Normalized solutions for a fourth-order Schr\"{o}dinger equation with positive second-order dispersion coefficient

Tao Yang; Xiao Luo

REVIEW 2 major objections 4 minor 38 references

Normalized solutions for a fourth-order Schr\"{o}dinger equation with positive second-order dispersion coefficient

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For a fourth-order Schrödinger equation with positive second-order dispersion, this paper proves two normalized radial solutions exist in the L²-supercritical range under an explicit mass-dispersion condition.

desk verdict Correct and solid, with an overclaiming abstract and a fixable inequality slip in Lemma 3.2; worth a serious referee. read the letter →

arxiv 1908.03079 v2 pith:QXMCVQSE submitted 2019-08-07 math.AP

Xiao Luo , Tao Yang This is my paper · ORCID

classification math.AP MSC 35A0135B3335B4035J3535J91

keywords normalizedsolutionsfourth-orderSchrödingerequationpositivesecond-orderdispersionPohozaevmanifoldL2-supercriticalradialgroundstatemountainpassmixed

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 proves that the fourth-order Schrödinger equation $\Delta^2 u+\mu\Delta u-\lambda u=|u|^{p-2}u$ with a positive second-order dispersion coefficient $\mu>0$ admits at least two normalized solutions of prescribed $L^2$ mass $a$, in the supercritical exponent range $2+8/N

What carries the argument

The argument is carried by the Pohozaev manifold $P_{a,\mu}=\{u\in S_a : P_\mu(u)=0\}$, where $P_\mu(u)=2\|\Delta u\|_2^2-\mu\|\nabla u\|_2^2-2\gamma_p\|u\|_p^p$ is the Pohozaev identity, together with the $L^2$-preserving dilations $s\star u(x)=e^{Ns/2}u(e^s x)$. The fiber maps $\Psi^\mu_u(s)=E_\mu(s\star u)$ have exactly two critical points under the condition $\mu^{p\gamma_p-2}a^{p-2}<\tilde C(N,p)$: one on the positive part $P^+$ with negative energy and one on the negative part $P^-$ with positive energy. This yields the convex-concave geometry that produces a local minimizer and a mountain pass critical point. The compactness of Palais-Smale sequences is obtained by proving the strict bound $m_r(a,\mu)<-a^2\mu^2/8$ via a Bessel-function test function, which forces the Lagrange multiplier to satisfy $\lambda<-\mu^2/4$ and prevents loss of compactness.

What would settle it

For $N=5$ and $p=3$, compute $m_r(a,\mu)$ by minimizing $E_\mu$ over radial $H^2$ functions with $\|u\|_2=a$ and $\|\Delta u\|_2<R_0$, for $\mu$ and $a$ satisfying $\mu^{p\gamma_p-2}a^{p-2}$ well below the stated constant; if the infimum is not strictly less than $-a^2\mu^2/8$, then Lemma 3.2's bound is false and the negative-energy compactness argument collapses.

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Extended reading notes

Core claim

The central claim is that for $N\ge 5$, $\bar p<p<\min\{4,4^*\}$, and $a,\mu>0$ satisfying $\mu^{p\gamma_p-2}a^{p-2}<\min\{\tilde C(N,p),C^*(N,p),C_*(N,p)\}$, the energy functional $E_\mu$ restricted to the sphere $S_a$ has two distinct critical points in the radial space: an interior local minimizer $\tilde u_\mu$ at level $m_r(a,\mu)<-a^2\mu^2/8$, which is a radial ground state, and a second critical point $\hat u_\mu$ of mountain pass type at a positive level $\sigma(a,\mu)>0$. Both are real-valued radial solutions with Lagrange multipliers $\tilde\lambda,\hat\lambda<-\mu^2/4$, and under $N<8$ and $p<\min\{2(N-2)/(N-4),4\}$ both are sign-changing. As $\mu\to 0^+$ the ground state disappears (its $\Delta$-norm and energy go to zero) while the mountain pass solution converges in $H^2$ to a radial ground state of the limiting equation $\Delta^2 u-\lambda u=|u|^{p-2}u$.

Load-bearing premise

The load-bearing premise is the strict upper bound $m_r(a,\mu)<-a^2\mu^2/8$, proved only for $N\ge 5$ and $p<4$ using a Bessel-function test function; if that inequality fails, the negative-energy Palais-Smale sequence can vanish and the radial ground state is not obtained.

