On a scaled abstract linking theorem with an application to the Schr\"{o}dinger--Poisson--Slater equation

Kanishka Perera; Kaye Silva

REVIEW 3 major objections 4 minor 16 references

On a scaled abstract linking theorem with an application to the Schr\"{o}dinger--Poisson--Slater equation

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves an abstract linking theorem that yields two solutions with opposite energy signs for the Schrödinger–Poisson–Slater equation once the perturbation is small, for every positive parameter λ.

desk verdict New scaled linking theorem with a real gap: the λ<λ1 case uses an empty linking set that Theorem 1.6 cannot handle. read the letter →

arxiv 2506.01165 v1 pith:W4R7ITXM submitted 2025-06-01 math.AP

Kanishka Perera , Kaye Silva This is my paper

classification math.AP MSC 58E0535A1549J2758E07

keywords scaledproblemsexistencemultiplicitylinkingtheoremSchrödinger–Poisson–Slaterequationcohomologicalindexvariationalellipticequationseigenvalueproblem

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

The paper develops an abstract critical point theorem for variational equations whose leading part is homogeneous under a nonlinear scaling of the space variable. It proves that, for any positive parameter $\lambda$, a small perturbation $\mu f(u)$ yields two solutions of the scaled operator equation $A_s(u)=\lambda B_s(u)+\mu f(u)+g(u)$, one with negative and one with positive energy. The main application is the Schrödinger–Poisson–Slater equation, for which the same two-solution statement is obtained in the radial Coulomb–Sobolev space. The interest is that the framework also covers $p$-Laplacian, Kirchhoff, and fractional $p$-Laplacian problems, and it handles values of $\lambda$ beyond the first eigenvalue where the usual mountain-pass geometry fails.

What carries the argument

The central object is the pair of scaled potential operators $(A_s,B_s)$, together with the scaling law $A_s(u_t)=t^sA_s(u)$; $A_s$ and $B_s$ are the Fréchet derivatives of even $C^1$ functionals $I_s,J_s$ with the same homogeneity. The load-bearing identity is the level-set index relation from the companion scaled-eigenvalue theorem, $i(\Psi_{\lambda_k})=i(M_s\setminus\Psi^{\lambda_{k+1}})=k$, where $i$ is the $\mathbb{Z}_2$-cohomological index of symmetric sets, a topological measure of the size of the set. This identity selects the two sets $A_0=\Psi_{\lambda_k}$ and $B_0=\Psi^{\lambda_{k+1}}$ that are linked in Theorem 1.6. The proof builds scaled tubes $A_*$, $A$, $B_*$, $B$ by applying the scaling $u_t$ and the radial projection $\pi_{\mathrm{rad}}$, and uses exact cohomology sequences to show the link forces two critical points. In the application, the scaling $u_t(x)=t^\sigma u(tx)$ and the Pohozaev identity verify condition (H11), the requirement that every solution of the scaled eigenvalue problem satisfy $I_s(u)=\lambda J_s(u)$.

What would settle it

For the scaled operator pair used in the paper's application, compute the first scaled eigenvalues $\lambda_k$ and the cohomological indices $i(\Psi_{\lambda_k})$ and $i(M_s\setminus\Psi^{\lambda_{k+1}})$; a single value different from the claimed integer $k$ would refute the imported theorem and break the linking proof.

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Extended reading notes

Core claim

The central claim is that the competition between the two nonlinearities can be organized by one scaling exponent $s$. If $A_s$ and $B_s$ are odd potential operators on a reflexive Banach space $W$, with a scaling $(u,t)\mapsto u_t$ satisfying the axioms and the homogeneity $A_s(u_t)=t^sA_s(u)$, $B_s(u_t)=t^sB_s(u)$, then the scaled eigenvalue problem $A_s(u)=\lambda B_s(u)$ has eigenvalues $\lambda_k\nearrow\infty$ and the level sets of the Rayleigh quotient $\widetilde{\Psi}=I_s/J_s$ on the manifold $M_s=\{I_s=1\}$ satisfy the index identities $i(\Psi_{\lambda_k})=i(M_s\setminus\Psi^{\lambda_{k+1}})=k$. Choosing $A_0=\Psi_{\lambda_k}$ and $B_0=\Psi^{\lambda_{k+1}}$ as the linking pair, Theorem 1.6 constructs a cohomological link in dimension $k$, and Theorem 1.7 concludes that for every $\lambda>0$ there is $\mu^*>0$ such that for all $\mu\in(0,\mu^*)$ the equation $A_s(u)=\lambda B_s(u)+\mu f(u)+g(u)$ has two solutions $u_1,u_2$ with $\Phi_\lambda(u_1)<0<\Phi_\lambda(u_2)$. The paper verifies the axioms for the radial Coulomb–Sobolev space with the scaling $u_t(x)=t^\sigma u(tx)$, $\sigma=(2+\alpha)/(2(p-1))$, using the Pohozaev identity to check the eigenvalue-homogeneity condition, and so obtains the two-solution theorem for the Schrödinger–Poisson–Slater equation.

