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Qualitative properties of positive solutions of a mixed order nonlinear Schr\"{o}dinger equation

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Pith's one-line read The paper proves that positive solutions of the mixed local/nonlocal Schrödinger equation are radially symmetric and decay at infinity like $|x|^{-(n+2s)}$.

desk verdict Solid mixed-order Schrödinger paper with a real self-containment gap: C^{2,α} regularity and the main qualitative theorem rest on an unpublished companion [31]. read the letter →

arxiv 2411.09941 v1 pith:FMJ35XIK submitted 2024-11-15 math.AP

Serena Dipierro , Xifeng Su , Enrico Valdinoci , Jiwen Zhang This is my paper · ORCID
classification math.AP MSC 35A0835B0635B0935B4035J10
keywords MixedorderoperatorsfractionalLaplaciannonlinearSchrödingerequationradialsymmetrypower-typedecayheatkernelBesselregularitytheory
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 studies positive solutions that vanish at infinity of the mixed-order Schrödinger equation $-\Delta u+(-\Delta)^s u+u=u^p$ on $\mathbb{R}^n$, where the diffusion is the sum of the classical Laplacian and the fractional Laplacian of order $2s$. The authors establish existence of a nontrivial nonnegative weak solution, then prove a regularity ladder from Hölder continuity up to $C^{2,\alpha}$, and finally show that every classical positive solution is radially symmetric and decays at infinity like a constant times $|x|^{-(n+2s)}$. The interest is that the operator has two different scaling invariances, so standard single-operator methods do not apply; the paper supplies kernel estimates that let the fractional part govern the far field. If the results are correct, the mixed equation inherits the symmetry and decay structure of the Bessel kernel for $-\Delta+(-\Delta)^s+1$.

What carries the argument

The load-bearing objects are the heat kernel $H(x,t)=\int_{\mathbb{R}^n} e^{-t(|\xi|^2+|\xi|^{2s})+2\pi i x\cdot\xi}\,d\xi$ and the Bessel kernel $K(x)=\int_0^\infty e^{-t}H(x,t)\,dt$ of $-\Delta+(-\Delta)^s+1$. The paper proves uniform asymptotic formulae $|x|^{n+2s}H(x,1,\eta)\to$ a positive constant, uniformly in $\eta\in(0,1)$, using Bessel-function representations and contour rotation; these yield two-sided bounds on $H$ and hence $K(x)\asymp |x|^{-(n+2s)}$ for large $|x|$, with $K\in L^1(\mathbb{R}^n)$. Convolving the characteristic function of a ball with $K$ (and with a rescaled kernel $K_{1/2}$) produces barriers $\omega$ and $v$ that force the solution between two multiples of $|x|^{-(n+2s)}$, while Fourier-multiplier $W^{2,p}$ theory, a localization trick, and a truncation and covering argument produce the Hölder, $C^{1,\alpha}$, and $C^{2,\alpha}$ regularity used by the moving-plane argument.

What would settle it

For $n=2$, $s=1/2$, take $t=2$ and $|x|=3$ and evaluate the integral $H(x,t)$ numerically; if $H(x,t) < C_2 t/|x|^{n+2s}$ with the constant from Theorem 3.1, then the subsolution barrier in Lemma 4.3 fails and the decay exponent is not established.

Watch

Extended reading notes

Core claim

The central assertion is Theorem 1.5: problem (1.1) admits a classical, positive, radially symmetric solution, and every classical positive solution is radially symmetric, with $C_1/|x|^{n+2s}\le u(x)\le C_2/|x|^{n+2s}$ for $|x|\ge 1$. The exponent $n+2s$ is exactly the far-field decay of the Bessel kernel $K$ of the operator, and the proof shows that this kernel supplies both the subsolution and the supersolution needed for the comparison argument. Radial symmetry is obtained by the method of moving planes, using the $C^2$ regularity and a quantitative estimate on the set where the reflected solution dominates.

Load-bearing premise

The load-bearing premise is a sharp lower bound on the mixed heat kernel in the regime $1<t<|x|^{2s}$; if it fails, the exponent $n+2s$ and the barrier construction built from the Bessel kernel collapse.

