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The existence of a nontrivial weak solution to a double critical problem involving fractional Laplacian in ${\R}^n$ with a Hardy term

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Pith's one-line read The paper proves that the doubly critical fractional Laplacian equation with a Hardy term admits at least one nontrivial weak solution in the parameter ranges of Theorem 1.1.

desk verdict A promising new double-critical fractional problem with a genuinely useful weighted Morrey inequality, but the main theorem overreaches because the Λ-minimizers for γ<0 are asserted, not proved. read the letter →

arxiv 1908.02536 v2 pith:VIJRS2ZB submitted 2019-08-07 math.AP

Gongbao Li , Tao Yang This is my paper · ORCID
classification math.AP MSC 35A0135A2335B3335R1135R70
keywords fractionalLaplacianHardytermdoublecriticalexponentsnontrivialweaksolutionweightedMorreyspaceimprovedSobolevinequalitymountainpasslemmaRieszpotentialnonlinearity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper aims to prove that a fractional Laplacian equation carrying two critical nonlinearities—a Hardy–Sobolev term and a Riesz-potential (convolution) term—has at least one nonzero weak solution whenever the Hardy coefficient $\gamma$ stays below the critical constant $\gamma_H$ (and, when both singular exponents $\alpha,\beta$ are positive, even for negative $\gamma$). Standard compactness fails because both nonlinearities sit at the critical exponent, so the paper introduces a weighted Morrey embedding chain and an improved Sobolev inequality to control the Palais–Smale sequences directly in $\dot H^s(\mathbb{R}^n)$, without truncation or extension to the upper half-space. If correct, this broadens the admissible parameters relative to earlier results and gives a simpler proof of known existence theorems. The same mechanism is claimed to transfer to the p-Laplacian analogue.

What carries the argument

The engine of the proof is the embedding chain $$\dot H^s(\mathbb{R}^n) \hookrightarrow $L^{{2^{*}}$_s(\$\alpha$)}(\mathbb{R}^n,|y|^{-\$\alpha$}) \hookrightarrow $L^{{p,\frac{n-2s}}${2}p+pr}(\mathbb{R}^n,|y|^{-pr}),$$ where the middle space is a weighted Morrey space: it controls, on every ball, the $p$-integral of $|u|^p|y|^{-pr}$ with a scale factor that is exactly dilation invariant. Proposition 1.3's improved Sobolev inequality, $$\left(\int_{\mathbb{R}^n}\frac{|u(y)|^{$2^{{*}}$_s(\$\alpha$)}}{|y|^\$\alpha$}dy\right)^{1/$2^{{*}}$_s(\$\alpha$)} \le C\|u\|^\theta_{\dot H^s(\mathbb{R}^n)}\|u\|^{1-\$\theta$}_{$L^{{p,\frac{n-2s}}${2}p+pr}(\mathbb{R}^n,|y|^{-pr})},$$ then bounds the critical Hardy–Sobolev norm by a product of the energy norm and a Morrey norm. This is what rules out the vanishing of (PS) sequences and yields the minimizers of $S_\mu$ and $\Lambda$ used to define the mountain-pass threshold $c^*$.

What would settle it

Compute the Hardy–Sobolev ratio $\Lambda(n,s,\gamma,\alpha)$ for a concrete negative $\gamma$ (for example $n=3$, $s=1/2$, $\alpha=1$, $\gamma=-1$). If a minimizing sequence concentrates and the infimum is not achieved by any function in $\dot H^s(\mathbb{R}^n)$, then the assumed compactness in Proposition 4.1(3) fails and the strict mountain-pass inequality $c<c^*$ collapses.

