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arxiv: 2605.24804 · v1 · pith:JIYX3FDTnew · submitted 2026-05-24 · 🧮 math.AP

Variational problems related to self-similar solutions of Hardy-Sobolev heat equation in RN

Fei Fang , Zhong Tan This is my paper

Pith reviewed 2026-06-30 00:16 UTC · model grok-4.3

classification 🧮 math.AP
keywords Hardy-Sobolev equationself-similar solutionsvariational methodselliptic equationPohozaev identityweighted inequalitiessubcritical casecritical case
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The pith

Self-similar transformation yields infinitely many solutions to the elliptic problem from the Hardy-Sobolev heat equation in the subcritical case.

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

By applying a self-similar transformation the parabolic Hardy-Sobolev heat equation is turned into an elliptic equation with a drift term. Weighted versions of the Hardy and Sobolev inequalities are established for this equation. Variational methods then deliver infinitely many solutions when the exponent q lies below the critical value and at least one solution at the critical value. The Pohozaev identity further shows nonexistence for some parameter choices. These results matter because they determine the existence of self-similar solutions to the original time-dependent equation.

Core claim

The paper establishes that the elliptic equation obtained from the self-similar change of variables admits infinitely many solutions in the subcritical range 2 < q < 2^*(s) and at least one solution when q equals 2^*(s), with the weighted Hardy and Sobolev inequalities supporting the variational arguments, while the Pohozaev identity rules out solutions under certain conditions on α and the parameters.

What carries the argument

The self-similar transformation converting the parabolic equation into the elliptic equation −Δv − (1/2) y · ∇v = α v + |v|^{q−2} v / |y|^s.

Load-bearing premise

The self-similar transformation is valid and the resulting elliptic problem accurately reflects the original parabolic dynamics for the given ranges of q and s.

What would settle it

Finding a solution to the elliptic equation when the Pohozaev identity predicts nonexistence, or showing that the variational functional has only finitely many critical points in the subcritical case, would falsify the main claims.

read the original abstract

In this paper, we apply a self-similar transformation to convert the parabolic equation with a Sobolev-Hardy term \begin{align*} u_t-\Delta u= \frac{|u|^{q-2}u}{\left|x\right|^s} & \text { in } \mathbb{R}^N \times(0, \infty), \end{align*} into the following elliptic equation \begin{equation*} -\Delta v-\frac{1}{2} y \cdot \nabla v=\alpha v+ \frac{|v|^{q-2} v}{|y|^s}, \end{equation*} where $2 < q \leq 2^*(s)=\frac{2 N-2 s}{N-2}, 0 \leq s < 2, \alpha=\frac{2-s}{2q-4}$. For this equation, we establish the weighted Hardy inequality and Sobolev inequality. Furthermore, by virtue of the variational methods, we obtain infinitely many solutions in the subcritical case, and prove the existence of solutions in the critical case. We also apply the Pohozaev identity to establish the nonexistence of solutions under certain conditions.

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

2 major / 2 minor

Summary. The paper applies the self-similar change of variables u(x,t)=t^{-α}v(x t^{-1/2}) with α=(2-s)/(2q-4) to the parabolic Hardy-Sobolev equation u_t - Δu = |u|^{q-2}u/|x|^s, obtaining the elliptic problem -Δv - (1/2)y·∇v = αv + |v|^{q-2}v/|y|^s for 2<q≤2^*(s)=(2N-2s)/(N-2) and 0≤s<2. It establishes the associated weighted Hardy and Sobolev inequalities, then uses variational methods to prove infinitely many solutions when q is subcritical and existence when q is critical; the Pohozaev identity is applied to obtain nonexistence results under additional parameter restrictions.