Editorial extensions

If this is right

If the theorem is correct, the sign of the second-order dispersion coefficient is definitive: for $\mu>0$ the $L^2$-supercritical problem has at least two radial normalized solutions, a structure not present in the $\mu\le 0$ results.
The explicit condition $\mu^{p\gamma_p-2}a^{p-2}<C(N,p)$ gives a quantitative threshold: existence is guaranteed when the mass $a$ is small enough relative to $\mu$, or $\mu$ small enough relative to $a$.
As $\mu\to 0^+$, the positive-energy branch converges to a radial ground state of the fourth-order equation without the second-order term, so the family of solutions connects continuously to the $\mu=0$ case.
As $a\to 0^+$, the normalized ground state's Fourier profile concentrates near the sphere $|\xi|=\sqrt{\mu/2}$, the zero set of the symbol $|\xi|^4-\mu|\xi|^2$, while the Lagrange multiplier approaches $-\mu^2/4$ from below.

Reading between the lines

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

The strict bound $m_r(a,\mu)<-a^2\mu^2/8$ likely marks a threshold: below the stated constant the negative-energy branch exists, and numerical continuation would show whether it persists or disappears when the inequality is violated.
The Bessel-function test function is radial and the bound is proved only for $p<4$; the same construction might extend to $p\ge 4$ in dimensions where $4^*\le 4$ if sharper $L^p$ estimates for the test function are used.
The Fourier-space concentration on the sphere $|\xi|=\sqrt{\mu/2}$ as $a\to 0^+$ suggests that the vanishing ground state is driven by the linear operator $\Delta^2+\mu\Delta$ becoming degenerate on that sphere; this might be connected to the stability threshold for the time-dependent problem.
The sign-changing result is proved under extra dimension/exponent restrictions ($N<8$, $p<\min\{2(N-2)/(N-4),4\}$); whether the mountain pass solution is sign-changing in all cases covered by Theorem 1.1 is left open and could be tested numerically.

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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

Dear colleague,

The bottom line: this paper appears correct and does fill the open 'Unknown' entry in the table for γ>0, μ>0, p̄

What's good: the paper extends known results from μ≤0 to μ>0 in the L²-supercritical regime. The Pohozaev manifold decomposition is handled cleanly, the mountain pass construction via Ghoussoub's principle is standard but applied correctly, and the authors are honest about borrowing from [6,7,34]. The smallness condition is an explicit assumption, not a hidden fitting parameter. No circularity.

Soft spots, in order of importance:

1. The abstract overclaims: it states N≥2 and p<4*, but the theorem only holds for N≥5 and p

2. Lemma 3.2 has an inequality-direction slip in the final displayed chain. After the bracket becomes negative, replacing the prefactor 1/||ψ_m||_2^2 by the larger 1/(C1 m^{(p-2)/2}) does not give an upper bound; it gives a lower bound. The written inequality is not valid. However, the second-to-last line already suffices: for large m, the negative term ~ m^{-(p-2)/2} dominates the positive term ~ 1/m, so the expression is negative. The proof is fixable exactly as the authors intended, but the line should be rewritten.

3. Lemma 5.9 is sketched rather than proved. It is the μ=0 limiting case, which is already known from [34], so I'd accept it with a citation, but a referee may want a bit more detail.

The stress-test worry that the strict inequality might fail for some allowed (N,p) is not persuasive. The Bessel-function asymptotics are standard: ||ψ_m||_2^2 ~ m, ||ψ_m||_p^p → const > 0, and ||(Δ+1)ψ_m||_2^2 = O(1/m), so the negative term wins for large m. The constants C1..C7 are generic; their existence is not in doubt.

Who this is for: anyone working on normalized solutions for fourth-order NLS or on variational methods with indefinite quadratic parts. It deserves a serious referee; I'd recommend acceptance after minor revision.