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Extended reading notes

Core claim

The central claim, Theorem 1.1, is that the equation $$(-\$\Delta$)^s u - \gamma \frac{u}{|x|^{2s}} = \frac{|u|^{$2^{{*}}$_s(\$\beta$)-2}u}{|x|^\$\beta$} + [I_\mu * F_\$\alpha$(\cdot,u)](x) f_\$\alpha$(x,u), \qquad u \in \dot H^s(\mathbb{R}^n),$$ has at least one nontrivial weak solution for $s\in(0,1)$, $0\le\alpha,\beta<2s<n$, $\mu\in(0,n)$, provided either $\alpha,\beta>0$ and $\gamma<\gamma_H$, or $\alpha\beta=0$ and $0\le\gamma<\gamma_H$. The supporting discovery is the embedding chain $$\dot H^s(\mathbb{R}^n) \hookrightarrow $L^{{2^{*}}$_s(\$\alpha$)}(\mathbb{R}^n,|y|^{-\$\alpha$}) \hookrightarrow $L^{{p,\frac{n-2s}}${2}p+pr}(\mathbb{R}^n,|y|^{-pr})$$ and the improved Sobolev inequality built on it, which are strong enough to prevent the Palais–Smale sequence from vanishing and to produce minimizers for the variational constants that set the mountain-pass threshold.

Load-bearing premise

The load-bearing premise is that the variational constants $\Lambda(n,s,\gamma,\alpha)$ and $\Lambda(n,s,\gamma,0)$ are actually attained in $\dot H^s(\mathbb{R}^n)$ for the full claimed range of $\gamma$, including negative values; the paper asserts this by analogy with a proven case rather than supplying the full proof.

Editorial extensions

If this is right

  • Every parameter set in case (I) of Theorem 1.1, including negative Hardy coefficients $\gamma<0$, yields a nonzero weak solution in $\dot H^s(\mathbb{R}^n)$.
  • Every parameter set in case (II), where at least one of $\alpha,\beta$ is zero, yields the same conclusion for $0\le\gamma<\gamma_H$; the fully unweighted case $\alpha=\beta=0$ is recovered through the Nehari-manifold route.
  • The improved Sobolev inequality (1.10) makes the earlier truncation and extension-space arguments unnecessary in the fractional setting, giving a direct proof of the corresponding existence results it extends.
  • Corollary 1.4 transfers the same machinery to the p-Laplacian equation with two critical exponents, producing Theorem 5.5.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the method relies only on dilation-invariant norm control, a natural testable extension is to systems or higher-order operators with several critical Riesz-potential terms; the p-Laplacian case is already sketched in the paper, but biharmonic or mixed-order analogues are not.
  • If the attainment of $\Lambda(n,s,\gamma,\alpha)$ fails for some negative $\gamma$, the existence claim could still be true through a different choice of mountain-pass path; the paper's proof, however, would need a new construction of $v_0$ that does not use the minimizer asserted in Proposition 4.1(3).
  • The sharp range of the exponent $\theta$ in the improved Sobolev inequality could be probed numerically; classifying equality cases would give explicit threshold functions and possibly identify the extremals of the critical problem.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The manuscript studies the existence of nontrivial weak solutions in the homogeneous fractional Sobolev space \dot{H}^s(\mathbb{R}^n) to the double critical equation (1.1), which combines a fractional Laplacian with a Hardy potential, a Hardy–Sobolev critical term, and a Choquard-type nonlocal critical term. The main theorem claims existence under either (I) 0<\alpha,\beta<2s<n, \mu\in(0,n) and \gamma<\gamma_H, or (II) \alpha\beta=0, 0\le \alpha,\beta<2s<n, \mu\in(0,n) and 0\le\gamma<\gamma_H. The proof introduces weighted Morrey-space embeddings and an improved Sobolev inequality, uses them to prove attainment of the variational constants S_\mu(n,s,\gamma,\alpha) and \Lambda(n,s,\gamma,\alpha), and then applies the mountain-pass lemma with a threshold c^* to produce a Palais-Smale sequence whose rescaled weak limit is shown to be a nontrivial solution.

