally via the even functional I_λ and symmetric mountain-pass or Krasnoselskii genus methods.

Significance. If the proofs are complete, the results extend classical bifurcation/multiplicity theory for critical nonlocal problems to the degenerate Grushin setting, where the homogeneous dimension N_γ and the adapted critical exponent 2^*_{γ,μ} govern the Sobolev embedding and the Hardy-Littlewood-Sobolev inequality. The variational construction is standard for this class of problems.

major comments (1)
  1. The multiplicity claim (at least 2·mult(λ*) solutions for λ slightly less than λ*) is obtained by producing critical values c_j(λ) via genus or symmetric mountain-pass on the even functional I_λ. Because the Choquard nonlinearity is critical with respect to N_γ, the Palais-Smale condition for I_λ holds only below a threshold determined by the best constant in the Grushin-adapted HLS inequality. The manuscript must supply an explicit uniform upper bound showing c_j(λ) lies strictly below this threshold for λ near λ* from the left; without this verification the compactness argument fails and the multiplicity statement does not follow from the variational geometry.
minor comments (1)
  1. Notation for the homogeneous norm d(·) and the precise definition of the critical exponent 2^*_{γ,μ} should be recalled at the beginning of the variational section for reader convenience.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough reading and the insightful comment on the compactness verification. We address the concern below and will revise the manuscript to incorporate the requested explicit bound.

read point-by-point responses

  1. Referee: The multiplicity claim (at least 2·mult(λ*) solutions for λ slightly less than λ*) is obtained by producing critical values c_j(λ) via genus or symmetric mountain-pass on the even functional I_λ. Because the Choquard nonlinearity is critical with respect to N_γ, the Palais-Smale condition for I_λ holds only below a threshold determined by the best constant in the Grushin-adapted HLS inequality. The manuscript must supply an explicit uniform upper bound showing c_j(λ) lies strictly below this threshold for λ near λ* from the left; without this verification the compactness argument fails and the multiplicity statement does not follow from the variational geometry.

    Authors: We agree that an explicit uniform upper bound for the critical values c_j(λ) is required to confirm that they lie strictly below the compactness threshold given by the best constant in the Grushin-adapted Hardy-Littlewood-Sobolev inequality. In the revised manuscript we will add a new lemma (placed immediately after the variational geometry section) that derives such a bound. Specifically, using the fact that I_λ converges to the quadratic functional associated with the eigenvalue problem as λ → λ*− and employing the characterization of the genus levels, we will show that there exists δ > 0 such that for all λ ∈ (λ* − δ, λ*), one has c_j(λ) < (1/2) S_{γ,μ}^{N_γ/(2^*_{γ,μ}−2)} for each j = 1, … , mult(λ*), where S_{γ,μ} denotes the optimal constant in the relevant embedding. This estimate is uniform in λ near λ* from the left and guarantees that the Palais-Smale condition holds at all these levels, thereby justifying the multiplicity statement. revision: yes

Circularity Check

0 steps flagged

No significant circularity; variational proof is self-contained

full rationale

The paper's central claims of bifurcation from any eigenvalue λ* of −Δ_γ and multiplicity at least 2·mult(λ*) in a left neighborhood are established via independent variational constructions on the even functional I_λ, employing the symmetric mountain-pass theorem and Krasnoselskii genus to produce critical values c_j(λ). These rely on the geometry of the functional, the Grushin-adapted Hardy-Littlewood-Sobolev inequality for the critical exponent, and standard compactness arguments below the threshold, without any reduction of the target statements to fitted parameters, self-definitions, or load-bearing self-citations that are themselves unverified. The derivation chain uses external mathematical tools (eigenvalue theory for the Grushin operator, genus theory) that do not presuppose the multiplicity result, making the proof self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the functional-analytic framework for the Grushin operator (including its homogeneous dimension and associated Sobolev embeddings) and on standard variational methods for nonlocal critical problems; these are drawn from prior literature rather than introduced here.

axioms (2)
  • domain assumption The Grushin operator generates a Hilbert space setting in which the critical exponent 2^*_{γ,μ} is well-defined via the homogeneous dimension N_γ
    Invoked throughout the problem formulation to justify the choice of nonlinearity and the functional space.
  • standard math Standard compactness and embedding properties hold for the weighted spaces associated with the Grushin operator on bounded domains intersecting the degeneracy set
    Required for the variational formulation and for the application of topological methods to count solutions.

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