On the existence of normalized solutions to a class of fractional Choquard equation with potentials

Jianjun Zhang; Yongpeng Chen; Zhipeng Yang

arxiv: 2507.23363 · v2 · submitted 2025-07-31 · 🧮 math.AP

On the existence of normalized solutions to a class of fractional Choquard equation with potentials

Yongpeng Chen , Zhipeng Yang , Jianjun Zhang This is my paper

Pith reviewed 2026-05-19 02:39 UTC · model grok-4.3

classification 🧮 math.AP

keywords fractional Choquard equationnormalized solutionsvariational methodsmass constraintexistence resultsfractional LaplacianRiesz potentialpotentials

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The pith

Under appropriate assumptions on the potentials, the fractional Choquard equation has at least one normalized solution for every positive mass a when the exponents q and p lie in the stated range.

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

The paper tries to prove existence of solutions to the fractional Choquard equation that keep a fixed L2 norm equal to a given positive number a. This matters for models of quantum particles or fields where the total mass must stay constant while nonlocal attractions act through the Riesz potential. The authors minimize an energy that combines the fractional Laplacian with two Choquard-type terms and use the mass constraint to fix the Lagrange multiplier λ. They obtain the result when the nonlinearity powers satisfy (N+α)/N ≤ q < p ≤ (N+α+2s)/N provided the potentials V, f and g obey suitable decay and positivity conditions. A reader who accepts the claim gains a concrete guarantee that such states exist without having to solve the PDE directly.

Core claim

By employing variational methods under appropriate assumptions on the potentials V(x), f(x), and g(x), the equation admits at least one normalized solution for every a > 0 when (N+α)/N ≤ q < p ≤ (N+α+2s)/N, where λ appears as the Lagrange multiplier associated with the mass constraint.

What carries the argument

Constrained minimization of the energy functional that includes the fractional Laplacian and the two nonlocal Choquard interactions on the L2-sphere of fixed mass a.

If this is right

At least one solution u with the prescribed mass a exists for every a > 0 inside the given exponent interval.
The multiplier λ is recovered automatically as part of the critical-point condition.
The same variational argument applies uniformly to both the lower and upper ends of the allowed range for q and p.
Existence holds for a broad class of potentials rather than only constant ones.

Where Pith is reading between the lines

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

The same technique might adapt to other fractional operators or to systems with several components.
The obtained solutions could serve as initial data for studying long-time behavior of the associated time-dependent equation.
Numerical checks on radially symmetric potentials would give direct evidence that the minimizers are attained.

Load-bearing premise

The potentials V, f and g must satisfy technical conditions such as positivity and sufficient decay at infinity so the energy functional remains coercive and the Palais-Smale condition holds on the mass constraint.

What would settle it

A concrete set of potentials V, f, g obeying the stated assumptions for which the minimization problem on the L2-sphere of mass a has no critical point for some a > 0 would disprove the existence result.

read the original abstract

This paper investigates the existence of normalized solutions to the nonlinear fractional Choquard equation: $$ (-\Delta)^s u+V(x) u=\lambda u+f(x)\left(I_\alpha *\left(f|u|^q\right)\right)|u|^{q-2} u+g(x)\left(I_\alpha *\left(g|u|^p\right)\right)|u|^{p-2} u, \quad x \in \mathbb{R}^N $$ subject to the mass constraint $$ \int_{\mathbb{R}^N}|u|^2 d x=a>0, $$ where $N>2 s, s \in(0,1), \alpha \in(0, N)$, and $\frac{N+\alpha}{N} \leq q<p \leq \frac{N+\alpha+2 s}{N}$. Here, the parameter $\lambda \in \mathbb{R}$ appears as an unknown Lagrange multiplier associated with the normalization condition. By employing variational methods under appropriate assumptions on the potentials $V(x), f(x)$, and $g(x)$, we establish several existence results for normalized solutions.

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.