Significance. If the proof is completed, the result is a genuine advance: it removes the restriction \mu\in(n-2s,n) appearing in [41], it allows negative \gamma when both \alpha and \beta are positive, and it offers a direct argument that avoids the extension-space and truncation machinery used in [2]. The improved Sobolev inequality in weighted Morrey spaces, Proposition 1.3 and Corollary 1.4, is a useful independent contribution. The manuscript is also transparent about the external origin of the inequalities it uses, and I found no circular dependence on the existence statement. However, a load-bearing part of the attainment argument for \Lambda is only asserted, not proved, so the current version is not yet acceptable.

major comments (3)
  1. [Section 4, Proposition 4.1(3)-(4) and Remark 4.2] The attainment of \Lambda(n,s,\gamma,\alpha) for 0<\alpha<2s and of \Lambda(n,s,\gamma,0) is stated as Proposition 4.1(3)-(4), but Remark 4.2 explicitly says that only items (1)-(2) are proved and that the strategy 'can be applied' to (3)-(4). This is not a proof. Proposition 5.2 then uses item (3) to construct the minimizer V_{\gamma,\beta} and to establish the strict mountain-pass inequality c<c^*, and Theorem 1.1(I) depends on this in the negative-\gamma regime, which is exactly the advertised improvement over [2]. Since [2] covers only 0\le\gamma<\gamma_H, the negative-\gamma case is not covered by the cited result. The missing proof should be supplied; the concentration argument used for Proposition 4.1(1) appears transferable by replacing B_\alpha with the weighted norm \int_{\mathbb{R}^n}|u|^{2^*_s(\alpha)}/|x|^\alpha\,dx and using the subadditivity exponent 2/2^*_s(\alpha)<1, but this must be written out.
  2. [Section 5, Proposition 5.3] The statement of Proposition 5.3 covers the two cases \alpha=0<\beta<2s and \beta=0<\alpha<2s, but its proof says that items (2) and (4) of Proposition 4.1 are used, which are the \alpha=0 items. For the case \alpha=0<\beta, the required constant is \Lambda(n,s,\gamma,\beta), i.e., item (3), not item (4); for the case \beta=0<\alpha, one needs items (1) and (4). The proof therefore does not match the statement and must be corrected, with the needed attainment results identified precisely.
  3. [Theorem 1.1(II), case \alpha=\beta=0] The final paragraph of the proof of Theorem 1.1 dismisses the case \alpha=\beta=0 by saying that the Nehari manifold method in [41] gives a nontrivial weak solution for 0\le\gamma<\gamma_H. However, the hypotheses of [41] as summarized in the introduction are 0<\beta<2s, \mu\in(n-2s,n), and 0<\gamma<\gamma_H; these do not directly cover \alpha=\beta=0 with \mu\in(0,n) and \gamma=0 or more generally \gamma\in[0,\gamma_H). This part of Theorem 1.1 therefore needs either a proof adapted to the present parameter range or a precise reference whose assumptions match the claim.
minor comments (6)
  1. [Equations (2.3) and surrounding text] The Hardy term in several displayed formulas is missing the division slash; it should read \int_{\mathbb{R}^n} u^2/|x|^{2s}\,dx rather than \int_{\mathbb{R}^n} u^2|x|^{2s}\,dx.
  2. [Equations (0.2), (1.9), and Section 3] The notation L^{p,\frac{n-2s}{2}p+pr}(\mathbb{R}^n,|y|^{-pr}) is ambiguous; it should be written as L^{p,\frac{n-2s}{2}p+pr}(\mathbb{R}^n,|y|^{-pr}) is ambiguous; it should be written as L^{p,((n-2s)/2)p+pr}(\mathbb{R}^n,|y|^{-pr}) to make the Morrey exponent clear.
  3. [Equation (1.12)] There is a stray parenthesis in the domain: it should read u\in\dot{H}^s(\mathbb{R}^n)\setminus\{0\}, not u\in\dot{H}^s(\mathbb{R}^n))\setminus\{0\}.
  4. [Proposition 5.2 and Theorem 1.1, definition of c^*] The coefficient in the second term of c^* should be parenthesized as (2s-\beta)/(2(n-\beta)), and the exponent as (n-\beta)/(2s-\beta), to avoid misreading.
  5. [Notation throughout] The symbol 2^\#_\mu(\alpha) is easily confused with a superscripted hash or with 2 times a number; introducing a separate symbol such as q_\mu(\alpha) for the Choquard exponent would improve readability.
  6. [Section 5, Theorem 5.5] Theorem 5.5 states a p-Laplace analogue without proof. If it is intended as a mathematical result, a proof or a precise reference is needed; otherwise it should be labeled as a conjecture or an extension remark.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the variational constants are infima, the improved Sobolev inequality is proved from external Sawyer-Wheeden/HLS inequalities, and the only author-overlapping citation [40] is background. The unproved attainment claim in Proposition 4.1(3)-(4) for negative gamma is a proof gap, not a circular reduction.

