Multiplicity of solutions with prescribed mass for a quasilinear critical Choquard equation driven by a local-nonlocal operator

J. Giacomoni; K. Sreenadh; Nidhi Nidhi

Multiplicity of solutions with prescribed mass for a quasilinear critical Choquard equation driven by a local-nonlocal operator

T0 review · 0 major / 3 minor · reviewed 2026-06-29 · grok-4.3

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Pith's one-line read A mixed local-nonlocal Choquard equation with critical growth has at least two distinct solutions of fixed mass when a subcritical perturbation is present.

desk verdict The paper shows at least two normalized solutions for this mixed local-nonlocal critical Choquard equation when the perturbation exponent q lies in (p, p + s p²/N). read the letter →

arxiv 2605.25787 v3 pith:G42466U4 submitted 2026-05-25 math.AP

J. Giacomoni , Nidhi Nidhi , K. Sreenadh This is my paper · ORCID

classification math.AP

keywords Choquardequationnormalizedsolutionsfractionalp-LaplaciancriticalexponentmultiplicityRieszpotentialquasilinearequations

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 authors study normalized solutions to a quasilinear equation that combines the p-Laplacian with its fractional version and includes a critical Choquard term driven by the Riesz potential. They prove that adding the lower-order term μ |u|^{q-2}u produces a second distinct solution provided the exponent q lies strictly between p and p + s p² / N. This matters for models that conserve mass, since the result shows how a controlled perturbation can create multiple steady states at the same total mass level. The argument adapts variational techniques to the mixed operator while preserving the necessary compactness at the critical level.

What carries the argument

The mass-subcritical perturbation μ |u|^{q-2}u, which restores mountain-pass geometry and allows extraction of a second critical point at the fixed-mass constraint.

What would settle it

A concrete counterexample in which, for some admissible p, N, s and q inside the stated interval, the second solution fails to appear.

Watch

Extended reading notes

Core claim

For the equation -Δ_p u + (-Δ_p)^s u = λ |u|^{p-2}u + μ |u|^{q-2}u + (I_α * |u|^{p*_α}) |u|^{p*_α-2}u in R^N with ||u||_p = τ, at least two distinct solutions exist when p < q < p + s p² / N, under suitable restrictions on p, N and s.

Load-bearing premise

The perturbation exponent q must lie strictly inside the open interval from p to p + s p² / N.

Editorial extensions

If this is right

The Lagrange multiplier λ is determined separately for each of the two solutions.
The existence holds on the whole space R^N without boundary restrictions.
The same interval condition on q guarantees that the perturbation stays below the critical threshold set by the fractional term.
Multiplicity occurs for any positive mass parameter τ and any positive μ.

Reading between the lines

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

The same perturbation strategy could be tested on other combinations of local and nonlocal operators beyond the p-Laplacian pair.
The result indicates that mass-subcritical terms can generically destroy uniqueness in critical Choquard problems.
Direct numerical minimization of the associated energy functional at fixed mass might locate the two solutions for concrete values of N, p and s.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The central point is that the authors obtain multiplicity of solutions with fixed L^p norm for the equation mixing the p-Laplacian and its fractional counterpart, plus a critical Choquard term and a lower-order perturbation. They do this under the stated restrictions on p, N, s and with q strictly between p and the upper bound p + s p²/N.

What stands out as new is the specific mix of local and nonlocal operators together with the critical Choquard nonlinearity under the mass constraint. Earlier work handled either local or nonlocal cases separately or without the critical term, so this combination fills a gap. The paper sets up the functional framework cleanly and identifies the parameter window where the mountain-pass geometry and compactness recovery both hold, which is the usual route for these problems.

The argument appears to hold together without obvious gaps in the abstract formulation. The interval on q is narrow by design to keep the perturbation subcritical enough for compactness, and the conditions on p, N, s are standard for the fractional p-Laplacian to behave well. No circularity or unjustified scaling shows up.

The main limitation is that the result stays inside a fairly narrow regime of exponents and operators, so it adds a data point rather than opening a new direction. Readers working on normalized solutions or mixed local-nonlocal problems will find it directly relevant; others in broader elliptic theory will see it as incremental.

I would send this to referees. The claim is concrete, the setting is well-posed, and the technical ingredients line up with what is needed for a solid existence paper in this area.

Referee Report

0 major / 3 minor

Summary. The manuscript proves the existence of at least two distinct normalized solutions (fixed L^p-norm) to the critical Choquard problem driven by the mixed operator −Δ_p u + (−Δ_p)^s u, with the critical nonlocal term (I_α * |u|^{p*_α})|u|^{p*_α−2}u and a mass-subcritical perturbation μ|u|^{q−2}u. The result is obtained when p < q < p + s p²/N together with auxiliary restrictions on p, N and s (N ≥ 3, 2 ≤ p < N, s ∈ (0,1), α in the admissible range).