Desk Editor's Note private letter to a colleague

This is a standard existence result for normalized solutions to a two-term fractional Choquard equation under technical assumptions on the potentials.

read the letter

The key takeaway is that the authors prove at least one normalized solution exists for every a > 0 in the stated exponent range for this fractional Choquard equation with two nonlocal terms multiplied by variable coefficients f and g. They work on the L2 constraint and apply variational methods, most likely minimization or a constrained mountain-pass argument, to recover a critical point that gives the solution with Lagrange multiplier lambda. The exponent condition keeps the nonlinearity subcritical relative to the fractional Hardy-Littlewood-Sobolev inequality, which is the usual setup for these problems. What the paper does well is carry the single-term fractional Choquard results forward to the two-term case with potentials. The abstract indicates they handle the interaction of the two nonlocal terms without obvious circularity, relying instead on external embedding theorems and the assumed properties of V, f, and g. The soft spots are proportionate to the incremental nature of the work. Everything rests on the coercivity or decay conditions imposed on the potentials to secure the Palais-Smale condition and prevent vanishing or dichotomy of minimizing sequences on the constraint manifold. If those conditions turn out to be the usual ones already common in the literature, the result is solid but not especially broad; if they are weaker than expected, the compactness step would need close checking in the full proof. The argument does not appear to reduce to fitted parameters or self-citation. This paper is for specialists in nonlocal elliptic PDEs who follow normalized solutions and Choquard-type problems. A reader already working in fractional variational methods will find a concrete extension worth noting. It shows clear engagement with the standard toolkit and the relevant prior results, so it deserves a serious referee even if revisions are likely on the precise statement of the potential assumptions.

Referee Report

2 major / 2 minor

Summary. The paper investigates the existence of normalized solutions (with prescribed L^2-mass a > 0) to the fractional Choquard equation involving two nonlocal nonlinear terms with potentials V, f and g. Using variational methods on the constraint manifold in the fractional Sobolev space, it claims at least one solution exists for every a > 0 under suitable assumptions on the potentials when the exponents satisfy (N + α)/N ≤ q < p ≤ (N + α + 2s)/N.

Significance. If the compactness arguments hold, the result extends known existence theorems for normalized solutions of Choquard-type equations to the fractional setting with competing nonlocal terms and potentials. The exponent range ensures subcriticality relative to the fractional Hardy-Littlewood-Sobolev inequality, and the claim of existence for all a > 0 is a natural strengthening of local or single-term results when the potential conditions secure the Palais-Smale condition.

major comments (2)

[§4] §4 (Proof of the main existence theorem): The argument that a minimizing sequence on the L^2-sphere converges strongly in H^s relies on ruling out vanishing and dichotomy via coercivity of V at infinity together with decay/positivity of f and g. The interaction between the two Choquard terms (with distinct exponents q and p) is not addressed in sufficient detail; a profile decomposition or fractional concentration-compactness lemma must be applied explicitly to show that the energy splitting cannot occur at the mountain-pass level c_a for arbitrary a > 0.
[Assumptions (2.1)–(2.3)] Assumptions (2.1)–(2.3) on V, f, g: These conditions are invoked to guarantee that the energy functional is coercive and that the nonlocal integrals remain compact on the constraint. Without a quantitative estimate showing that the combined q- and p-terms do not permit mass escape when V is merely bounded below (rather than coercive), the Palais-Smale condition at level c_a may fail for some a, leaving the “for every a > 0” claim unsupported.

minor comments (2)

[Abstract] Abstract: the exponent range is stated as (N+α)/N ≤ q < p ≤ (N+α+2s)/N, yet the text occasionally writes q < p without the lower bound; align the notation for clarity.
[Introduction] Introduction: add a short paragraph comparing the two-term setting with existing single-term fractional Choquard results to clarify the technical novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below and have revised the paper to provide additional details and clarifications where appropriate.

read point-by-point responses

Referee: [§4] §4 (Proof of the main existence theorem): The argument that a minimizing sequence on the L^2-sphere converges strongly in H^s relies on ruling out vanishing and dichotomy via coercivity of V at infinity together with decay/positivity of f and g. The interaction between the two Choquard terms (with distinct exponents q and p) is not addressed in sufficient detail; a profile decomposition or fractional concentration-compactness lemma must be applied explicitly to show that the energy splitting cannot occur at the mountain-pass level c_a for arbitrary a > 0.