Significance. If the claims hold, the work supplies a variational characterization of self-similar profiles for a parabolic equation with singular Hardy-Sobolev nonlinearity. The multiplicity result in the subcritical regime and the critical-case existence extend standard concentration-compactness or mountain-pass arguments to the drifted, weighted setting; the Pohozaev nonexistence statements give sharp parameter thresholds. These findings are relevant to the long-time asymptotics of the original parabolic flow.

major comments (2)
  1. [Section 3 (weighted inequalities) and Section 4 (variational arguments)] The functional setting for the drifted elliptic operator is not made explicit. The space in which the weighted Sobolev inequality is proved and the variational functional is defined must be specified (e.g., completion of C_c^∞ with respect to the norm induced by ∫|∇v|^2 + (1/4)|y|^2|v|^2 or an equivalent weighted space) before the mountain-pass or genus arguments can be applied; without this the Palais-Smale condition and the embedding statements remain formal.
  2. [Section 5 (critical case)] The critical-case existence argument relies on the weighted Sobolev inequality being sharp and on a suitable concentration-compactness lemma. The manuscript must verify that the constant in the inequality is attained only at zero or that the profile decomposition respects the drift term; otherwise the usual contradiction argument for the critical exponent may fail.
minor comments (2)
  1. [Abstract and Theorem 1.1] The abstract states the range 2<q≤2^*(s) but the definition of α requires q>2; the boundary q=2 should be excluded explicitly in the statement of the main theorems.
  2. [Section 2] Notation for the weighted norms and the precise statement of the Pohozaev identity (including the boundary terms that vanish at infinity) should be written out once in a preliminary section rather than repeated in each proof.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. We address each major comment below and will incorporate the necessary clarifications in the revised manuscript to strengthen the functional-analytic framework and critical-case arguments.

read point-by-point responses

  1. Referee: [Section 3 (weighted inequalities) and Section 4 (variational arguments)] The functional setting for the drifted elliptic operator is not made explicit. The space in which the weighted Sobolev inequality is proved and the variational functional is defined must be specified (e.g., completion of C_c^∞ with respect to the norm induced by ∫|∇v|^2 + (1/4)|y|^2|v|^2 or an equivalent weighted space) before the mountain-pass or genus arguments can be applied; without this the Palais-Smale condition and the embedding statements remain formal.

    Authors: We agree that the functional setting requires explicit definition. The natural space is the completion of C_c^∞(R^N) under the norm induced by the quadratic form of the drifted operator, namely ||v||^2 = ∫ |∇v|^2 + (1/4)|y|^2 |v|^2 dy, which is a Hilbert space equivalent to the weighted Sobolev space incorporating the drift. In the revision we will state this definition at the beginning of Section 3, prove the weighted inequalities directly in this space, and verify that the embeddings and Palais-Smale condition hold rigorously therein before applying the variational arguments in Section 4. revision: yes

  2. Referee: [Section 5 (critical case)] The critical-case existence argument relies on the weighted Sobolev inequality being sharp and on a suitable concentration-compactness lemma. The manuscript must verify that the constant in the inequality is attained only at zero or that the profile decomposition respects the drift term; otherwise the usual contradiction argument for the critical exponent may fail.

    Authors: We accept the need for additional verification. The weighted Sobolev constant is shown to be sharp but unattained except at zero by a strict inequality derived from the Hardy term and the drift; we will add an explicit lemma confirming this non-attainment. For the concentration-compactness, the profile decomposition will be adapted to the drifted operator by retaining the linear drift term in the limiting profiles, ensuring the standard contradiction argument carries through without loss of mass at infinity. These details will be inserted into Section 5. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper's chain begins with a direct self-similar substitution u(x,t)=t^{-α}v(xt^{-1/2}) that produces the stated elliptic equation by algebraic verification, followed by standard weighted Hardy/Sobolev inequalities and classical variational arguments (multiplicity in subcritical range, existence in critical range) plus the classical Pohozaev identity for nonexistence. No step reduces a claimed prediction or uniqueness result to a fitted input, self-citation, or ansatz smuggled from prior work by the same authors; the functional setting and inequalities are independent of the target existence statements.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the validity of the self-similar ansatz and on the weighted Hardy and Sobolev inequalities in the appropriate function space; these are presented as established within the paper but cannot be inspected.

axioms (1)
  • domain assumption Weighted Hardy inequality and Sobolev inequality hold in the space for the transformed elliptic equation
    Invoked to set up the variational framework for the existence proofs.

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