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arxiv: 2605.23810 · v1 · pith:3H4D3U5Knew · submitted 2026-05-22 · 🧮 math.AP

Asymptotic behavior of solutions for the nonlinear Hartree equation involving the fractional Laplacian

Natalino Borgia , Silvia Cingolani , Minbo Yang , Shunneng Zhao This is my paper

Pith reviewed 2026-05-25 03:20 UTC · model grok-4.3

classification 🧮 math.AP
keywords fractional LaplacianHartree nonlinearityblow-up analysisasymptotic behaviormoving planes methodnonlocal elliptic systemsconcentration phenomena
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The pith

Solutions to the slightly subcritical fractional Hartree equation blow up at exactly one interior point whose location is characterized by the Green's function as epsilon approaches zero.

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

The paper studies positive solutions of a nonlocal equation driven by the fractional Laplacian with a Hartree-type nonlinearity that is perturbed slightly below the critical exponent. It reduces the problem to a coupled system and first obtains uniform L1 bounds away from the boundary together with uniform L^infty bounds near the boundary by means of the moving planes method and convolution estimates. With these bounds in hand, the authors analyze the limit as the perturbation parameter epsilon tends to zero and prove that the solutions develop a single blow-up point inside the domain. The location of this point, the precise blow-up profile, and the exact rate are all identified. The same concentration result is established for the associated fractional Brezis-Nirenberg problem with critical Hartree nonlinearity.

Core claim

As epsilon approaches zero, every family of positive solutions to the reduced system blows up at precisely one point x0 inside Omega. The point x0 is characterized as a critical point of a functional built from the Green's function of the fractional Laplacian with zero exterior condition. The rescaled profile around x0 converges to the standard bubble for the limiting critical equation, and the blow-up rate is expressed explicitly in terms of epsilon and the value of the Green's function at x0.

What carries the argument

Reduction of the perturbed Hartree equation to the subcritical fractional system A_s u = u^{2_s^sharp-2-epsilon} v, A_s v = u^{2_s^sharp-1-epsilon}, followed by moving-planes arguments that produce uniform L1 bounds away from the boundary and uniform L^infty bounds near the boundary.

If this is right

  • Blow-up occurs only at interior points and never on the boundary.
  • The exact blow-up rate is determined by the Green's function evaluated at the concentration point.
  • The same single-point concentration holds for the fractional Brezis-Nirenberg problem with critical Hartree nonlinearity.
  • The location of the blow-up point maximizes a functional involving the regular part of the Green's function.

Where Pith is reading between the lines

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

  • The concentration mechanism appears robust enough to apply to other nonlocal Hartree-type nonlinearities that admit a similar system reduction.
  • The uniform bounds near the boundary suggest that boundary blow-up is prevented by the exterior Dirichlet condition even for fractional operators.

Load-bearing premise

The moving planes method together with integral estimates on the convolution term produces uniform L1 bounds away from the boundary and uniform L^infty bounds near the boundary.

What would settle it

A sequence of solutions that either remains bounded in L^infty or develops blow-up at two or more distinct points inside Omega as epsilon tends to zero would falsify the single-point concentration claim.

read the original abstract

In this paper, we investigate the nonlocal problem \begin{equation*}\left\lbrace \begin{aligned} &A_{s} u=(|x|^{-(n-2s)}\ast u^{2_{s}^{\sharp}-1-\epsilon})u^{2_{s}^{\sharp}-2-\epsilon} \quad\quad\hspace{3.5mm} \mbox{in}\hspace{2mm}\Omega,\\ &u>0\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\hspace{2mm}\mbox{in}\hspace{2mm}\Omega,\\ &u=0\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\hspace{2mm}\mbox{on}\hspace{2mm}\mathbb{R}^n\setminus\Omega, \end{aligned} \right.\end{equation*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^n$, $0<s<1$, $n\in(2s,\min\{6s,n+2s\})$, $\epsilon>0$ small, $2_{s}^{\sharp}-1=(n+2s)/(n-2s)$ and $A_{s}$ stands for the fractional Laplace operator $(-\Delta)^{s}$ in $\Omega$ with outside zero Dirichlet boundary condition. The above problem is reduced to the subcritical fractional system $$ A_{s}u=u^{2_{s}^{\sharp}-2-\epsilon}v,\hspace{2mm}A_{s}v=u^{2_{s}^{\sharp}-1-\epsilon},\hspace{2mm}u,v>0\hspace{2mm}\mbox{in}\hspace{2mm}\Omega\hspace{2mm}\mbox{and}\hspace{2mm}u=(-\Delta)^sv=0\hspace{2mm}\mbox{on}\hspace{2mm}\mathbb{R}^n\setminus\Omega.$$ For a general domain $\Omega$ or domains with convexity, we first prove a uniform $L^1$ bound away from the boundary and a uniform $L^{\infty}$ bound near the boundary for positive solutions to the general fractional Hartree-type PDEs by applying the moving planes method and integral estimates for the convolution term.Among these results, we study the asymptotic behavior of solutions as $\epsilon\rightarrow0$.These solutions are shown to blow-up at exactly one point $x_0$ and location of this point is characterized. In addition, the shape and exact rates for blowing-up are studied.Finally,we also establish the corresponding main results for solutions of the fractional Brezis-Nirenberg problem involving critical Hartree-type nonlinearity.