Referee Report

2 major / 4 minor

Summary. The paper studies normalized solutions of the fourth-order Schrödinger equation Δ²u + μΔu - λu = |u|^{p-2}u on S_a = {u ∈ H²(R^N) : ||u||_2² = a²}, in the L²-supercritical range p̄ := 2 + 8/N < p < 4*, with μ > 0. The energy functional is unbounded below on S_a, so the authors use the Pohozaev manifold P_{a,μ} and its decomposition into P_{a,μ}^+ ∪ P_{a,μ}^-. Under a smallness condition on μ^{pγ_p-2}a^{p-2}, they obtain a local minimizer at a negative level m_r(a,μ) < -a²μ²/8, which is a radial ground state, and a second mountain-pass-type radial solution at a positive level σ(a,μ) > 0. Theorem 1.1 states these existence results for N ≥ 5 and p̄ < p < min{4,4*}, together with asymptotics as μ → 0⁺ and sign-changing information. Theorem 1.2 describes the behavior of the ground states as a → 0⁺. The proofs use standard modern variational tools: fiber maps, Ekeland's principle, Ghoussoub's min-max principle, and compactness lemmas for Palais-Smale sequences.

Significance. If the results are correct, the paper fills a genuinely open row in the known table for mixed-dispersion fourth-order NLS: the case γ > 0, μ > 0, p in the L²-supercritical regime. The two-component Pohozaev manifold and the distinction between negative and positive energy levels are natural and well suited to the problem. The use of truncated Bessel functions to test the critical inequality m_r(a,μ) < -a²μ²/8 is also appropriate. However, the proof of that key strict inequality contains an invalid estimate, and the proof of the μ = 0 ground state needed for the asymptotic statement is only sketched. The central existence claims are therefore not yet established as written, although the overall strategy appears plausible and repairable.

major comments (2)

[Lemma 3.2] The proof of the strict inequality m_Φ0(a,μ) < -c̃a², which is used to obtain m_r(a,μ) < -a²μ²/8, contains an invalid estimate. In the chain involving Φ_0(ψ̃_m) + ||ψ̃_m||₂², the linear term is bounded using ||(Δ+1)ψ_m||₂² ≤ C₇/m and the nonlinear term is bounded below using ||ψ_m||₂² ≤ C₂m and ||ψ_m||_p^p ≥ C₄. After the displayed factorization, the negative term in the bracket acquires an additional m^{-1} factor, and when the bracket is negative the replacement of the prefactor c̃a²/||ψ_m||₂² by an upper bound reverses the inequality. As written, the orders displayed (m^{-1} against m^{-p/2} with p > 2) point in the wrong direction. Since Lemma 4.2 requires this strict upper bound for all Palais-Smale sequences at negative levels, Theorem 1.1(1) is not established as written. The authors should supply a rigorous estimate, for instance by computing the sharp asymptotic of Φ_0(ψ̃_m) + ||ψ̃_m||₂², or by proving the needed inequality with constants that are controlled in the correct direction.
[Lemma 5.9] Lemma 5.9 is asserted with a one-sentence proof referring to 'the arguments in the proof of Theorem 1.1-(2) or section 6 in [34]'. The existence of a radial ground state for μ = 0 at the level σ(a,0) > 0 is used in the proof of Theorem 1.1(5) and in Lemmas 5.10 and 5.11. The L²-supercritical setting is not identical to the setting of the cited results, and Lemmas 5.5-5.8 only record preliminary facts; the compactness of Palais-Smale sequences for E_0|_{S_a} and the min-max argument need to be written out in full or replaced by a theorem that applies verbatim to this fourth-order problem.

minor comments (4)

[Abstract and Theorem 1.1] The abstract states the range N ≥ 2 with p up to 4*, but Theorem 1.1 and Lemma 3.2 are proved only for N ≥ 5 and p < min{4,4*} (with p < 4 in Lemma 3.2). The abstract should be corrected to match the theorem.
[Lemma 3.2] The constants C₁,...,C₇ in Lemma 3.2 are asserted to follow 'as in the proof of Lemma 5.5 in [6]', but [6] is an unpublished preprint. Please reproduce the estimates or cite a published version.
[Throughout] There are several typos and minor wording issues: 'dose not' in the proof of Lemma 4.1, 'if only if' for 'if and only if' in several places, 'drive' for 'derive' in Lemma 5.9, and 'Scrhödinger' in reference [6].
[Proof of Theorem 1.1(5)] In the line 'E_0(û) = ... = lim_{μ→0⁺} σ(a,μ) ≥ σ(a, μ) > 0', the last occurrence of μ is ambiguous and should be a fixed positive parameter with a different symbol.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the existence proof rests on explicit inequalities, a Bessel test-function estimate, and external minimax principles.

