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arxiv: 1907.11303 · v1 · pith:4JP6QLSXnew · submitted 2019-07-25 · 🧮 math.AP

A perturbative approach to non-degeneracy of the Lane-Emden system

Seunghyeok Kim , Angela Pistoia This is my paper

Pith reviewed 2026-05-24 15:49 UTC · model grok-4.3

classification 🧮 math.AP
keywords Lane-Emden systemnon-degeneracyground statescritical hyperbolasemilinear elliptic systemsperturbative methodsentire solutions
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The pith

Ground state solutions to the critical Lane-Emden system are non-degenerate near two points on the critical hyperbola.

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

The paper proves that positive decaying ground state solutions to the Lane-Emden system remain non-degenerate when the exponents (p,q) lie on the critical hyperbola and are sufficiently close to either (1, (n+4)/(n-4)) for n at least 5, or to the equal-exponent point ((n+2)/(n-2), (n+2)/(n-2)) for n at least 3. Non-degeneracy here means the linearized operator at these solutions has trivial kernel in suitable weighted spaces. A reader would care because this property is typically the starting point for applying implicit-function arguments to nearby problems, for establishing uniqueness, or for tracking how the solution set changes with the exponents.

Core claim

We prove that ground state solutions of the critical Lane-Emden system are non-degenerate when (p,q) is close to (1,(n+4)/(n-4)) if n≥5 or close to ((n+2)/(n-2),(n+2)/(n-2)) if n≥3.

What carries the argument

Perturbative continuation of the linearized operator from the two known base cases on the critical hyperbola, where non-degeneracy is already established.

If this is right

  • The non-degeneracy permits local continuation of the solutions via the implicit function theorem when the exponents vary slightly.
  • It supplies a starting point for studying the local structure of the solution set near those points on the hyperbola.
  • The result extends known non-degeneracy statements from the exact base cases to open neighborhoods around them.

Where Pith is reading between the lines

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

  • The same perturbative technique might apply at other points on the hyperbola if suitable base cases with known non-degeneracy can be identified.
  • Non-degeneracy in these regimes could be used to compute the Morse index of the ground states for nearby subcritical exponents.
  • The method may connect to classification questions for entire solutions when one exponent is fixed and the other varies.

Load-bearing premise

Ground state solutions continue to exist, stay positive, and decay at infinity when the exponents are moved slightly away from the two base points along the hyperbola.

What would settle it

An explicit nontrivial bounded solution to the linearized system at some (p,q) arbitrarily close to one of the two base points on the hyperbola.

read the original abstract

We consider ground state solutions of the critical Lane-Emden system \[\begin{cases} -\Delta u = v^p &\text{in } \mathbb{R}^n,\\ -\Delta v = u^q &\text{in } \mathbb{R}^n,\\ u,v >0\ &\text{in } \mathbb{R}^n, \end{cases}\] where $n \ge 3$ and $p,q>0$ and $(p,q)$ belongs to the critical hyperbola $\frac{1}{p+1} + \frac{1}{q+1} = \frac{n-2}{n}.$ We prove that they are non-degenerate when either $(p,q)$ is close to $(1,{n+4\over n-4})$ (if $n\ge5$) or $(p,q)$ is close to $({n+2\over n-2},{n+2\over n-2})$ (if $n\ge3$).

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 / 0 minor

Summary. The paper considers positive ground-state solutions (u,v) of the critical Lane-Emden system on the hyperbola 1/(p+1) + 1/(q+1) = (n-2)/n in R^n (n≥3). It claims to prove, via a perturbative argument, that these solutions are non-degenerate (linearized operator has trivial kernel modulo translations) whenever (p,q) lies sufficiently close to either (1,(n+4)/(n-4)) for n≥5 or ((n+2)/(n-2),(n+2)/(n-2)) for n≥3.

Significance. Non-degeneracy results for this system are useful for subsequent bifurcation or uniqueness arguments. A perturbative proof that extends known non-degeneracy from the two base points on the hyperbola would be a modest but concrete contribution, provided the continuation of the ground states themselves is secured.

major comments (1)
  1. [Abstract] Abstract (first paragraph): the statement that “they” (the ground states) are non-degenerate for (p,q) near the two base points presupposes the existence of positive, radially decreasing, integrable solutions in a whole neighborhood on the hyperbola. The perturbative non-degeneracy argument (implicit-function continuation from a base solution whose linearized operator is already known to be invertible) is conditional on this continuation; the manuscript must either prove or cite a prior existence/continuation result for the perturbed exponents, otherwise the claim remains conditional.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need to secure the existence/continuation of the ground states. We address the comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract (first paragraph): the statement that “they” (the ground states) are non-degenerate for (p,q) near the two base points presupposes the existence of positive, radially decreasing, integrable solutions in a whole neighborhood on the hyperbola. The perturbative non-degeneracy argument (implicit-function continuation from a base solution whose linearized operator is already known to be invertible) is conditional on this continuation; the manuscript must either prove or cite a prior existence/continuation result for the perturbed exponents, otherwise the claim remains conditional.

    Authors: We agree that the non-degeneracy claim is conditional on the existence of the solutions in a neighborhood along the hyperbola. The perturbative argument in the paper proceeds from the known non-degenerate base solutions at the two indicated points via the implicit-function theorem, but the manuscript does not contain a self-contained existence/continuation proof. In the revised version we will add a short paragraph (with appropriate citations to prior existence results for the critical Lane-Emden system on the hyperbola) clarifying that positive radially symmetric solutions exist and can be continued in a neighborhood of each base point. The abstract will be rephrased to reflect this. revision: yes

Circularity Check

0 steps flagged

No circularity: perturbative non-degeneracy is independent of fitted inputs or self-citations

full rationale

The paper states a direct theorem on non-degeneracy of ground states for the Lane-Emden system near two explicit base points on the critical hyperbola, using a perturbative argument. No equation or claim reduces by construction to a fitted parameter, renamed result, or load-bearing self-citation; the abstract and claim are self-contained existence/non-degeneracy statements whose validity rests on standard implicit-function continuation from known base solutions rather than any definitional loop. This matches the default expectation of no significant circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are stated. The result rests on standard elliptic PDE theory and existence of ground states, which are not detailed here.

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