Editorial extensions

If this is right

  • Positive ground states of the mixed operator have a universal far-field shape, decaying at the same rate as the Bessel kernel and not, for example, like the purely Laplacian kernel $|x|^{2-n}$.
  • All classical positive solutions are radially symmetric, so the search for ground states reduces to a one-dimensional problem.
  • The heat-kernel and Bessel-kernel estimates give a ready-made $L^p$ and $W^{2,p}$ theory for the linear mixed operator.
  • The $C^{2,\alpha}$ regularity puts the mixed equation within reach of classical elliptic methods such as maximum principles and moving planes.
  • The mountain-pass solution found in Theorem 1.1 is in fact a classical positive solution with all the qualitative features above.

Reading between the lines

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

  • The same barrier construction should work for slightly more general nonlinearities or for mixed operators with different weights on the fractional terms, predicting the same $n+2s$ far-field exponent whenever the fractional part has order $s$.
  • A direct numerical check of the heat-kernel lower bound in the regime $1<t<|x|^{2s}$ would settle the most delicate step without needing the full Bessel asymptotic machinery.
  • If the decay rate is sharp, it suggests the ground state is nondegenerate, which would open the door to uniqueness and stability arguments for the mixed equation.
  • The radial symmetry result may extend to other sign-changing or nonlocal settings, but the current proof relies essentially on positivity and on the maximum principle.
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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 / 3 minor

Summary. The paper studies positive solutions of the mixed local/nonlocal Schrödinger equation -Δu+(-Δ)^s u+u=u^p in R^n. It proves existence of nontrivial nonnegative weak solutions by a mountain-pass/Ekeland argument, establishes C^0,μ, C^1,α, and C^2,α regularity for bounded weak solutions, and then derives the qualitative properties announced in Theorem 1.5: existence of a classical positive radially symmetric solution with two-sided power decay C_1/|x|^{n+2s} ≤ u(x) ≤ C_2/|x|^{n+2s} for |x|≥1, and radial symmetry of all classical positive solutions by the moving-plane method. The appendices provide Fourier-analysis proofs of heat-kernel and Bessel-kernel bounds, including the sharp far-field lower bounds used in the barriers.

Significance. If the results are fully established, the paper gives a complete qualitative picture for this mixed-order problem: positive ground states decay like the Bessel kernel and are radially symmetric in every direction. The self-contained treatment of the heat and Bessel kernels associated with -Δ+(-Δ)^s is a useful contribution in its own right. However, the C^2,α regularity theorem is not proved in this manuscript; it is imported from an unpublished companion preprint, and the sketch provided omits a key absorption estimate. Since the later qualitative results use pointwise second-order information, this gap is load-bearing and must be resolved before the main claims can be considered verified.