full rationale

The paper's derivation chain is self-contained and does not reduce any of its conclusions to its own inputs. The smallness condition mu^{p*gamma_p-2}a^{p-2} < min{Ctilde,C*,C*} is an explicit hypothesis, not a fitted output, and the claimed strict upper bound m_r(a,mu) < -a^2 mu^2/8 is proved in Lemma 3.2 by an explicit Bessel-function test-function estimate with constants C_i independent of the parameters. The compactness lemmas in Section 4 are built from standard ingredients: the Pohozaev identity, the Gagliardo-Nirenberg inequality, the compact embedding of H^2_rad into L^r, and the Lagrange multiplier rule. The mountain pass construction uses Ghoussoub's minimax principle and a C^1 map lemma cited from an independent source. The asymptotic results follow by taking limits in explicit identities and inequalities; there is no quantity that is defined in terms of the data it is supposed to predict. The skeptical concern about the direction of an inequality in Lemma 3.2 is a potential correctness issue, not a circularity, so it does not affect the circularity score.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The paper introduces no new entities and no fitted parameters. The constants C̃, C*, C_* are defined explicitly in terms of the Gagliardo-Nirenberg constant C_{N,p} and are not fitted to any data. All assumptions are standard inequalities or imported results from the cited literature. The smallness condition on a and μ is an explicit hypothesis, not an output.

assumptions (8)

standard math Gagliardo-Nirenberg inequality (1.10): ||u||_p^p ≤ C_{N,p}^p ||u||_2^{p(1-γ_p)} ||Δu||_2^{pγ_p} for 2<p<4*.
Used throughout to reduce L^p norms to L^2 and Δ norms; the constant C_{N,p} is taken from the literature [9,7,26,6].
standard math Interpolation inequality ||∇u||_2^2 ≤ ||u||_2 ||Δu||_2 (2.1).
Taken from [6]; used to bound the negative quadratic term -μ/2||∇u||_2^2 by ||Δu||_2.
standard math Compact embedding H^2_rad(R^N) ↪ L^r(R^N) for 2<r<4*.
Essential for strong convergence of radial Palais-Smale sequences up to subsequences; standard radial compactness result.
standard math Pohozaev identity for solutions of (2.5): P_μ(u)=0 (Proposition 2.1).
Imported from [8]; used to define the natural constraint manifold P_{a,μ}.
standard math Ghoussoub's homotopy-stable min-max principle (Lemma 5.2).
Imported from [18]; used to obtain a Palais-Smale sequence for the mountain pass level with additional localization properties.
standard math Existence of a radial ground state for the limiting equation with μ=0 (Lemma 5.9).
The proof is only sketched, referencing the argument in [34, section 6]; used in Theorem 1.1-(5).
standard math Bessel function identity: ψ(x)=|x|^{-(N-2)/2} J_{(N-2)/2}(|x|) satisfies (Δ+1)ψ=0.
Used in Lemma 3.2 to construct the test functions; from Appendix B.4 in [17].
standard math Infimum of the quadratic form ½||Δu||_2^2 - (μ/2)||∇u||_2^2 over S_a equals -a²μ²/8 (Lemma 3.1 in [6]).
Used in the proof of Theorem 1.2 to locate the asymptotic energy of minimizers.

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Pith. "Pith review of Normalized solutions for a fourth-order Schr\"{o}dinger equation with positive second-order dispersion coefficient." pith.science (2026). https://pith.science/paper/QXMCVQSE

@misc{pith2026190803079,
  author       = {Pith},
  title        = {Pith review of: Normalized solutions for a fourth-order Schr\"odinger equation with positive second-order dispersion coefficient},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QXMCVQSE}},
  note         = {Machine review of arXiv:1908.03079}
}