full rationale

The derivation chain is not circular. Proposition 1.3 is proved from the external Sawyer-Wheeden weighted Riesz-potential inequality (Lemma 3.1) and the HLS inequality, not from the existence of a solution to (1.1). The constants S_mu and Lambda are infima of variational quotients; they are not fitted parameters. Proposition 4.1 proves attainment for cases (1)-(2) and refers to [2] for case (3), while case (4) is said to imitate case (2); the mountain-pass threshold c* is a function of these infima, and the strict inequality c<c* is obtained by choosing v0 as a minimizer of one of the two limiting problems and estimating sup_t I(tU) < sup_t f1(t), which is a standard strictness argument, not an identity. The nonvanishing argument uses c<c* and the improved Sobolev inequality, again without assuming the conclusion. The only author-overlapping citation is [40] (Kang-Li), cited in the introductory survey and never used in the proofs, so it is not load-bearing. The honest limitation appears in Remark 4.2: only (1)-(2) are proved, while (3) is attributed to [2] and (4) is promised by imitation; for gamma<0, [2] covers only 0<=gamma<gamma_H, so Theorem 1.1(I) in the negative-gamma regime rests on an unproved attainment claim. This is an omitted proof or correctness risk, not a circular step, because no equation or constant is defined in terms of the target weak solution and no fitted quantity is relabeled as a prediction. Hence the circularity score is low.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The paper's central claim rests on standard external inequalities (Hardy, Hardy-Sobolev, HLS, Sawyer-Wheeden) and standard critical point tools (mountain pass, Brezis-Lieb). No free parameters are fitted to data, and no new entities are invented. The main unproven assumption is the attainment of the variational constants Lambda(n,s,gamma,alpha) for the full gamma < gamma_H range, which is analogous to a standard lemma but not fully demonstrated in the text.

assumptions (8)
  • standard math Fractional Hardy inequality with best constant gamma_H (Lemma 2.1)
    Imported from Frank-Lieb-Seiringer [27]; used to define the equivalent norm ||u|| and the parameter range gamma < gamma_H in the whole paper.
  • standard math Fractional Hardy-Sobolev inequality (Lemma 2.2)
    Imported from Ghoussoub-Shakerian [2]; gives the critical weighted Lebesgue embedding dot H^s into L^{2^*_s(alpha)}(|x|^{-alpha}) used throughout.
  • standard math Hardy-Littlewood-Sobolev inequality (Lemma 2.4)
    Imported from Lieb-Loss [35]; bounds the nonlocal bilinear form B_alpha and puts the Choquard term in the energy.
  • standard math Sawyer-Wheeden weighted norm inequality for fractional integrals (Lemma 3.1)
    Imported from Sawyer-Wheeden [29] and Muckenhoupt-Wheeden [8]; is the external tool behind the improved Sobolev inequality, Proposition 1.3.
  • standard math Brezis-Lieb type lemmas for the nonlocal term (Lemmas 2.5, 2.7, 2.8, 2.9)
    Variants of the classical Brezis-Lieb lemma from Moroz-van Schaftingen [38] and Singh [45]; used to decompose energies of weakly convergent sequences and pass the convolution term to the limit.
  • standard math Mountain pass lemma (Lemma 5.1)
    Ambrosetti-Rabinowitz [37]; produces the Palais-Smale sequence at the level c used in the proof of Theorem 1.1.
  • standard math Fractional Polya-Szego inequality and rearrangement bounds (used in Proposition 4.1(2))
    Imported from Park [34] and Servadei-Valdinoci [32]; used only in the alpha = 0 attainment proof for S_mu(n,s,gamma,0).
  • standard math Local compact embedding of dot H^s into L^q_loc for q < 2^*_s (Corollary 7.2 of [21])
    Used in Lemma 2.3 to obtain strong local L^2 convergence of rescaled sequences under the weight |x|^{-2r}.

