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arxiv: 2503.15142 · v3 · submitted 2025-03-19 · 🧮 math.AP

Non-uniqueness of normalized NLS ground states on polygons with homogeneous Neumann boundary conditions

Simone Dovetta , Enrico Serra , Lorenzo Tentarelli This is my paper

Pith reviewed 2026-05-22 23:41 UTC · model grok-4.3

classification 🧮 math.AP
keywords nonlinear Schrödinger equationnormalized ground statesnon-uniquenesspolygonsNeumann boundary conditionsL2-critical exponentenergy minimizationvariational methods
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The pith

For nonlinearity powers slightly below the L²-critical exponent, normalized ground states of the NLS equation on polygons with homogeneous Neumann boundary conditions are not unique for at least one mass value.

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

The paper proves a non-uniqueness result for normalized ground states of nonlinear Schrödinger equations with pure power nonlinearity. These states are global minimizers of the energy functional among functions with a prescribed L² mass. On polygonal domains with homogeneous Neumann boundary conditions, when the power is slightly smaller than the L²-critical exponent, there is always at least one mass for which multiple distinct minimizers exist. This matters because many analyses of stability and long-time behavior assume uniqueness of ground states, and its failure changes what can be concluded about possible solution profiles.

Core claim

On polygons with homogeneous Neumann boundary conditions, for nonlinearity powers slightly smaller than the L²-critical exponent, there always exists at least one value of the mass for which normalized ground states are not unique.

What carries the argument

Constrained minimization of the energy functional at fixed L² mass on polygonal domains with Neumann boundary conditions, where the geometry and boundary conditions permit multiple distinct minimizers for certain parameters.

If this is right

  • Multiple distinct functions achieve the same minimal energy at that mass.
  • The non-uniqueness occurs in a range of masses that depends on how close the power is to the critical value.
  • The result is specific to polygonal domains and does not claim to hold on smooth domains or with other boundary conditions.

Where Pith is reading between the lines

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

  • Similar multiplicity might appear on domains with corners even if they are not strictly polygons.
  • When multiple ground states exist, orbital stability questions become more involved because different profiles may have different stability properties.
  • The non-uniqueness could be used to construct solutions that switch between different concentration patterns under small perturbations.

Load-bearing premise

The result holds specifically for polygons with homogeneous Neumann boundary conditions and for powers slightly below the L2-critical exponent; if the domain or power range changes, uniqueness may hold.

What would settle it

An explicit example of a polygon and a power slightly below critical where numerical or analytical computation shows a unique minimizer (up to the natural symmetries) for every mass value.

read the original abstract

We provide a non-uniqueness result for normalized ground states of nonlinear Schr\"odinger equations with pure power nonlinearity on polygons with homogeneous Neumann boundary conditions, defined as global minimizers of the associated energy functional among functions with prescribed mass. Precisely, for nonlinearity powers slightly smaller than the $L^2$-critical exponent, we prove that there always exists at least one value of the mass for which normalized ground states are not unique.

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

0 major / 2 minor

Summary. The manuscript proves a non-uniqueness result for normalized ground states of the nonlinear Schrödinger equation with pure power nonlinearity on polygonal domains with homogeneous Neumann boundary conditions. For nonlinearity exponents p slightly smaller than the L²-critical exponent, it shows that there exists at least one mass value such that the global minimizers of the associated energy functional subject to the fixed-mass constraint are not unique.

Significance. If the result holds, it provides a targeted contribution to the literature on multiplicity versus uniqueness of ground states for mass-constrained variational problems in NLS equations. The result is scoped to a regime just below criticality and to polygonal domains, where the Neumann condition and corner geometry may permit symmetry-breaking or multiple minimizers; this complements existing uniqueness theorems in smoother or different settings and relies on a direct variational construction without reduction to fitted parameters.

minor comments (2)
  1. The introduction would benefit from an explicit statement of the precise range of p (e.g., an interval (p*, 2) or similar) immediately after the abstract claim.
  2. Notation for the energy functional E(u) and the mass constraint should be introduced with a displayed equation in §1 for immediate reference.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript. No major comments were raised.