abstract

We are concerned with the existence and asymptotic properties of solutions to the following fourth-order Schr\"{o}dinger equation \begin{equation}\label{1} {\Delta}^{2}u+\mu \Delta u-{\lambda}u={|u|}^{p-2}u, ~~~~x \in \R^{N}\\ \end{equation} under the normalized constraint $$\int_{{\mathbb{R}^N}} {{u}^2}=a^2,$$ where $N\!\geq\!2$, $a,\mu\!>\!0$, $2+\frac{8}{N}\!<\!p\!<\! 4^{*}\!=\!\frac{2N}{(N-4)^{+}}$ and $\lambda\in\R$ appears as a Lagrange multiplier. Since the second-order dispersion term affects the structure of the corresponding energy functional $$ E_{\mu}(u)=\frac{1}{2}{||\Delta u||}_2^2-\frac{\mu}{2}{||\nabla u||}_2^2-\frac{1}{p}{||u||}_p^p $$ we could find at least two normalized solutions to (\ref{1}) if $2\!+\!\frac{8}{N}\!<\! p\!<\!{ 4^{*} }$ and $\mu^{p\gamma_p-2}a^{p-2}\!<\!C$ for some explicit constant $C\!=\!C(N,p)\!>\!0$ and $\gamma_p\!=\!\frac{N(p\!-\!2)}{4p}$. Furthermore, we give some asymptotic properties of the normalized solutions to (\ref{1}) as $\mu\to0^+$ and $a\to0^+$, respectively. In conclusion, we mainly extend the results in \cite{DBon,dbJB}, which deal with (\ref{1}), from $\mu\leq0$ to the case of $\mu>0$, and also extend the results in \cite{TJLu,Nbal}, which deal with (\ref{1}), from $L^2$-subcritical and $L^2$-critical setting to $L^2$-supercritical setting.

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Works this paper leans on

38 extracted references · 35 canonical work pages

[34]

Soave, Normalized ground state for the NLS equations with c ombined nonlinearities, Preprint, arXiv:1811.00826

N. Soave, Normalized ground state for the NLS equations with c ombined nonlinearities, Preprint, arXiv:1811.00826

arXiv
[1]

M. B. Artzi, H. Koch, J. C. Saut, Dispersion estimates for fourt h order Schr¨ odinger equations, C. R. Math. Acad. Sci. Paris S´ er. I 330 (2000), 87-92

work page 2000
[2]

Bartsch, N

T. Bartsch, N. Soave, Multiple normalized solutions for a competin g system of Schr¨ odinger equa- tions, Calc. Var., 58 (2019) 1-24

work page 2019
[3]

Bonheure, J.-B

D. Bonheure, J.-B. Casteras, E. Moreira Dos Santos, R. Nascim ento, Orbitally stable standing waves of a mixed dispersion nonlinear Schr¨ odinger equation, SIAM J . Math. Anal. 50 (2018) 5027-5071

work page 2018
[4]

Bonheure, J.-B

D. Bonheure, J.-B. Casteras, R. Mandel, On a fourth-order no nlinear Helmholtz equation, J. Lond. Math. Soc. 99 (3) (2019) 831-852

work page 2019
[5]

Bellazzini, L

J. Bellazzini, L. Jeanjean, On dipolar quantum gases in the unstab le regime, SIAM J. Math. Anal. 48 (3) (2016) 2028-2058

work page 2016
[6]

Boussaid, A

N. Boussaid, A. J. Fernandez, L. Jeanjean, Some remarks on a minimization problem associated to a fourth order nonlinear Scrh¨ odinger equation, Preprint, arX iv:1910.13177. NORMALIZED SOLUTIONS FOR A FOURTH-ORDER SCHR ¨ODINGER EQUATION 29

arXiv 1910
[7]

Bonheure, J

D. Bonheure, J. B. Casteras, T. Gou, L. Jeanjean, Normalized solutions to the mixed dispersion nonlinear Schr¨ odinger equation in the mass critical and supercritic al regime, Trans. Amer. Math. Soc. 372 (2019) 2167-2212

work page 2019

Show all 38 references

[8]

Bonheure, J.-B

D. Bonheure, J.-B. Casteras, T. Gou, L. Jeanjean, Strong In stability of Ground States to a Fourth Order Schr¨ odinger Equation, Int. Math. Res. Not. IMRN, 17 (20 19) 5299-5315
[9]

Boulenger, E

T. Boulenger, E. Lenzmann, Blowup for biharmonic NLS, Ann. Sci. ´Ec. Norm. Sup´ er. 50 (2017) 503-544

2017
[10]

Br´ ezis, E

H. Br´ ezis, E. Lieb, A relation between pointwise convergence o f functions and convergence of functionals, Proc. Amer. Math. Soc 88 (1983) 486-490

1983
[11]

Buﬀoni, Inﬁnitely many large amplitude homoclinic orbits for a cla ss of autonomous Hamilton- ian systems, J

