Normalized solutions to the mixed dispersion nonlinear Schrödinger equation in the mass critical and supercritical regime

Bonheure, Denis; Casteras, Jean-Baptiste; Gou, Tianxiang; Jeanjean, Louis

doi:10.1090/tran/7769

Cited by 66 publications

(56 citation statements)

References 53 publications

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“…(2) If Q(u) > 0 and u 2 L 2 = c, then v n → u in L 2 as n → ∞. This implies that v n → u in L q as n → ∞, for any q ∈ [2,6) . On the other hand, we deduce from Q(u) > 0 that Q(v n − u) ≤ 0 for sufficiently large n. Thus, we can obtain v n → u in H 1 r as n → ∞.…”

Section: Normalized Solutions Without Partial Confinementioning

confidence: 99%

“…In fact, many known physical models enjoy this property. For example, NLS or Davey-Stewartson system with combined focusing mass-supercritical and defocusing mass-subcritical nonlinearities (see [35]), the mixed dispersion NLS (see [6,7]) and so on. Moreover, we think that our method is robust and can be applied to many other models, only if the corresponding function f (λ) = S ω (v λ ) has an unique critical point on (0, ∞).…”

Section: Introductionmentioning

confidence: 99%

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Optimal Bilinear Control for the X-Ray Free Electron Laser Schrödinger Equation

Feng¹,

Zhao²

2019

SIAM J. Control Optim.

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In this paper, we are concerned with the standing waves for the following nonlinear Schrödinger equationwhere 0 < p < 4. This equation arises as an effective single particle model in X-ray Free ElectronLasers. We mainly study the existence and stability/instability properties of standing waves for this equation, in two cases: the first one is that no magnetic potential is involved, (i.e. b = 0 in the equation) and the second one is that b = 0. To be precise, in the first case, by considering a minimization problem on a suitable Pohozaev manifold, we prove the existence of radial solutions, and show further that the corresponding standing waves are strongly unstable by blow-up in finite time. Moreover, by making use of the ideas of these proofs, we are able to prove the existence and instability of normalized solutions, whose proofs seem to be new, compared with the studies of normalized solutions in the existing literature. This study also indicates that there is a close connection between the study of the strong instability and the one of the existence of normalized solutions. In the second case, the situation is more difficult to be treated, due to the additional term of the partial harmonic potential. We manage to prove the existence of stable standing waves for p ∈ (0, 4) and with some assumptions on the coefficients, solutions are obtained as global minimizers if p ∈ (0, 4 3 ], and as local minimizers if p ∈ [ 4 3 , 4). In the mass-critical and supercritical cases p ∈ [ 4 3 , 4), we also establish the variational characterization of the ground states on a new manifold which is different from the one neither of the Nehari type nor of the Pohozaev type, and then prove the existence of ground states. Finally under some assumptions on the coefficients, we prove that the ground state standing waves are strongly unstable.2010 Mathematics Subject Classification. 35Q55, 35J50, 37K45.for all t ∈ [0, T * ). This completes the proof.Proof of Theorem 1.11. When 4 3 ≤ p < 2, let u be the ground state related to (1.4), under the assumptionfor all λ > 1. By the same argument as Theorem 1.4, we can obtain the initial data χ M 0 u λ 0 ∈ Σ∩K ω such that the corresponding solution ψ λ 0 (t) of (1.3) blows up in finite time. Hence, we can prove this theorem.When 2 ≤ p < 4, we see from Lemma 6.2 that S b,ω (λu) < S b,ω (u), K b,ω (λu) < 0 and Q b (λu) < 0, for all λ > 1. Therefore, we can prove the strong instability along the above lines by replacing u λ 0 by λ 0 u. This completes the proof.

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Section: Normalized Solutions Without Partial Confinementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Optimal Bilinear Control for the X-Ray Free Electron Laser Schrödinger Equation

Feng¹,

Zhao²

2019

SIAM J. Control Optim.