major comments (3)
  1. [§3.3, Theorem 1.4] Theorem 1.4 is not self-contained. The text states that it 'can be obtained by appropriately modifying [31, Theorem 1.6]', and then imports [31, Lemma 5.4] as Lemma 3.6 and [31, Proposition 4.3] as Proposition 3.7 without proof. Reference [31] is listed as an unpublished 2023 preprint. This is not a stylistic issue: Theorem 1.4 is used to conclude that the weak solution is classical in Theorem 4.1, and the pointwise computations in Lemma 4.7 and the maximum-principle comparisons in Theorem 4.2 require u∈C^2. Without a complete proof of Theorem 1.4, or a reference to a published version of [31] containing the needed results, the central claims of Theorem 1.5 are unsupported.
  2. [§3.3, estimate (3.25)] Even accepting the imported lemmas, the key absorption step leading to (3.25) is not demonstrated. The fractional term R^2|(-Δ)^s v_ε|'_{0,α;B_R} is first bounded by C|u_ε|'_{2,α0;B_{2R}} with α0<α, and then replaced by δ|u_ε|'_{2,α;B_{2R}}+C_δ‖u_ε‖_{L∞}. The interpolation inequality behind this replacement is not stated, and it is not shown that the constants can be chosen uniformly in ε and R. Since (3.25) is the sole input to Proposition 3.7, an omission here breaks the uniform C^2,α bound needed for the Arzelà-Ascoli step.
  3. [§4.1.2, Lemmas 4.4 and 4.5] Lemma 4.4 is stated for the kernel K_a with parameter a>0, but no proof is given; the surrounding text only says it is obtained by using Theorem 3.2 'with a parameter a>0 in place of 1'. Because the operator is not scale invariant, this reduction is not immediate. Lemma 4.5 relies on K_{1/2} and is used to construct the supersolution in the proof of the upper bound in Theorem 4.2. The estimates in Lemma 4.4 should either be proved or reduced explicitly to Theorem 3.2 by a displayed change of variables.
minor comments (3)
  1. [Definition 2.2] The double integral in the weak formulation contains a typographical parenthesis error: '(u(x)-u(y)(v(x)-v(y))' should read '(u(x)-u(y))(v(x)-v(y))'.
  2. [§3.3, notation] In the estimate preceding (3.25), the definition of α0 is introduced but its role in the interpolation is not explained; a one-sentence clarification that the α0-norm is absorbed by interpolation would improve readability.
  3. [Appendix A.2, Lemma A.3] The one-dimensional integral representation (A.4) is introduced as 'immediate'; since the subsequent asymptotic analysis depends on it, a short derivation or a precise citation would be helpful.

Circularity Check

1 steps flagged · score 4.0 of 10

C^{2,α} regularity—and hence the classical-solution premise of Theorem 1.5—is imported from the authors' own unpublished [31].

  1. self citation load bearing [Section 3.3 (C^{2,α}-regularity; Theorem 1.4, Lemmas 3.6 and 3.7)]
    "We point out that Theorem 1.4 can be obtained by appropriately modifying [31, Theorem 1.6]. For the convenience of the reader, we sketch the proof in the following subsections. ... Lemma 3.6. ([31, Lemma 5.4]) ... Proposition 3.7. ([31, Proposition 4.3])"

    Theorem 1.4 is stated as a new result of this paper, but its proof is not given here: the paper says it follows by modifying [31, Theorem 1.6], and the two key ingredients, Lemma 3.6 and Proposition 3.7, are quoted verbatim from [31]. Reference [31] is an unpublished preprint by three of the four present authors (X. Su, E. Valdinoci, J. Zhang). The later arguments depend essentially on Theorem 1.4: Theorem 4.1 uses it to obtain a classical solution, Lemma 4.7 uses u ∈ C^2 for the pointwise moving-plane identity, and Theorem 4.2 applies the maximum principle to functions in C^2. Thus the classical-regularity premise of Theorem 1.5 is not independently proved in this manuscript; it rests on the authors' own unpublished work.

full rationale

The decay and symmetry results are not circular: Theorem 3.1 (heat kernel bounds) and Theorem 3.2 (Bessel kernel bounds) are proved in Appendices A and B by Fourier and Bessel-function analysis, with no appeal to the target decay or symmetry. Theorem 4.2 compares solutions to explicit barriers K * χ and K_{1/2} * χ, whose decay is obtained from those kernel estimates, and the moving-plane argument in Section 4.2 uses the radial symmetry and monotonicity of K plus the kernel representation u = K * u^p. None of these steps assumes the conclusion. The only load-bearing self-citation is Theorem 1.4: it is presented as a new theorem, but its proof is only a sketch that imports [31, Lemma 5.4] and [31, Proposition 4.3] and refers to [31, Theorem 1.6], an unpublished preprint by three of the four authors. Since Theorem 4.1, Lemma 4.7, and Theorem 4.2 all require u ∈ C^2, the classical regularity needed for the main qualitative theorem is not established independently here. This is a significant self-citation burden, but it is not a reduction-by-construction of the decay or symmetry claims, so the appropriate score is 4.