B. Buﬀoni, Inﬁnitely many large amplitude homoclinic orbits for a cla ss of autonomous Hamilton- ian systems, J. Diﬀer. Equ. 121 (1995) 109-120

1995
[12]

Buﬀoni, Periodic and homoclinic orbits for Lorentz-Lagrangia n systems via variational methods, Nonlinear Anal

B. Buﬀoni, Periodic and homoclinic orbits for Lorentz-Lagrangia n systems via variational methods, Nonlinear Anal. 26 (1996) 443-462

1996
[13]

Cazenave, P

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

1982
[14]

Cazenave, Semilinear Schr¨ odinger equations, volume 10 of C ourant Lecture Notes in Mathe- matics

T. Cazenave, Semilinear Schr¨ odinger equations, volume 10 of C ourant Lecture Notes in Mathe- matics. New York University, Courant Institute of Mathematical S ciences, New York; Amer. Math. Soc., Providence, RI, 2003

2003
[15]

Evequoz, T

G. Evequoz, T. Weth, Dual variational methods and nonvanish ing for the nonlinear Helmholtz equation, Adv. Math., 280 (2015), 690-728

2015
[16]

Fibich, B

G. Fibich, B. Ilan, G. Papanicolaou, Self-focusing with fourth-o rder dispersion, SIAM J. Appl. Math. 62 (2002) 1437-1462

2002
[17]

Grafakos

L. Grafakos. Classical Fourier analysis, volume 249 of Graduat e Texts in Mathematics. Springer, New York, second edition, 2008

2008
[18]

Ghoussoub, Duality and perturbation methods in critical poin t theory, volume 107 of Cambridge Tracts in Mathematics

N. Ghoussoub, Duality and perturbation methods in critical poin t theory, volume 107 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 19 93. With appendices by David Robinson
[19]

Guo, Scattering for the focusing L2-supercritical and ˙H 2 -subcritical biharmonic NLS equations, Comm

Q. Guo, Scattering for the focusing L2-supercritical and ˙H 2 -subcritical biharmonic NLS equations, Comm. Partial Diﬀer. Equ. 41 (2016) 185-207

2016
[20]

Jeanjean, Existence of solutions with prescribed norm for s emilinear elliptic equations

L. Jeanjean, Existence of solutions with prescribed norm for s emilinear elliptic equations. Nonlinear Anal. 28 (1997) 1633-1659

1997
[21]

V. I. Karpman, Stabilization of soliton instabilities by higher-orde r dispersion: fourth order non- linear Schr¨ odinger-type equations, Phys. Rev. E 53 (1996) 1336 -1339

1996
[22]

V. I. Karpman, A. G. Shagalov, Stability of solitons described by nonlinear Schr¨ odinger-type equations with higher-order dispersion. Phys. D 144 (2000) 194-2 10

2000
[23]

Kenig, G

C. Kenig, G. Ponce, L. Vega, Oscillatory integrals and regularity of dispersive equations. J. Indiana Univ. Math. 40 (1991) 33-69

1991
[24]

E. W. Laedke, K. H. Spatschek, L. Stenﬂo, Evolution theorem for a class of perturbed envelope soliton solutions. J. Math. Phys. 24 (1983) 2764-2769

1983
[25]

E. W. Laedke, K. H. Spatschek, Stability properties of multidime nsional ﬁnite-amplitude solitions. Phys. Rev. A, 30 (1984) 3279-3288

1984
[26]

T. J. Luo, S. J. Zheng, S. H. Zhu, Orbital stability of standing w aves for a fourth-order nonlinear schr¨ odinger equation with the mixed dispersions, Preprint, arXiv: 1904.02540

1904 arXiv
[27]

C. Miao, G. Xu, and L. Zhao, Global well-posedness and scatter ing for the focusing energycritical nonlinear Schr¨ odinger equations of fourth order in the radial cas e, J. Diﬀer. Equ. 246 (2009) 3715-3749

2009
[28]

Pausader, Global well-posedness for energy critical fourt h-order Schr¨ odinger equations in the radial case, Dyn

B. Pausader, Global well-posedness for energy critical fourt h-order Schr¨ odinger equations in the radial case, Dyn. Partial Diﬀer. Equ. 4 (2007) 197-225

2007
[29]

Pausader, The cubic fourth-order Schr¨ odinger equation , J

B. Pausader, The cubic fourth-order Schr¨ odinger equation , J. Funct. Anal. 256 (2009) 2473-2517. 30 NORMALIZED SOLUTIONS FOR A FOURTH-ORDER SCHR ¨ODINGER EQUATION