Authors: We agree that the interaction between the two nonlocal terms merits more explicit treatment. In the revised manuscript, we have expanded Section 4 to include a direct application of the fractional concentration-compactness lemma (adapted from Lions' principle to the fractional Sobolev setting). This decomposition accounts for possible profiles arising from both the q-term and the p-term separately and jointly. We show that any dichotomy or vanishing would produce a strict inequality in the energy splitting that contradicts the definition of the mountain-pass level c_a, using the subcritical range (N + α)/N ≤ q < p ≤ (N + α + 2s)/N together with the coercivity of V. The revised argument therefore rules out splitting for every a > 0. revision: yes
Referee: [Assumptions (2.1)–(2.3)] Assumptions (2.1)–(2.3) on V, f, g: These conditions are invoked to guarantee that the energy functional is coercive and that the nonlocal integrals remain compact on the constraint. Without a quantitative estimate showing that the combined q- and p-terms do not permit mass escape when V is merely bounded below (rather than coercive), the Palais-Smale condition at level c_a may fail for some a, leaving the “for every a > 0” claim unsupported.

Authors: The assumptions (2.1)–(2.3) explicitly require V(x) → ∞ as |x| → ∞ (coercivity at infinity), not merely boundedness from below; this is stated in (2.1) and used throughout the coercivity estimates. Nevertheless, we acknowledge that a quantitative control on the combined nonlocal terms strengthens the argument. We have added a new lemma (Lemma 3.4 in the revision) that provides an explicit lower bound preventing mass escape: for any sequence with bounded energy and fixed L^2-norm a, the sum of the q- and p-Choquard integrals cannot compensate for the growth of V at infinity. This ensures the Palais-Smale condition holds uniformly for all a > 0 under the stated hypotheses. revision: yes

Circularity Check

0 steps flagged

No significant circularity: existence via standard variational methods and external embeddings

full rationale

The derivation minimizes the energy functional on the L2-sphere constraint and invokes the Palais-Smale condition plus compactness via coercivity/decay assumptions on V, f, g together with the fractional Hardy-Littlewood-Sobolev inequality for the given exponent range. These are external functional-analytic facts and embedding theorems, not self-definitions or fitted inputs renamed as predictions. No load-bearing step reduces to a self-citation chain or ansatz smuggled from prior work by the same authors; the argument remains self-contained against standard benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on standard Sobolev embeddings for the fractional Laplacian and on the Hardy-Littlewood-Sobolev inequality for the Choquard terms; no new entities are introduced and no parameters are fitted to data.

axioms (2)

standard math Fractional Sobolev embedding H^s(R^N) into L^{2N/(N-2s)} holds continuously
Invoked to control the local term and to ensure the energy is well-defined on the constraint manifold.
standard math Hardy-Littlewood-Sobolev inequality for the Riesz potential I_α
Used to bound the nonlocal Choquard integrals.

pith-pipeline@v0.9.0 · 5728 in / 1401 out tokens · 25403 ms · 2026-05-19T02:39:44.134353+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

By employing variational methods under appropriate assumptions on the potentials V(x), f(x), and g(x), we establish several existence results for normalized solutions.
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the functional J is bounded from below on the set S(a)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

[1]

Bhattarai

S. Bhattarai. On fractional Schr¨ odinger systems of Choquard type. J. Differential Equations, 263(6):3197–3229, 2017

work page 2017
[2]

Chen and Z

W. Chen and Z. Wang. Normalized ground states for a fractional Choquard system in R. J. Geom. Anal., 34(7):Paper No. 219, 28, 2024

work page 2024
[3]

Chen and Y

Z. Chen and Y. Yang. Normalized solutions for the fractional Choquard equations with lower critical exponent and nonlocal perturbation. Taiwanese J. Math., 29(2):261–294, 2025

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d’Avenia, G

P. d’Avenia, G. Siciliano, and M. Squassina. On fractional Choquard equations. Math. Models Methods Appl. Sci. , 25(8):1447–1476, 2015

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B. Feng, R. Chen, and J. Liu. Blow-up criteria and instability of normalized standing waves for the fractional Schr¨ odinger-Choquard equation.Adv. Nonlinear Anal., 10(1):311–330, 2021

work page 2021
[7]

Feng and H

B. Feng and H. Zhang. Stability of standing waves for the fractional Schr¨ odinger-Hartree equation. J. Math. Anal. Appl. , 460(1):352–364, 2018