Circularity Check

0 steps flagged

No significant circularity; direct variational proof of non-uniqueness

full rationale

The paper states a direct existence result: for p slightly below the L²-critical exponent on polygonal domains with Neumann BC, there exists at least one mass value at which normalized ground states (global energy minimizers) are non-unique. The abstract and reader's summary describe a variational construction establishing this existence without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. No equations or steps are quoted that equate a claimed prediction back to its own inputs by construction. The result is scoped to a specific regime and domain class and is presented as a theorem proved from standard variational methods, making the derivation self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no details on free parameters, axioms, or invented entities used in the proof.

pith-pipeline@v0.9.0 · 5590 in / 910 out tokens · 46302 ms · 2026-05-22T23:41:41.334297+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages

  1. [1]

    Adami R., Serra E., Tilli P., Negative energy ground states for theL 2-critical NLSE on metric graphs, Comm. Math. Phys.352(2017), 387-406

    work page 2017

  2. [2]

    Adami R., Serra E., Tilli P., NLS ground states on graphs,Calc. Var. PDE54(2015), 743-761

    work page 2015

  3. [3]

    Adami R., Serra E., Tilli P., Threshold phenomena and existence results for NLS ground states on metric graphs,J. Funct. Anal.271(2016), no. 1, 201-223

    work page 2016

  4. [4]

    2, 025005

    Agostinho F., Correia S., Tavares H., Classification and stability of positive solutions to the NLS equation on theT-metric graph,Nonlinearity37(2024), no. 2, 025005

    work page 2024

  5. [5]

    Bandle C., Isoperimetric inequalities and applications, Pitman, London, 1980

    work page 1980

  6. [6]

    Math.100 (2013), no

    Bartsch T., de Valeriola S., Normalized solutions of nonlinear Schr¨ odinger equations,Arch. Math.100 (2013), no. 1, 75-83

    work page 2013

  7. [7]

    Royal Soc

    Bartsch T., Jeanjean L., Normalized solutions for nonlinear Schr¨ odinger systems,Proc. Royal Soc. Edinb. Section A: Math.148(2018), no. 2, 225-242. NON-UNIQUENESS OF NORMALIZED NLS GROUND STATES 21

    work page 2018

  8. [8]

    Bartsch T., Jeanjean L., Soave N., Normalized solutions for a system of coupled cubic Schr¨ odinger equations onR 3,J. Math. Pures Appl.106(2016), no. 4, 583-614

    work page 2016

  9. [9]

    Bartsch T., Soave N., A natural constraint approach to normalized solutions of nonlinear Schr¨ odinger equations and systems,J. Funct. Anal.272(2017), no. 12, 4998-5037

    work page 2017

  10. [10]

    Bartsch T., Soave N., Multiple normalized solutions for a competing system of Schr¨ odinger equations, Calc. Var. PDE58(2019), no. 1, art. 22, 24

    work page 2019

  11. [11]

    Rational Mech

    Berestycki H., Lions P.-L., Existence of solutions for nonlinear scalar field equations, I existence of a ground state,Arch. Rational Mech. Anal.82(1983), no. 4, 313-345

    work page 1983

  12. [12]

    Borthwick J., Chang X., Jeanjean L., Soave N., Normalized solutions ofL2-supercritical NLS equations on noncompact metric graphs with localized nonlinearities,Nonlinearity36(2023), 3776-3795

    work page 2023

  13. [13]

    Brezis H., Lieb E.H., A relation between pointwise convergence of functions and convergence of func- tionals,Proc. Amer. Math. Soc.88(3) (1983), 486-490

    work page 1983

  14. [14]