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

No free parameters or invented entities appear. The paper proves the needed kernel estimates from Fourier analysis, and the only external dependence is the unpublished preprint [31] used for part of the C^{2,α} regularity argument.

assumptions (5)
  • domain assumption Subcritical exponent range p in (1, (n+2)/(n-2)) for n>2 and p in (1, +infinity) for n=2
    Assumed in (1.2); the bootstrap iteration in Theorem 1.2 and the Sobolev embedding require p to be below the critical exponent.
  • standard math Calderon-Zygmund W^{2,p} estimates for the linear equation -Δu + u = g
    Invoked in Lemma 3.3 through [30, Theorem 3, page 135] to transfer regularity from the forcing term to the solution.
  • standard math Concentration-compactness lemma [7, Lemma 2.18]
    Used in the proof of Theorem 1.1 to rule out vanishing of the Palais-Smale sequence when local L^2 mass goes to zero.
  • standard math Maximum principle for the mixed operator with lower order term in exterior domains
    Used in Theorem 4.2 to compare u with the barriers ω and v, relying on C^2 regularity and pointwise signs of Δ and (-Δ)^s at an interior minimum.
  • standard math Moving plane machinery: strong comparison, Bessel kernel integral representation, and periodicity exclusion
    Used in Section 4.2; assumes classical positive solutions with sufficient decay and integrability of the Bessel kernel K.

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Pith. "Pith review of Qualitative properties of positive solutions of a mixed order nonlinear Schr\"{o}dinger equation." pith.science (2026). https://pith.science/paper/FMJ35XIK

@misc{pith2026241109941,
  author       = {Pith},
  title        = {Pith review of: Qualitative properties of positive solutions of a mixed order nonlinear Schr\"odinger equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FMJ35XIK}},
  note         = {Machine review of arXiv:2411.09941}
}
abstract

In this paper, we deal with the following mixed local/nonlocal Schr\"{o}dinger equation \begin{equation*} \left\{ \begin{array}{ll} - \Delta u + (-\Delta)^s u+u = u^p \quad \hbox{in $\mathbb{R}^n$,} u>0 \quad \hbox{in $\mathbb{R}^n$,} \lim\limits_{|x|\to+\infty}u(x)=0, \end{array} \right. \end{equation*} where $n\geqslant2$, $s\in (0,1)$ and $p\in\left(1,\frac{n+2}{n-2}\right)$. The existence of positive solutions for the above problem is proved, relying on some new regularity results. In addition, we study the power-type decay and the radial symmetry properties of such solutions. The methods make use also of some basic properties of the heat kernel and the Bessel kernel associated with the operator $- \Delta + (-\Delta)^s$: in this context, we provide self-contained proofs of these results based on Fourier analysis techniques.

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Reference graph

Works this paper leans on

34 extracted references · 25 canonical work pages · cited by 1 Pith paper

  1. [31]

    X. Su, E. V aldinoci, Y . Wei, and J. Zhang. On some regular ity properties of mixed local and nonlocal elliptic equations. 2023. Preprint

    work page 2023

  2. [1]

    C. J. Amick and J. F. Toland. Uniqueness and related analy tic properties for the Benjamin-Ono equation—a nonlinear Neumann problem in the plane. Acta Math. , 167(1-2):107–126, 1991. ISSN 0001-5962. URL https://doi. org/10.1007/BF02392447

    work page doi:10.1007/bf02392447 1991

  3. [2]

    N. H. Bingham, C. M. Goldie, and J. L. Teugels. Regular variation. Number 27. Cambridge university press, 1989

    work page 1989

  4. [3]

    R. M. Blumenthal and R. K. Getoor. Some theorems on stable processes. Trans. Amer . Math. Soc., 95:263–273,

    work page

  5. [4]

    Bochner and K

    S. Bochner and K. Chandrasekharan. F ourier Transforms. Annals of Mathematics Studies, No. 19. Princeton Uni- versity Press, Princeton, NJ; Oxford University Press, Lon don, 1949. ix+219 pp

    work page 1949

  6. [5]

    Bogdan, T

    K. Bogdan, T. Grzywny, and M. Ryznar. Density and tails of unimodal convolution semigroups. J. Funct. Anal., 266 (6):3543–3571, 2014. ISSN 0022-1236

    work page 2014

  7. [6]