Load-bearing premise

The proof takes as given, from a companion preprint, that the scaled eigenvalue problem produces level sets with exactly the topological sizes used to build the link; if that result were wrong, the geometry that yields the two solutions would not hold.

Editorial extensions

If this is right

For the Schrödinger–Poisson–Slater equation (1.1), under the growth assumptions (1.2)–(1.3), every $\lambda>0$ admits a threshold $\mu^*>0$ such that for all $\mu\in(0,\mu^*)$ there are two radial solutions $u_1,u_2$ with $\Phi(u_1)<0<\Phi(u_2)$.
The abstract theorem covers parameter values $\lambda\ge\lambda_1$, where the mountain-pass geometry of the functional fails; the cohomological linking construction replaces that geometry.
By the paper's Remark 1.8, the same result applies to equations involving the $p$-Laplacian, the Kirchhoff operator, and the fractional $p$-Laplacian, since these fit the scaling $u_t=tu$.
The proof handles every positive $\lambda$ through the scaled eigenvalue sequence, splitting only according to whether $\lambda$ lies below or at/above the first scaled eigenvalue $\lambda_1$.

Reading between the lines

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

The scaling exponent $s$ is the organizing parameter of the whole argument; this suggests a general recipe for any problem whose two nonlinear terms have different homogeneity degrees under a dilation, with the sign of the two critical energies inherited from the linking levels.
The only imported ingredient is the companion preprint's eigenvalue-index theorem; re-proving it inside the present framework, or replacing it by a direct construction of $A_0$ and $B_0$, would make the linking argument self-contained.
The exchange-correlation type nonlinearity $h(u)=u^{5/3}-u^{7/3}$ mentioned as motivation does not have the exact form treated here, but the framing points toward an extension to combined nonlinearities with two different scaling rates.
A practical next step would be to quantify $\mu^*$ from the constants in conditions (F1)–(G3) and to test numerically how the two radial solutions and their energy signs behave as $\lambda$ crosses each scaled eigenvalue $\lambda_k$.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things to know. First, the paper proves a genuinely new abstract linking theorem for scaled functionals, and the application to the Schrödinger–Poisson–Slater equation is a serious test of that machinery. Second, there is a gap in the proof for λ<λ1: the argument sets A0=∅ and invokes Theorem 1.6, but with an empty linking set the conclusion of Theorem 1.6 is nonsense. The stress-test note is right.

What is actually new: The scaled linking theorem (Theorems 1.6 and 2.4) extends Perera's earlier constant-scaling linking theorem to the general scaling u_t. The cohomological index machinery is standard, but the statement is useful and the proof is careful. The application verifies all the hypotheses from known embeddings and the authors' prior results; Lemma 3.1's PS-condition proof is compressed but looks internally consistent. The paper reads well and the abstract framework is reusable.

Where it is soft: First, the empty-linking-set gap. In the proof of Theorem 1.7, the case λ∈(0,λ1) is dispatched by taking A0=∅ and B0=Ms. That is not covered by Theorem 1.6 as stated or proved. The proof of Theorem 1.6 starts with a nonemptiness assertion that fails, and the claimed bound E(u1) ≤ sup_A E would read E(u1) ≤ -∞. The introduction notes that λ<λ1 should be handled by a local minimizer plus mountain pass, but that argument is not actually carried out in the proof of Theorem 1.7. This is a genuine, repairable gap. Second, the paper leans on two unreviewed preprints by the same authors: Theorem 1.5 comes from [12] and the compactness argument in Lemma 3.1 uses [15]. That is not automatically disqualifying, but a referee should verify those imported results or ask the authors to include the needed statements.

Overall, the central idea is sound and the gap is fixable. The paper deserves a serious referee, but it should not be accepted as-is. I would send it to review with a request to repair the λ<λ1 case and to clarify the dependency on the preprints.

Referee Report

3 major / 4 minor

Summary. The paper proves an abstract linking theorem (Theorem 1.6) using the Z2-cohomological index and a scaling framework for operators, and applies it to the Schrödinger--Poisson--Slater equation. The main abstract result (Theorem 1.7) asserts that, under hypotheses (H1)--(H13), (F1)--(F3), (G1)--(G3), and a Palais--Smale condition, equation (1.13) has two solutions for sufficiently small μ>0. The application (Theorem 1.2) proves two solutions of the radial Schrödinger--Poisson--Slater equation (1.1) for every λ>0 when μ is small. The proof of Theorem 1.7 constructs the linking sets from the scaled eigenvalue sublevel/superlevel sets and verifies the energy geometry of Theorem 1.6; the application verifies the hypotheses using known embedding theorems, a Pohozaev identity, and a PS-condition proof in Lemma 3.1.