    Reine Angew

    Brothers J.E., Ziemer W.P., Minimal rearrangements of Sobolev functions,J. Reine Angew. Math. 384(1988), 153-179

    work page 1988

  15. [15]

    Cazenave T., Lions P.L., Orbital stability of standing waves for some nonlinear Schr¨ odinger equations, Comm. Math. Phys.85(1982), 549-561

    work page 1982

  16. [16]

    Cazenave T., Semilinear Schr¨ odinger Equations, Courant Lecture Notes, vol.10, American Mathemat- ical Society, Providence, RI, 2003

    work page 2003

  17. [17]

    Chang X., Jeanjean L., Soave N., Normalized solutions ofL 2-supercritical NLS equations on compact metric graphs,Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire41(2023), no. 4, 933-959

    work page 2023

  18. [18]

    Pure Appl

    Coti Zetati V., Rabinowitz P., Homoclinic Type Solutions for a Semilinear Elliptic PDE onRn,Comm. Pure Appl. Math.XL V(1992), 1217-1269

    work page 1992

  19. [19]

    De Coster C., Dovetta S., Galant D., Serra E., An action approach to nodal and least energy nor- malized solutions for nonlinear Schr¨ odinger equations,Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 10.4171/AIHPC/160

    work page doi:10.4171/aihpc/160

  20. [20]

    London Math

    Dovetta S., Non-uniqueness of normalized ground states for nonlinear Schr¨ odinger equations on metric graphs,Proc. London Math. Soc.130(2025), no. 2, e70025

    work page 2025

  21. [21]

    Math.374(2020), 107352, 41pp

    Dovetta S., Serra E., Tilli P., Uniqueness and non-uniqueness of prescribed mass NLS ground states on metric graphs,Adv. Math.374(2020), 107352, 41pp

    work page 2020

  22. [22]

    Ann.385(2023), no

    Dovetta S., Serra E., Tilli P., Action versus energy ground states in nonlinear Schr¨ odinger equations, Math. Ann.385(2023), no. 3-4, 1545-1576

    work page 2023

  23. [23]

    Gou T., Jeanjean L., Multiple positive normalized solutions for nonlinear Schr¨ odinger systems,Non- linearity31(2018), no. 5, 2319

    work page 2018

  24. [24]

    Anal.28(1997), no

    Jeanjean L., Existence of solutions with prescribed norm for semilinear elliptic equations,Nonlin. Anal.28(1997), no. 10, 1633-1659

    work page 1997

  25. [25]

    Jeanjean L., Lu S.-S., On global minimizers for a mass constrained problem,Calc. Var. PDE61 (2022), no. 6, art. n. 214, 18 pp

    work page 2022

  26. [26]

    Kosaka A., Condensation phenomena of a least-energy solution to semilinear Neumann problems on piecewise smooth domains,J. Diff. Eq.265(2018), 1-32

    work page 2018

  27. [27]

    Rational Mech

    Kwong M.K., Uniqueness of positive solutions of ∆u−u+u p = 0 inR n,Arch. Rational Mech. Anal. 105(1989), no. 3, 243-266

    work page 1989

  28. [28]

    The locally compact case, part 2,Ann

    Lions P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case, part 2,Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire1(1984), 223-283

    work page 1984

  29. [29]

    Li X., Liu L., Zhang G., Ground states for the NLS equation with combined local nonlinearities on noncompact metric graphs,J. Math. Anal. Appl.530(2024), no. 1, 127672

    work page 2024

  30. [30]

    J.70(1993), 247-281

    Ni W.-M., Takagi I., Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J.70(1993), 247-281

    work page 1993

  31. [31]

    Pure Appl

    Ni W.-M., Takagi I., On the shape of least-energy solutions to a semilinear Neumann problem,Comm. Pure Appl. Math.44(1991), no. 7, 819-851

    work page 1991

  32. [32]