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“…with parameters β 0 and ω > 0 provides solitary wave solutions for the fourth-order NLS (see e. g. [4,6,14] and references given there) and the corresponding time-dependent equation arises as a model equation in nonlinear optics. The case of vanishing β = 0 is referred to as the biharmonic NLS.…”

Section: 2mentioning

confidence: 99%

A Sharp Rearrangement Principle in Fourier Space and Symmetry Results for PDEs with Arbitrary Order

Lenzmann

Sok

2020

International Mathematics Research Notices

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We prove sharp inequalities for the symmetric-decreasing rearrangement in Fourier space of functions in R d . Our main result can be applied to a general class of (pseudo-)differential operators in R d of arbitrary order with radial Fourier multipliers. For example, we can take any positive power of the Laplacian (−∆) s with s > 0 and, in particular, any polyharmonic operator (−∆) m with integer m 1. As applications, we prove radial symmetry and real-valuedness (up to trivial symmetries) of optimizers for: i) Gagliardo-Nirenberg inequalities with derivatives of arbitrary order, ii) ground states for bi-and polyharmonic NLS, and iii) Adams-Moser-Trudinger type inequalities for H d/2 (R d ) in any dimension d 1. As a technical key result, we solve a phase retrieval problem for the Fourier transform in R d . To achieve this, we classify the case of equality in the corresponding Hardy-Littlewood majorant problem for the Fourier transform in R d . Applications: Radial Symmetry of OptimizersIn this section, we discuss some applications of Theorem 1 to show radial symmetry of optimizers. More precisely, we will consider the following examples.• Gagliardo-Nirenberg type inequalities with derivatives of arbitrary order.• Ground states for generalized NLS type (e. g. fourth-order NLS).

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“…They also show that their solution has a sign, is radially symmetric if in addition

β ⩾ 2 \sqrt{α}

and, in , that it is exponentially decreasing (if

β > - 2 \sqrt{α}

). The case

α = 0

and

β > 0

has been considered in where the existence of solutions, belonging to

X : = {u \in D^{1, 2} false(R^{N} false) | ∥ Δ u {false∥}_{L^{2} false(R^{N} false)} < \infty}

, has been obtained provided that

2 N / (N - 2) ⩽ p

N = 3, 4

and

2 N / (N - 2) ⩽ p < 2 N / (N - 4)

N > 4

. Moreover, this solution has a sign and belongs to

L^{2} false(R^{N} false)

if and only if

N ⩾ 5

suggesting that it decays only polynomially.…”

Section: Introductionmentioning

confidence: 99%

“…Despite being less studied than the classical (2NLS), an increasing attention has been given to (4NLS). We refer to the works of Pausader [35][36][37][38][39], Miao et al [33], Ruzhansky et al [40], Segata [41,42] concerning global well-posedness and scattering, to [6,8] for finite-time blow-up and to [4,5,34] for the stability of standing wave solutions. We also mention that (4NHE) also appears in the theory of water waves [9] and as a model to study travelling waves in suspension bridges [24,32] (see also [10,26]).…”

Section: Introductionmentioning

confidence: 99%

On a fourth‐order nonlinear Helmholtz equation

Bonheure

Casteras

Mandel

2018

J. London Math. Soc.

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In this paper, we study the mixed dispersion fourth‐order nonlinear Helmholtz equation normalΔ2u−βΔu+αu=Γ|u|p−2uindouble-struckRN,for positive, bounded and ZN‐periodic functions normalΓ in the following three cases: false(afalse)α<0,β∈Rorfalse(bfalse)α>0,β<−2αorfalse(cfalse)α=0,β<0.Using the dual method of Evéquoz and Weth, we find solutions to this equation and establish some of their qualitative properties.

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Normalized solutions to the mixed dispersion nonlinear Schrödinger equation in the mass critical and supercritical regime

Cited by 66 publications

References 53 publications

Optimal Bilinear Control for the X-Ray Free Electron Laser Schrödinger Equation

Optimal Bilinear Control for the X-Ray Free Electron Laser Schrödinger Equation

A Sharp Rearrangement Principle in Fourier Space and Symmetry Results for PDEs with Arbitrary Order

On a fourth‐order nonlinear Helmholtz equation

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