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 / 2 minor

Summary. The paper studies the asymptotic behavior as ε→0 of positive solutions to the fractional Hartree equation A_s u = (|x|^{-(n-2s)} * u^{2_s^♯-1-ε}) u^{2_s^♯-2-ε} in a smooth bounded domain Ω ⊂ R^n (with exterior Dirichlet condition), reducing it to the subcritical system A_s u = u^{2_s^♯-2-ε} v, A_s v = u^{2_s^♯-1-ε}. For general Ω (or convex domains), it claims uniform L^1 bounds away from ∂Ω and L^∞ bounds near ∂Ω via the moving-planes method plus convolution integral estimates; it then proves single-point blow-up at a characterized point x_0 with explicit shape and rates. Analogous results are stated for the fractional Brezis-Nirenberg problem with critical Hartree nonlinearity.

Significance. If the claimed uniform bounds hold, the work supplies the first detailed single-point blow-up analysis for this class of nonlocal fractional Hartree systems, including location characterization and exact rates; this would extend classical concentration results to a setting with both fractional diffusion and nonlocal convolution, providing a template for related nonlocal problems.

major comments (1)
  1. [Abstract / uniform-bounds section] Abstract (and the section establishing the uniform bounds): the assertion that moving planes plus integral estimates for the convolution term yield a uniform L^1 bound away from ∂Ω for the coupled system on a general (possibly non-convex) bounded Ω is load-bearing for the subsequent single-point blow-up claim. The standard moving-planes comparison for the difference u - u_λ requires that the linear inequality satisfied by the difference preserves sign under the nonlocal kernel; for non-convex caps this is sensitive to the interaction between the exterior Dirichlet condition and the support of the convolution term, and no explicit remainder estimate or maximum-principle verification for the system is indicated to close the gap.
minor comments (2)
  1. [Abstract] The abstract states the result holds 'for a general domain Ω or domains with convexity'; this phrasing is ambiguous and should be clarified to indicate whether the general-domain case is fully proved or requires additional geometric assumptions.
  2. [Introduction / preliminaries] Notation for the critical exponent 2_s^♯ and the range n ∈ (2s, min{6s, n+2s}) should be cross-checked against the system reduction to ensure consistency with the subcritical regime.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed reading and for identifying a point that requires clarification in the justification of the uniform bounds. We address the concern directly below and will revise the manuscript to strengthen the presentation of the moving-planes argument for the coupled system.

read point-by-point responses

  1. Referee: [Abstract / uniform-bounds section] Abstract (and the section establishing the uniform bounds): the assertion that moving planes plus integral estimates for the convolution term yield a uniform L^1 bound away from ∂Ω for the coupled system on a general (possibly non-convex) bounded Ω is load-bearing for the subsequent single-point blow-up claim. The standard moving-planes comparison for the difference u - u_λ requires that the linear inequality satisfied by the difference preserves sign under the nonlocal kernel; for non-convex caps this is sensitive to the interaction between the exterior Dirichlet condition and the support of the convolution term, and no explicit remainder estimate or maximum-principle verification for the system is indicated to close the gap.

    Authors: We agree that the moving-planes method for the fractional system must be justified carefully when the domain is non-convex, because the exterior Dirichlet condition and the nonlocal kernel can affect sign preservation in reflected caps. In the current manuscript the argument proceeds by applying the standard moving-planes comparison to the difference of the pair (u,v) and controlling the resulting integral terms via positivity of the convolution kernel together with the subcritical exponents. However, the manuscript does not supply an explicit remainder estimate or a self-contained verification of the maximum principle for the linearized nonlocal system. We will therefore add a dedicated subsection (or short appendix) that (i) records the precise linear inequality satisfied by the difference, (ii) verifies that the nonlocal operator preserves the sign under the exterior condition for any smooth bounded Ω, and (iii) derives the necessary integral remainder bounds that close the comparison. With these additions the uniform L¹ bound away from the boundary will be rigorously established for general smooth domains, as claimed. The subsequent single-point blow-up analysis remains unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses standard external techniques

full rationale

The paper reduces the Hartree problem to a subcritical system and obtains the key uniform L1/L∞ bounds by applying the moving planes method together with convolution integral estimates. These steps invoke established analytical tools from the nonlocal PDE literature rather than any self-definitional loop, fitted-input prediction, or load-bearing self-citation chain internal to the present work. The subsequent single-point blow-up characterization and rate analysis are built on those bounds without reducing any claimed prediction to a tautological renaming or ansatz smuggled via prior author work. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper builds on standard mathematical tools from the literature on fractional operators without introducing new free parameters or entities.

axioms (1)
  • standard math Standard properties of the fractional Laplacian and applicability of the moving planes method in this setting
    These are invoked to derive the uniform bounds and asymptotic behavior.

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