Significance. If the central proofs are completed, the paper offers a broadly applicable abstract critical point theorem for problems with combined nonlinearities, using a scaling that is natural for nonlocal elliptic equations. The cohomological index arguments are standard and presented in detail, and the application to the Schrödinger--Poisson--Slater equation addresses a problem of current interest. The paper demonstrates command of the existing literature and connects to the authors' prior work on scaled functionals. However, the current manuscript has a clear gap in the proof of Theorem 1.7 for λ<λ1, and the energy bound conclusions in Theorem 1.6 are not fully justified, so the main claims are not yet established as written.

major comments (3)

[Section 2, proof of Theorem 1.7, case λ∈(0,λ1)] The proof dispatches the case λ∈(0,λ1) by taking A0=∅ and B0=M_s and then applying Theorem 1.6. This is not a valid application. The proof of Theorem 1.6 begins with the assertion that B*∩A and B∩A* are nonempty, which is false when A0=∅ (then A, A*, and A1 are empty). Moreover, the conclusion of Theorem 1.6 would require a critical point u1 with inf_{B*}E ≤ E(u1) ≤ sup_A E, but sup_A E = -∞ when A=∅, making the claim impossible. Theorem 2.4, used in the proof of Theorem 1.6, relies on A2 being contractible; the empty set is not contractible, and the contraction to w̃ is ill-defined. Therefore Theorem 1.7 is not proved for λ∈(0,λ1), and consequently Theorem 1.2 is not proved for that range. The introduction mentions that standard local-minimizer and mountain-pass arguments handle this case, but no such argument is provided in the manuscript.
[Section 2, proof of Theorem 1.7, case λ≥λ1] The proof chooses k with λ∈[λ_k,λ_{k+1}) and sets A0=eΨ_{λ_k}, B0=eΨ^{λ_{k+1}}, citing Theorem 1.5 for the index equalities i(A0)=i(M_s\B0)=k. However, Theorem 1.5(iii) states these equalities for λ strictly between λ_k and λ_{k+1}. When λ is a repeated eigenvalue (λ_k=λ_{k+1}) the interval is empty, the index equalities may fail, and the coefficient 1-λ/λ_{k+1} in (2.2) vanishes, so the proof of inf_B Φ_λ > 0 does not work. The paper does not provide an alternative argument for eigenvalues of multiplicity greater than one, which is a load-bearing omission for the claim that Theorem 1.2 holds for every λ>0.
[Section 2, proof of Theorem 1.6] The proof opens with 'Since B* ∩ A and B ∩ A* are nonempty we have inf_{B*}E ≤ sup_A E and inf_B E ≤ sup_{A*}E,' but this nonemptiness is not an explicit hypothesis of Theorem 1.6 and is not proved. The conclusion of Theorem 1.6 asserts the energy bounds inf_{B*}E ≤ E(u1) ≤ sup_A E and inf_B E ≤ E(u2) ≤ sup_{A*}E, yet the proof only establishes nonvanishing cohomology groups H^k(E_β,E_α) and H^{k+1}(E_γ,E_β), which imply the existence of critical points with levels in (α,β) and (β,γ) but do not imply those sharper bounds unless [inf_{B*}E, sup_A E] is nonempty. In the application, 0∈B*, so inf_{B*}Φ_λ ≤ 0, while sup_A Φ_λ < 0 is proved, but the inequality inf_{B*}Φ_λ ≤ sup_A Φ_λ is not verified. The authors need to either add the missing nonemptiness assumptions or revise the conclusion and the verification of the linking geometry.

minor comments (4)

[Section 1.2, Theorem 1.5] The paper relies on Theorem 1.5 from the authors' preprint [12] without proof. Since this theorem is central to the linking construction, the referee requests that the authors provide a proof or refer to a published version.
[Section 3, Lemma 3.1] Equation (3.2) contains the expression 'I(un) σ(r−q) s r', which is ambiguous; the intended exponent σ(r-q)/s should be written with clear parentheses.
[Section 1, notation] The definitions of eΨ_a and eΨ^a are given after Theorem 1.1; consider moving them to just after the first use in (1.7) to avoid confusion.
[References] Items [12] and [15] are preprints by the same authors; the reader would benefit from updated publication data if available.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; the derivation is conditional on prior results with independent content.