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Pith. "Pith review of The existence of a nontrivial weak solution to a double critical problem involving fractional Laplacian in ${\R}^n$ with a Hardy term." pith.science (2026). https://pith.science/paper/VIJRS2ZB

@misc{pith2026190802536,
  author       = {Pith},
  title        = {Pith review of: The existence of a nontrivial weak solution to a double critical problem involving fractional Laplacian in $\R^n$ with a Hardy term},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VIJRS2ZB}},
  note         = {Machine review of arXiv:1908.02536}
}
abstract

In this paper, we consider the existence of nontrivial weak solutions to a double critical problem involving fractional Laplacian with a Hardy term: \begin{equation} \label{eq0.1} (-\Delta)^{s}u-{\gamma} {\frac{u}{|x|^{2s}}}= {\frac{{|u|}^{ {2^{*}_{s}}(\beta)-2}u}{|x|^{\beta}}}+ \big [ I_{\mu}* F_{\alpha}(\cdot,u) \big](x)f_{\alpha}(x,u), \ \ u \in {\dot{H}}^s(\R^{n}) \end{equation} where $s \in(0,1)$, $0\leq \alpha,\beta<2s<n$, $\mu \in (0,n)$, $\gamma<\gamma_{H}$, $I_{\mu}(x)=|x|^{-\mu}$, $F_{\alpha}(x,u)=\frac{ {|u(x)|}^{ {2^{\#}_{\mu} }(\alpha)} }{ {|x|}^{ {\delta_{\mu} (\alpha)} } }$, $f_{\alpha}(x,u)=\frac{ {|u(x)|}^{{ 2^{\#}_{\mu} }(\alpha)-2}u(x) }{ {|x|}^{ {\delta_{\mu} (\alpha)} } }$, $2^{\#}_{\mu} (\alpha)=(1-\frac{\mu}{2n})\cdot 2^{*}_{s} (\alpha)$, $\delta_{\mu} (\alpha)=(1-\frac{\mu}{2n})\alpha$, ${2^{*}_{s}}(\alpha)=\frac{2(n-\alpha)}{n-2s}$ and $\gamma_{H}=4^s\frac{\Gamma^2(\frac{n+2s}{4})} {\Gamma^2(\frac{n-2s}{4})}$. We show that problem (\ref{eq0.1}) admits at least a weak solution under some conditions. To prove the main result, we develop some useful tools based on a weighted Morrey space. To be precise, we discover the embeddings \begin{equation} \label{eq0.2} {\dot{H}}^s(\R^{n}) \hookrightarrow {L}^{2^*_{s}(\alpha)}(\R^{n},|y|^{-\alpha}) \hookrightarrow L^{p,\frac{n-2s}{2}p+pr}(\R^{n},|y|^{-pr}) \end{equation} where $s \in (0,1)$, $0<\alpha<2s<n$, $p\in[1,2^*_{s}(\alpha))$, $r=\frac{\alpha}{ 2^*_{s}(\alpha) }$; We also establish an improved Sobolev inequality. By using mountain pass lemma along with an improved Sobolev inequality, we obtain a nontrivial weak solution to problem (\ref{eq0.1}) in a direct way. It is worth while to point out that the improved Sobolev inequality could be applied to simplify the proof of the main results in \cite{NGSS} and \cite{RFPP}.

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Reviewed August 14, 2026 · model on record in the stance chip above.