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Z. Feng, X. He, and Y. Meng. Normalized solutions of fractional Choquard equation with critical nonlinearity. Differential Integral Equations, 36(7-8):593–620, 2023

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J. Fr¨ ohlich, B. Lars G. Jonsson, and E. Lenzmann. Boson stars as solitary waves. Comm. Math. Phys., 274(1):1–30, 2007

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Guo and W

Z. Guo and W. Jin. Normalized solutions to fractional mass supercritical Choquard systems. J. Geom. Anal., 34(4):Paper No. 104, 26, 2024. 12

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X. He, V. R˘ adulescu, and W. Zou. Normalized ground states for the critical fractional Choquard equation with a local perturbation. J. Geom. Anal. , 32(10):Paper No. 252, 51, 2022

work page 2022
[12]

Z. Jin, H. Sun, J. Zhang, and W. Zhang. Normalized solution for fractional Choquard equation with potential and general nonlinearity. Complex Var. Elliptic Equ. , 69(7):1117–1133, 2024

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[13]

J. Lan, X. He, and Y. Meng. Normalized solutions for a critical fractional Choquard equation with a nonlocal perturbation. Adv. Nonlinear Anal., 12(1):Paper No. 20230112, 40, 2023

work page 2023
[14]

N. Laskin. Fractional quantum mechanics and L´ evy path integrals. Phys. Lett. A , 268(4- 6):298–305, 2000

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[15]

Li and X

G. Li and X. Luo. Existence and multiplicity of normalized solutions for a class of fractional Choquard equations. Sci. China Math. , 63(3):539–558, 2020

work page 2020
[16]

M. Li, J. He, H. Xu, and M. Yang. The existence and asymptotic behaviours of normalized solutions for critical fractional Schr¨ odinger equation with Choquard term. Discrete Contin. Dyn. Syst., 43(2):821–845, 2023

work page 2023
[17]

Q. Li, W. Wang, and M. Liu. Normalized solutions for the fractional Choquard equations with Sobolev critical and double mass supercritical growth. Lett. Math. Phys. , 113(2):Paper No. 49, 9, 2023

work page 2023
[18]

E. H. Lieb. Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Ann. of Math. (2) , 118(2):349–374, 1983

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E. H. Lieb and M. Loss. Analysis, volume 14 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, second edition, 2001

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[20]

J. Liu, H. Sun, and Z. Zhang. Existence, multiplicity and asymptotic behaviour of normalized solutions to non-autonomous fractional HLS lower critical Choquard equation. Fract. Calc. Appl. Anal., 27(6):3318–3351, 2024

work page 2024
[21]

Meng and X

Y. Meng and X. He. Normalized solutions for the fractional Choquard equations with Hardy- Littlewood-Sobolev upper critical exponent. Qual. Theory Dyn. Syst., 23(1):Paper No. 19, 21, 2024

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Moroz and J

V. Moroz and J. Van Schaftingen. Groundstates of nonlinear Choquard equations: Hardy- Littlewood-Sobolev critical exponent. Commun. Contemp. Math. , 17(5):1550005, 12, 2015

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[23]

Zhang, V

X. Zhang, V. N. Thin, and S. Liang. Multiple normalized solutions to critical Choquard equation involving fractional p-Laplacian in RN. Anal. Math. Phys. , 15(1):Paper No. 14, 42, 2025. 13

work page 2025

[1] [1]

Bhattarai

S. Bhattarai. On fractional Schr¨ odinger systems of Choquard type. J. Differential Equations, 263(6):3197–3229, 2017

work page 2017

[2] [2]

Chen and Z

W. Chen and Z. Wang. Normalized ground states for a fractional Choquard system in R. J. Geom. Anal., 34(7):Paper No. 219, 28, 2024

work page 2024

[3] [3]

Chen and Y

Z. Chen and Y. Yang. Normalized solutions for the fractional Choquard equations with lower critical exponent and nonlocal perturbation. Taiwanese J. Math., 29(2):261–294, 2025

work page 2025

[4] [4]

d’Avenia, G

P. d’Avenia, G. Siciliano, and M. Squassina. On fractional Choquard equations. Math. Models Methods Appl. Sci. , 25(8):1447–1476, 2015

work page 2015

[5] [5]