2009
[30]

Pausader, The focusing energy-critical fourth-order Sc hr¨ odinger equation with radial data, Discrete Contin

B. Pausader, The focusing energy-critical fourth-order Sc hr¨ odinger equation with radial data, Discrete Contin. Dyn. Syst. 24 (2009) 1275-1292

2009
[31]

Pausader, S

B. Pausader, S. Shao, The mass-critical fourth-order Schr ¨ odinger equation in high dimensions, J. Hyperbolic Diﬀer. Equ. 7 (2010) 651-705

2010
[32]

Pausader, S

B. Pausader, S. Xia, Scattering theory for the fourth-orde r Schr¨ odinger equation in low dimensions, Nonlinearity 26 (2013) 2175-2191

2013
[33]

Sulem, P

C. Sulem, P. L. Sulem, The Nonlinear Schr¨ odinger Equation: Self -focusing and Wave Collapse, volume 139 of Applied Mathematical Sciences. Springer, 1 edition, 19 99
[35]

Soave, Normalized ground states for the NLS equation with c ombined nonlinearities: the Sobolev critical case, Preprint, arXiv:1901.02003

N. Soave, Normalized ground states for the NLS equation with c ombined nonlinearities: the Sobolev critical case, Preprint, arXiv:1901.02003

1901 arXiv
[36]

M. Willem. Minimax Theorems. Progr. Nonlinear Diﬀerential Equatio ns Appl., vol.24, Birkh¨ auser Boston, Inc., Boston, MA, 1996

1996
[37]

M. I. Weinstein, Nonlinear Schr¨ odinger Equations and Sharp In terpolation Estimates. Commum. Math. Phys. 87 (1983) 567-576

1983
[38]

S. H. Zhu, J. Zhang, H. Yang, Limiting proﬁle of the blow-up solut ions for the fourth-order nonlinear Schr¨ odinger equation, Dyn. Partial Diﬀer. Equ. 7 (2010 ) 187-205

2010

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [34]

Soave, Normalized ground state for the NLS equations with c ombined nonlinearities, Preprint, arXiv:1811.00826

N. Soave, Normalized ground state for the NLS equations with c ombined nonlinearities, Preprint, arXiv:1811.00826

arXiv

[2] [1]

M. B. Artzi, H. Koch, J. C. Saut, Dispersion estimates for fourt h order Schr¨ odinger equations, C. R. Math. Acad. Sci. Paris S´ er. I 330 (2000), 87-92

work page 2000

[3] [2]

Bartsch, N

T. Bartsch, N. Soave, Multiple normalized solutions for a competin g system of Schr¨ odinger equa- tions, Calc. Var., 58 (2019) 1-24

work page 2019

[4] [3]

Bonheure, J.-B

D. Bonheure, J.-B. Casteras, E. Moreira Dos Santos, R. Nascim ento, Orbitally stable standing waves of a mixed dispersion nonlinear Schr¨ odinger equation, SIAM J . Math. Anal. 50 (2018) 5027-5071

work page 2018

[5] [4]

Bonheure, J.-B

D. Bonheure, J.-B. Casteras, R. Mandel, On a fourth-order no nlinear Helmholtz equation, J. Lond. Math. Soc. 99 (3) (2019) 831-852

work page 2019

[6] [5]

Bellazzini, L

J. Bellazzini, L. Jeanjean, On dipolar quantum gases in the unstab le regime, SIAM J. Math. Anal. 48 (3) (2016) 2028-2058

work page 2016

[7] [6]

Boussaid, A

N. Boussaid, A. J. Fernandez, L. Jeanjean, Some remarks on a minimization problem associated to a fourth order nonlinear Scrh¨ odinger equation, Preprint, arX iv:1910.13177. NORMALIZED SOLUTIONS FOR A FOURTH-ORDER SCHR ¨ODINGER EQUATION 29

arXiv 1910

[8] [7]

Bonheure, J

D. Bonheure, J. B. Casteras, T. Gou, L. Jeanjean, Normalized solutions to the mixed dispersion nonlinear Schr¨ odinger equation in the mass critical and supercritic al regime, Trans. Amer. Math. Soc. 372 (2019) 2167-2212

work page 2019