Significance. If the proof is complete, the paper supplies a multiplicity result for normalized solutions of a mixed local-nonlocal critical Choquard equation. The restriction on q is the natural mass-subcritical window that restores compactness while preserving the mountain-pass geometry; the combination of two distinct p-Laplacian-type operators adds technical novelty to the existing literature on constrained Choquard problems.

minor comments (3)

The abstract states the result holds “under some conditions on p, N and s” but does not list them explicitly; the introduction or the statement of the main theorem should give the precise parameter regime (e.g., the lower bound on N relative to p and s).
Notation for the fractional p-Laplacian (−Δ_p)^s and the critical exponent p*_α should be recalled once in the introduction with a forward reference to the precise definitions in §2.
The equation is written with λ appearing as a Lagrange multiplier; it would be clearer to state explicitly that λ is determined a posteriori once the constrained critical points are found.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript. The recommendation for minor revision is noted. No specific major comments appear in the report, so we have no individual points to rebut or revise at this stage. We remain available to address any minor suggestions or clarifications that may arise during the revision process.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The paper is a standard variational existence proof establishing multiplicity of normalized solutions for the given mixed local-nonlocal critical Choquard problem. The claimed result follows from direct analysis of the energy functional, mountain-pass geometry, and compactness in the mass-subcritical interval p < q < p + s p² / N, with no reduction of any existence statement to a fitted parameter, self-referential definition, or load-bearing self-citation. All steps remain independent of the target multiplicity conclusion.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The claim rests on standard Sobolev embeddings, Hardy-Littlewood-Sobolev inequality, and variational geometry that are assumed to hold under the listed parameter restrictions; no new entities or fitted constants are introduced in the abstract.

assumptions (2)

standard math Sobolev and fractional Sobolev embeddings hold for the given range of p, s, N
Invoked implicitly to define the functional setting and critical exponent p^*_α
domain assumption The interval p < q < p + s p² / N guarantees the required compactness or geometry
Stated as the condition under which the multiplicity result holds

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Pith. "Pith review of Multiplicity of solutions with prescribed mass for a quasilinear critical Choquard equation driven by a local-nonlocal operator." pith.science (2026). https://pith.science/paper/G42466U4

@misc{pith2026260525787,
  author       = {Pith},
  title        = {Pith review of: Multiplicity of solutions with prescribed mass for a quasilinear critical Choquard equation driven by a local-nonlocal operator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G42466U4}},
  note         = {Machine review of arXiv:2605.25787}
}

abstract

In this paper we study the normalized solutions of the following critical growth Choquard equation with mixed local and non-local operators: \begin{equation*} \begin{array}{rcl} -\Delta_p u +(-\Delta_p)^s u & = & \lambda |u|^{p-2}u +\mu |u|^{q-2}u +(I_{\alpha}*|u|^{p^*_{\alpha}})|u|^{p^*_{\alpha}-2}u \text{ in } \mathbb{R}^N; \left\| u \right\|_p & = & \tau. \end{array} \end{equation*} Here, $N\geq 3$, $2 \le p<N$, $\tau>0$, $I_{\alpha}$ is the Riesz potential of order $\alpha\in (\max\{0,N-2p\}, N)$, $p^*_{\alpha}=\frac{p}{2}\left(\frac{N+\alpha}{N-p}\right)$ is the critical exponent corresponding to the Hardy Littlewood Sobolev inequality, $(-\Delta_p)^s$ is the non-local fractional p-Laplacian operator with $s\in (0,1)$, $\mu>0$ is a parameter and $\lambda$ appears as a Lagrange multiplier. We show the existence of at least two distinct solutions in the presence of a mass subcritical perturbation, $\mu |u|^{q-2}u$ with $p<q<p+\frac{sp^2}{N}$ under some conditions on $p,N$ and $s$.

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Pith tools

Reviewed June 29, 2026 · model on record in the stance chip above.

[1] [1]

Alves, Y.H

C.O. Alves, Y.H. Ding, Multiplicity of positive solutions to a p-Laplacian equation involving critical nonlinearity , J. Math. Anal. Appl. 279 (2003), no. 2, 508-521

2003

[2] [2]

Anthal and P

G.C. Anthal and P. Garain, Pohozaev-type identities for classes of quasilinear el- liptic local and nonlocal equations and systems, with applications , arXiv preprint arXiv:2506.08667 (2025)

work page arXiv 2025

[3] [3]

Baernstein II, D

A. Baernstein II, D. Drasin and R. S. Laugesen, Symmetrization in analysis , Cambridge University Press (2019)

2019

[4] [4]

Bartsch and L

T. Bartsch and L. Jeanjean, Normalized solutions for nonlinear Schrödinger systems , Proc. Roy. Soc. Edinburgh Sect. A 148 (2018), no. 2, 225-242

2018

[5] [5]

Bartsch, L

T. Bartsch, L. Jeanjean and N. Soave, Normalized solutions for a system of coupled cubic Schrödinger equations on R3 , J. Math. Pures Appl. (9) 106 (2016), no. 4, 583- 614

2016

[6] [6]

Bartsch and N

T. Bartsch and N. Soave, Multiple normalized solutions for a competing system of Schrödinger equations, Calc. Var. Partial Diﬀerential Equations 58 (2019), 1-24

2019

[7] [7]

Biswas and S

R. Biswas and S. Tiwari, Regularity results for Choquard equations involving fractional p-Laplacian, Math. Nachr. 296 (2023), no. 09, 4060-4085

2023

[8] [8]

Brasco and E

L. Brasco and E. Parini, The second eigenvalue of the fractional p-Laplacian , Adv. Calc. Var. 9 (2016), no. 04, 323-355. 40

2016