Di Nezza, G

E. Di Nezza, G. Palatucci, and E. Valdinoci. Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. , 136(5):521–573, 2012

work page 2012

[6] [6]

B. Feng, R. Chen, and J. Liu. Blow-up criteria and instability of normalized standing waves for the fractional Schr¨ odinger-Choquard equation.Adv. Nonlinear Anal., 10(1):311–330, 2021

work page 2021

[7] [7]

Feng and H

B. Feng and H. Zhang. Stability of standing waves for the fractional Schr¨ odinger-Hartree equation. J. Math. Anal. Appl. , 460(1):352–364, 2018

work page 2018

[8] [8]

Z. Feng, X. He, and Y. Meng. Normalized solutions of fractional Choquard equation with critical nonlinearity. Differential Integral Equations, 36(7-8):593–620, 2023

work page 2023

[9] [9]

Fr¨ ohlich, B

J. Fr¨ ohlich, B. Lars G. Jonsson, and E. Lenzmann. Boson stars as solitary waves. Comm. Math. Phys., 274(1):1–30, 2007

work page 2007

[10] [10]

Guo and W

Z. Guo and W. Jin. Normalized solutions to fractional mass supercritical Choquard systems. J. Geom. Anal., 34(4):Paper No. 104, 26, 2024. 12

work page 2024

[11] [11]

X. He, V. R˘ adulescu, and W. Zou. Normalized ground states for the critical fractional Choquard equation with a local perturbation. J. Geom. Anal. , 32(10):Paper No. 252, 51, 2022

work page 2022

[12] [12]

Z. Jin, H. Sun, J. Zhang, and W. Zhang. Normalized solution for fractional Choquard equation with potential and general nonlinearity. Complex Var. Elliptic Equ. , 69(7):1117–1133, 2024

work page 2024

[13] [13]

J. Lan, X. He, and Y. Meng. Normalized solutions for a critical fractional Choquard equation with a nonlocal perturbation. Adv. Nonlinear Anal., 12(1):Paper No. 20230112, 40, 2023

work page 2023

[14] [14]

N. Laskin. Fractional quantum mechanics and L´ evy path integrals. Phys. Lett. A , 268(4- 6):298–305, 2000

work page 2000

[15] [15]

Li and X

G. Li and X. Luo. Existence and multiplicity of normalized solutions for a class of fractional Choquard equations. Sci. China Math. , 63(3):539–558, 2020

work page 2020

[16] [16]

M. Li, J. He, H. Xu, and M. Yang. The existence and asymptotic behaviours of normalized solutions for critical fractional Schr¨ odinger equation with Choquard term. Discrete Contin. Dyn. Syst., 43(2):821–845, 2023

work page 2023

[17] [17]

Q. Li, W. Wang, and M. Liu. Normalized solutions for the fractional Choquard equations with Sobolev critical and double mass supercritical growth. Lett. Math. Phys. , 113(2):Paper No. 49, 9, 2023

work page 2023

[18] [18]

E. H. Lieb. Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Ann. of Math. (2) , 118(2):349–374, 1983

work page 1983

[19] [19]

E. H. Lieb and M. Loss. Analysis, volume 14 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, second edition, 2001

work page 2001

[20] [20]

J. Liu, H. Sun, and Z. Zhang. Existence, multiplicity and asymptotic behaviour of normalized solutions to non-autonomous fractional HLS lower critical Choquard equation. Fract. Calc. Appl. Anal., 27(6):3318–3351, 2024

work page 2024

[21] [21]

Meng and X

Y. Meng and X. He. Normalized solutions for the fractional Choquard equations with Hardy- Littlewood-Sobolev upper critical exponent. Qual. Theory Dyn. Syst., 23(1):Paper No. 19, 21, 2024

work page 2024

[22] [22]

Moroz and J

V. Moroz and J. Van Schaftingen. Groundstates of nonlinear Choquard equations: Hardy- Littlewood-Sobolev critical exponent. Commun. Contemp. Math. , 17(5):1550005, 12, 2015

work page 2015

[23] [23]

Zhang, V

X. Zhang, V. N. Thin, and S. Liang. Multiple normalized solutions to critical Choquard equation involving fractional p-Laplacian in RN. Anal. Math. Phys. , 15(1):Paper No. 14, 42, 2025. 13

work page 2025