    H. Brezis. Functional analysis, Sobolev spaces and partial different ial equations. Universitext. Springer, New Y ork,

    work page

  8. [7]

    Coti Zelati and P

    V . Coti Zelati and P . H. Rabinowitz. Homoclinic type solu tions for a semilinear elliptic PDE on Rn. Comm. Pure Appl. Math. , 45(10):1217–1269, 1992. ISSN 0010-3640. URL https://doi.org/10.1002/cpa. 3160451002

    doi:10.1002/cpa 1992

Show all 34 references
  1. [8]

    Cygan, T

    W . Cygan, T. Grzywny, and B. Trojan. Asymptotic behaviorof densities of unimodal convolution semigroups. Trans. Amer . Math. Soc. , 369(8):5623–5644, 2017. ISSN 0002-9947. URL https://doi.org/10.1090/tran/ 6830

    2017 doi

  2. [9]

    D´ avila, M

    J. D´ avila, M. del Pino, S. Dipierro, and E. V aldinoci. Concentration phenomena for the nonlocal Schr¨ odinger equa- tion with Dirichlet datum. Anal. PDE, 8(5):1165–1235, 2015. ISSN 2157-5045. URL https://doi.org/10. 2140/apde.2015.8.1165

    2015

  3. [10]

    D´ avila, M

    J. D´ avila, M. del Pino, and J. Wei. Concentrating standing waves for the fractional nonlinear Schr¨ odinger equation. J. Differential Equations, 256(2):858–892, 2014. ISSN 0022-0396. URL https://doi.org/10.1016/j.jde. 2013.10.006

    2014 doi

  4. [11]

    Dipierro and E

    S. Dipierro and E. V aldinoci. Elliptic Partial Differential Equations from an Elementar y Viewpoint. A Fresh Glance at the Classical Theory . World Scientific, 2024

    2024

  5. [12]

    Erd´ elyi, W

    A. Erd´ elyi, W . Magnus, F. Oberhettinger, and F. G. Tricomi. Higher transcendental functions. Vols. I, II . McGraw- Hill Book Co., Inc., New Y ork-Toronto-London, 1953. xxvi+302, xvii+396 pp. Based, in part, on notes left by Harry Bateman

    1953

  6. [13]

    M. M. Fall and E. V aldinoci. Uniqueness and nondegenera cy of positive solutions of (−∆)su +u = up in RN when s is close to 1. Comm. Math. Phys. , 329(1):383–404, 2014. ISSN 0010-3616. URL https://doi.org/10.1007/ s00220-014-1919-y

    2014

  7. [14]

    Felmer, A

    P . Felmer, A. Quaas, and J. Tan. Positive solutions of th e nonlinear Schr¨ odinger equation with the fractional Lapla- cian. Proc. Roy. Soc. Edinburgh Sect. A , 142(6):1237–1262, 2012. ISSN 0308-2105. URL https://doi.org/ 10.1017/S0308210511000746

    2012 doi

  8. [15]

    Feynman and A

    R. Feynman and A. Hibbs. Quantum mechanics and path inte grals. McGraw-Hill, New Y ork, 1965

    1965

  9. [16]

    Floer and A

    A. Floer and A. Weinstein. Nonspreading wave packets fo r the cubic Schr¨ odinger equation with a bounded potential. J. Funct. Anal. , 69(3):397–408, 1986. ISSN 0022-1236. URL https://doi.org/10.1016/ 0022-1236(86)90096-0

    1986

  10. [17]

    R. L. Frank and E. Lenzmann. Uniqueness of non-linear gr ound states for fractional Laplacians in R. Acta Math. , 210(2):261–318, 2013. ISSN 0001-5962. URL https://doi.org/10.1007/s11511-013-0095-9

    2013 doi

  11. [18]

    R. L. Frank, E. Lenzmann, and L. Silvestre. Uniqueness o f radial solutions for the fractional Laplacian. Comm. Pure Appl. Math., 69(9):1671–1726, 2016. ISSN 0010-3640. URL https://doi.org/10.1002/cpa.21591