full rationale

The paper's derivation is conditional on prior results rather than circular. The main abstract theorem (Theorem 1.7) is proved here, but its linking geometry uses Theorem 1.5, imported from the authors' preprint [12], for the index identities i(Ψ_{λ_k}) = i(M_s \ Ψ_{λ_{k+1}}) = k. This is a load-bearing self-citation, but it is not circular: Theorem 1.5 is a separate eigenvalue/index theorem with stated hypotheses (H1)-(H11), and its conclusion does not contain the two-solution claim of Theorem 1.7, nor is it derived from Theorem 1.7 in this manuscript. Similarly, the SPS application cites [11] for embeddings and [15] for compactness; these are prior results with independent content. There are no fitted parameters, no data subsets, and no prediction that reduces by construction to an input. The reviewer-flagged concern that the case λ ∈ (0, λ1) uses A0 = ∅ in Theorem 1.6 is a possible correctness gap in the linking proof, not a circularity, and it does not change the circularity score. Overall, the self-citations are notable but the derivation chain is not self-referential.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free numeric parameters are fit to data; the abstract framework is conditional on hypotheses but the SPS application verifies them from known embeddedness results and prior work, some by the same authors. No new entities are introduced.

assumptions (5)

domain assumption Hypotheses (H1)-(H13), (F1)-(F3), (G1)-(G4) of the abstract theorems.
These are the explicit conditions under which Theorems 1.6 and 1.7 are proved; for the SPS application they are verified using [5], [11], and [15].
standard math Theorem 1.5 (scaled eigenvalue theorem) from [12], a preprint by the authors, is imported without proof.
Quoted as Theorem 1.5 and used to construct the linking sets A0 and B0 in the proof of Theorem 1.7.
standard math Radial Coulomb-Sobolev embedding theorems of Mercuri-Moroz-Van Schaftingen [11], including compact embeddings and lower bounds on L^ell mass over weakly compact subsets.
Used to verify (F1), (F3), (G1), (G3), and the boundedness in Lemma 3.1.
standard math The (S)_+ property (H7) and the PS condition for the SPS functional from [15, Lemma 2.10].
Imported from the authors' preprint; needed for the compactness of the abstract theory and for Lemma 3.1.
standard math Pohozaev identity for solutions of I'(u)=lambda J'(u) (from Ianni-Ruiz [5] or [11]) is used to verify (H11).
Quoted in the proof of Theorem 1.2.

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Cite this review

Pith. "Pith review of On a scaled abstract linking theorem with an application to the Schr\"{o}dinger--Poisson--Slater equation." pith.science (2026). https://pith.science/paper/W4R7ITXM

@misc{pith2026250601165,
  author       = {Pith},
  title        = {Pith review of: On a scaled abstract linking theorem with an application to the Schr\"odinger--Poisson--Slater equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W4R7ITXM}},
  note         = {Machine review of arXiv:2506.01165}
}

read the original abstract

We prove an abstract linking theorem that can be used to show existence of solutions to various types of variational elliptic equations, including Schr\"{o}dinger--Poisson--Slater type equations.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [12]

Variational methods for scaled functionals with applications to the Schr\"{o}dinger-Poisson-Slater equation

Carlo Mercuri and Kanishka Perera. Variational methods for scaled functionals with applications to the schr¨ odinger–poisson–slater equation. preprint,arXiv:2411.15887

work page Pith review arXiv

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Kanishka Perera and Kaye Silva.On theN-dimensional Schr¨ odinger–Poisson–Slater equation ` a la Brezis–Nirenberg, arXiv:2505.07146, 2025

work page Pith review arXiv 2025

[3] [1]

Existence of minimizers for Kohn-Sham models in quantum chemistry.Ann

Arnaud Anantharaman and Eric Canc` es. Existence of minimizers for Kohn-Sham models in quantum chemistry.Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 26(6):2425–2455, 2009

work page 2009

[4] [2]

Bao Weizhu, N. J. Mauser and H. P. Stimming. Effective one particle quantum dynamics of electrons: a numerical study of the Schr¨ odinger–Poisson-Xαmodel.Commun. Math. Sci., 1(4):809–828, 2003

work page 2003

[5] [3]

Fadell and Paul H

Edward R. Fadell and Paul H. Rabinowitz. Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems. Invent. Math., 45(2):139–174, 1978

work page 1978

[6] [4]

Frank, Phan Th` anh Nam, and Hanne Van Den Bosch

Rupert L. Frank, Phan Th` anh Nam, and Hanne Van Den Bosch. The ionization conjecture in Thomas-Fermi-Dirac–von Weizs¨ acker theory.Comm. Pure Appl. Math., 71(3):577–614, 2018

work page 2018

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Ground and bound states for a static Schr¨ odinger-Poisson- Slater problem.Commun

Isabella Ianni and David Ruiz. Ground and bound states for a static Schr¨ odinger-Poisson- Slater problem.Commun. Contemp. Math., 14(1):1250003, 22, 2012

work page 2012

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From atoms to crystals: a mathematical journey

Claude Le Bris and Pierre-Louis Lions. From atoms to crystals: a mathematical journey. Bull. Amer. Math. Soc. (N.S.), 42(3):291–363, 2005

work page 2005