    Noja D., Pelinovsky D.E., Standing waves of the quintic NLS equation on the tadpole graph,Calc. Var. PDE59(2020), no. 5, art. n. 173

    work page 2020

  33. [33]

    PDE7(2014), no

    Noris B., Tavares H., Verzini G., Existence and orbital stability of the ground states with prescribed mass for theL 2-critical and supercritical NLS on bounded domains,Anal. PDE7(2014), no. 8, 1807-1838

    work page 2014

  34. [34]

    Noris B., Tavares H., Verzini G., Stable solitary waves with prescribedL 2-mass for the cubic Schr¨ odinger system with trapping potentials,Discrete Contin. Dyn. Syst.35(2015), no. 12, 6085-6112

    work page 2015

  35. [35]

    3, 1044-1072

    Noris B., Tavares H., Verzini G., Normalized solutions for nonlinear Schr¨ odinger systems on bounded domains,Nonlinearity32(2019), no. 3, 1044-1072

    work page 2019

  36. [36]

    Mat.65(2021), no

    Pacella F., Tralli G., Isoperimetric cones and minimal solutions of partial overdetermined problems, Publ. Mat.65(2021), no. 1, 61-81. 22 S. DOVETTA, E. SERRA, AND L. TENTARELLI

    work page 2021

  37. [37]

    Pellacci B., Pistoia A., Vaira G., Verzini G., Normalized concentrating solutions to nonlinear elliptic problems,J. Diff. Eq.275(2021), 882-919

    work page 2021

  38. [38]

    Pierotti D., Soave N., Ground states for the NLS equation with combined nonlinearities on non- compact metric graphs,SIAM J. Math. Anal.54(2022), no. 1, 768-790

    work page 2022

  39. [39]

    Royal Soc

    Pierotti D., Soave N., Verzini G., Local minimizers in absence of ground states for the critical NLS energy on metric graphs,Proc. Royal Soc. Edinb. Section A: Math.151(2021), no. 2, 705-733

    work page 2021

  40. [40]

    Pierotti D., Verzini G., Normalized bound states for the nonlinear Schr¨ odinger equation in bounded domains,Calc. Var. PDE56(2017), no. 5, art. n. 133, 27

    work page 2017

  41. [41]

    Pierotti D., Verzini G., Yu J., Normalized solutions for Sobolev critical Schr¨ odinger equations on bounded domains,SIAM J. Math. Anal.57(2025), no. 1, 262-285

    work page 2025

  42. [42]

    Serra E., Tentarelli L., Bound states of the NLS equation on metric graphs with localized nonlinearities, J. Diff. Eq.260(2016), no. 7, 5627-5644

    work page 2016

  43. [43]

    Soave N., Normalized ground states for the NLS equation with combined nonlinearities,J. Diff. Eq. 269(2020), no. 9, 6941-6987

    work page 2020

  44. [44]

    Soave N., Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case,J. Funct. Anal.279(2020), no. 6, 108610

    work page 2020

  45. [45]

    Zeitschrift309(2025), art

    Song L., Zou W., On multiple sign-changing normalized solutions to the Brezis-Nirenberg problem, Math. Zeitschrift309(2025), art. 61

    work page 2025

  46. [46]

    Song L., Zou W., Two Positive Normalized Solutions on Star-shaped Bounded Domains to the Brezis- Nirenberg Problem, I: Existence, (2024) arXiv:2404.11204

    work page arXiv 2024

  47. [47]

    G.L. Lagrange

    Tentarelli L., NLS ground states on metric graphs with localized nonlinearities,J. Math. Anal. Appl. 433(2016), no. 1, 291-304. S. Dovetta: Politecnico di Torino, Dipartimento di Scienze Matematiche “G.L. Lagrange”, Corso Duca degli Abruzzi, 24, 10129, Torino, Italy Email address:simone.dovetta@polito.it E. Serra: Politecnico di Torino, Dipartimento di Sc...

    work page 2016