    2016 doi

  12. [19]

    Gilbarg and N

    D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order . Classics in Mathematics. Springer-V erlag, Berlin, 2001. ISBN 3-540-41160-7. xiv+517 pp. Reprint of the 1998 edition. MIXED ORDER SCHR ¨ODINGER EQUA TION 53

    2001

  13. [20]

    M. K. Kwong. Uniqueness of positive solutions of ∆u − u + up = 0 in Rn. Arch. Rational Mech. Anal. , 105(3): 243–266, 1989. ISSN 0003-9527. URL https://doi.org/10.1007/BF00251502

    1989 doi

  14. [21]

    N. Laskin. Fractional quantum mechanics and L´ evy path integrals. Phys. Lett. A , 268(4-6):298–305, 2000. ISSN 0375-9601. URL https://doi.org/10.1016/S0375-9601(00)00201-2

    2000 doi

  15. [22]

    N. Laskin. Fractional Schr¨ odinger equation. Phys. Rev. E (3) , 66(5):056108, 7, 2002. ISSN 1539-3755. URL https://doi.org/10.1103/PhysRevE.66.056108

    2002 doi

  16. [23]

    N. Laskin. Principles of fractional quantum mechanics . In Fractional dynamics, pages 393–427. World Sci. Publ., Hackensack, NJ, 2012

    2012

  17. [24]

    Mawhin and M

    J. Mawhin and M. Willem. Critical point theory and Hamiltonian systems , volume 74 of Applied Mathematical Sciences. Springer-V erlag, New Y ork, 1989. ISBN 0-387-96908-X. xiv+277 pp. URL https://doi.org/10. 1007/978-1-4757-2061-7

    1989

  18. [25]

    K. McLeod. Uniqueness of positive radial solutions of ∆u + f (u) = 0 in Rn. II. Trans. Amer . Math. Soc., 339(2): 495–505, 1993. ISSN 0002-9947. URL https://doi.org/10.2307/2154282

    1993 doi

  19. [26]

    McLeod and J

    K. McLeod and J. Serrin. Uniqueness of positive radial s olutions of ∆u + f (u) = 0 in Rn. Arch. Rational Mech. Anal., 99(2):115–145, 1987. ISSN 0003-9527. URL https://doi.org/10.1007/BF00275874

    1987 doi

  20. [27]

    P . H. Rabinowitz. On a class of nonlinear Schr¨ odinger equations. Z. Angew. Math. Phys., 43(2):270–291, 1992. ISSN 0044-2275. URL https://doi.org/10.1007/BF00946631

    1992 doi

  21. [28]

    M. Rao, R. Song, and Z. V ondraˇ cek. Green function estim ates and Harnack inequality for subordinate Brow- nian motions. Potential Anal. , 25(1):1–27, 2006. ISSN 0926-2601. URL https://doi.org/10.1007/ s11118-005-9003-z

    2006

  22. [29]

    Song and Z

    R. Song and Z. V ondraˇ cek. Parabolic Harnack inequality for the mixture of Brownian motion and stable process. T o- hoku Math. J. (2) , 59(1):1–19, 2007. ISSN 0040-8735. URL http://projecteuclid.org/euclid.tmj/ 1176734744

    2007

  23. [30]

    E. M. Stein. Singular integrals and differentiability properties of fu nctions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, NJ, 1970. xiv+290 p p

    1970

  24. [32]

    X. Wang. On concentration of positive bound states of no nlinear Schr¨ odinger equations.Comm. Math. Phys. , 153 (2):229–244, 1993. ISSN 0010-3616. URL http://projecteuclid.org/euclid.cmp/1104252679. DEPARTMENT OF MATHEMATICS AND STATISTICS , U NIVERSITY OF WESTERN AUSTRALIA ,...

    1993

  25. [1960]

    URL https://doi.org/10.2307/1993291

    ISSN 0002-9947. URL https://doi.org/10.2307/1993291

    doi

  26. [2011]

    xiv+599 pp

    ISBN 978-0-387-70913-0. xiv+599 pp

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