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arxiv: 1906.09147 · v1 · pith:CHNFTS65new · submitted 2019-06-20 · 🧮 math.AP

Existence, regularity, asymptotic decay and radiality of solutions to some extension problems

Hamilton Bueno , Aldo H. S. Medeiros , G. A. Pereira This is my paper

Pith reviewed 2026-05-25 19:14 UTC · model grok-4.3

classification 🧮 math.AP
keywords extension problemradial symmetrymoving plane methodAmbrosetti-Rabinowitz conditionexponential decayground state solutionHartree problem
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The pith

Solutions to the extension problem are radially symmetric in R^N under weak growth conditions on f.

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

The paper proves that if a nonlinearity f satisfies lim t to 0 of f(t)/t equals 0 and lim t to infinity of f(t)/t to the p equals 0 for p between 1 and (N+1)/(N-1), then solutions to the given extension problem in the upper half-space are radially symmetric. It also establishes regularity and exponential decay of solutions to the standard extension problem and existence of ground state solutions when the Ambrosetti-Rabinowitz condition holds. The same symmetry result is shown for a related extension Hartree problem. A sympathetic reader would care because the results apply to problems that arise from fractional operators via the extension technique, showing symmetry persists under growth slower than the critical power.

Core claim

Supposing only that lim t to 0 f(t)/t = 0 and lim t to infinity f(t)/t^p = 0 for some p in (1, (N+1)/(N-1)), solutions to the extension problem {-Delta u + m^2 u = 0 in R^{N+1}_+, - partial u / partial x (0,y) = f(u(0,y)) in R^N} and to the extension Hartree problem are radially symmetric in R^N. Under the same hypotheses, regularity and exponential decay of solutions to the first problem are proved, and supposing the traditional Ambrosetti-Rabinowitz condition, existence of a ground state solution is also shown.

What carries the argument

The moving plane method applied to the extended equation in the half-space, made possible by the stated growth limits on f.

If this is right

  • Solutions to the extension problem are radially symmetric in R^N.
  • Solutions to the extension problem are regular and decay exponentially.
  • Existence of a ground state solution holds under the Ambrosetti-Rabinowitz condition.
  • Solutions to the extension Hartree problem are also radially symmetric in R^N.

Where Pith is reading between the lines

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

  • The boundary trace u(0,y) being radial may allow reduction of the problem to an ODE in the radial variable for further study.
  • The results suggest that similar symmetry conclusions could hold for other nonlocal boundary conditions realizable by extensions.
  • Exponential decay implies solutions remain localized away from infinity, which may aid analysis of related variational problems.

Load-bearing premise

The growth limits on f at zero and infinity must hold so the moving plane method applies without stronger hypotheses on f.

What would settle it

A concrete non-radial solution to the extension problem that satisfies the growth conditions on f at zero and infinity would disprove the radial symmetry claim.

read the original abstract

Supposing only that $\displaystyle\lim_{t \to 0} \frac{f(t)}{t} = 0$ and $\displaystyle\lim_{t \to \infty} \frac{f(t)}{t^{p}} = 0$, for some $p \in \left(1,\frac{N+1}{N-1}\right)$, we prove that solutions to the extension problem \begin{equation*}\left\{ \begin{array}{rcll} -\Delta u+ m^2u &=& 0, &\mbox{in} \ \ \mathbb{R}^{N+1}_{+} \\ -\frac{\partial u}{\partial{x}} (0,y)& =& f(u(0,y)), & y \in \mathbb{R}^{N}, \end{array}\right. \end{equation*} and also to the extension Hartree problem \begin{equation*} \left\{\begin{aligned} -\Delta u +m^2u&=0, &&\mbox{in} \ \mathbb{R}^{N+1}_+,\\ -\displaystyle\frac{\partial u}{\partial x}(0,y)&=-V_\infty u(0,y)+\left(\frac{1}{|y|^{N-\alpha}}*F(u(0,y))\right)f(u(0,y)) &&\mbox{in} \ \mathbb{R}^{N}\end{aligned}\right. \end{equation*} are radially symmetric in $\mathbb{R}^N$. In the last problem, $V_\infty>0$ is a constant and $F$ the primitive of $f$. Under the same hypotheses, regularity and exponential decay of solutions to the first problem is also proved and, supposing the traditional Ambrosetti-Rabinowitz condition, also existence of a ground state solution.

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

0 major / 4 minor

Summary. The manuscript proves that, under the sole assumptions lim_{t→0} f(t)/t = 0 and lim_{t→∞} f(t)/t^p = 0 for p ∈ (1, (N+1)/(N-1)), positive solutions of the extension problem −Δu + m²u = 0 in the half-space with nonlinear boundary condition −∂_x u(0,y) = f(u(0,y)) are radially symmetric about some point in R^N. The same symmetry conclusion is obtained for the corresponding Hartree-type extension problem involving the convolution term (1/|y|^{N-α} * F(u))f(u). Under the same growth hypotheses the authors establish C^{1,α} regularity up to the boundary and exponential decay at infinity for solutions of the first problem; existence of a ground-state solution is shown when the Ambrosetti–Rabinowitz condition is added.

Significance. If the claims hold, the work supplies a clean extension of the moving-plane method to a class of nonlocal boundary-value problems that arise in the study of fractional Laplacians and Hartree equations. The hypotheses on f are minimal and parameter-free, the symmetry result is stated for both the local and nonlocal cases, and the decay and existence statements are obtained under standard additional assumptions. These features make the paper a useful reference for symmetry and qualitative properties in extension problems.

minor comments (4)
  1. [§2] §2, after (2.3): the statement that the comparison principle holds for the extension operator under the given growth on f would benefit from an explicit citation to the precise version of the maximum principle used (e.g., the reference to Berestycki–Nirenberg or a self-contained lemma).
  2. [Theorem 1.2] Theorem 1.2 (Hartree case): the range of α is not restated in the theorem statement even though it appears in the problem formulation; adding the interval for α would improve readability.
  3. [§4] §4, proof of exponential decay: the constant m appears in the decay rate but its dependence on the parameters of f is not tracked; a short remark clarifying whether the rate is uniform under the stated limits on f would be helpful.
  4. [References] Reference list: the paper cites several works on the moving-plane method for nonlocal equations but omits the original Caffarelli–Silvestre extension paper; adding it would place the setting in clearer context.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the clear summary of our results, and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper applies standard PDE techniques (moving planes for radial symmetry, regularity theory, and variational methods under Ambrosetti-Rabinowitz) to the extension problem under explicit growth hypotheses on f. These hypotheses are independent inputs that enable the comparison principle and integrability estimates; the symmetry, decay, and existence conclusions follow from them without any self-definitional reduction, fitted-parameter renaming, or load-bearing self-citation chain. The derivation chain is self-contained against external benchmarks and does not reduce any claimed result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard background results from elliptic PDE theory such as maximum principles and Sobolev embeddings; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Standard elliptic regularity theory and maximum principle apply to the extension operator in the half-space.
    Implicit in claims of regularity and symmetry.
  • domain assumption Moving plane method or equivalent symmetry technique can be applied under the given growth conditions on f.
    Central to the radiality proof.

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