Waveguide solutions for a nonlinear Schrödinger equation with mixed dispersion

Bonheure, Denis; Nascimento, Robson

doi:10.1007/978-3-319-19902-3_4

Cited by 22 publications

(35 citation statements)

References 22 publications

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“…Assuming σ > 2 when N = 3, σ > 1 when N = 4, and 4 ≤ σN < 4 * when N ≥ 5, we see from (2.2) that X embeds continuously in L 2σ+2 (R N ) and therefore the functional E defined by (1.3) is well-defined in X and we will show that it has critical points inside that space. On the other hand, using Pohozaev identity and multiplying (5.9) by u and integrating, we get To prove the existence of a minimizer, we proceed as in [23,Remark 3.2]. Let (u n ) n ⊂ M be a minimizing sequence.…”

Section: Some Properties Of the Function C → γ(C)mentioning

confidence: 99%

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

Bonheure

Casteras

Gou

et al. 2019

Trans. Amer. Math. Soc.

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In this paper, we study the existence of solutions to the mixed dispersion nonlinear Schrödinger equationWe assume γ > 0, N ≥ 1, 4 ≤ σN < 4N (N−4) + , whereas the parameter α ∈ R will appear as a Lagrange multiplier. Given c ∈ R + , we consider several questions including the existence of ground states, of positive solutions and the multiplicity of radial solutions. We also discuss the stability of the standing waves of the associated dispersive equation.

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Section: Some Properties Of the Function C → γ(C)mentioning

confidence: 99%

“…A possible choice is to consider that α > 0 is given and to look for solutions u ∈ H 2 (R N ) of (1.2). Such solutions correspond to critical points of the functional This point of view is adopted in the paper [23], see also [21].…”

Section: Introductionmentioning

confidence: 99%

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

Bonheure

Casteras

Gou

et al. 2019

Trans. Amer. Math. Soc.

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“…is achieved by some u ∈ M , then v = m 1 2σ u is a least energy critical point of A. The following result is proved in [10]. √ γα, then any ground state u is such that |u| is positive, radially symmetric around some point and strictly radially decreasing.…”

mentioning

confidence: 91%

Orbitally Stable Standing Waves of a Mixed Dispersion Nonlinear Schrödinger Equation

Bonheure¹,

Casteras²,

Santos³

et al. 2018

SIAM J. Math. Anal.

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We study the mixed dispersion fourth order nonlinear Schrödinger equationwhere γ, σ > 0 and β ∈ R. We focus on standing wave solutions, namely solutions of the form ψ(x, t) = e iαt u(x), for some α ∈ R. This ansatz yields the fourth-order elliptic equationWe consider two associated constrained minimization problems: one with a constraint on the L 2 -norm and the other on the L 2σ+2 -norm. Under suitable conditions, we establish existence of minimizers and we investigate their qualitative properties, namely their sign, symmetry and decay at infinity as well as their uniqueness, nondegeneracy and orbital stability.

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“…. Note that G 1 satisfies, qualitatively, the same bounds as the function Φ 1 in [16], see (4.4) and the estimates (26), (7) in [14,16], respectively. The proof of [16,Lemma 3.4] and [14,Claim 3,p.…”

Section: Moreover For Any Bounded and Measurable Setmentioning

confidence: 59%

“…First, observe that using the scaling

u false(x false) = w false(γ^{- 1 / 4} x false)

, we see that is equivalent to

{normalΔ}^{2} u - β Δ u + α u = {| u |}^{p - 2} u in {double-struckR}^{N},

where

β = γ^{- 1 / 2}

. Bonheure and Nascimento considered the following minimization problem

m : = \underset{u \in M}{trueprefixinf} J_{α, β} false(u false),

where

J_{α, β} false(u false) : = \int_{R^{N}} {(| Δ u |}^{2} + {β | \nabla u |}^{2} + α u^{2} false) d x

and

M : = \{u \in H^{2} ({double-struckR}^{N}) : \int_{{double-struckR}^{N}} {false| u false|}^{p} d x = 1\} .

Note that if

u \in M

achieves the infimum

m

, then

u

is a solution to

{normalΔ}^{2} u - β Δ u + α u = m {| u |}^{p - 2} u . ...

…”

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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Waveguide solutions for a nonlinear Schrödinger equation with mixed dispersion

Cited by 22 publications

References 22 publications

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

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

Orbitally Stable Standing Waves of a Mixed Dispersion Nonlinear Schrödinger Equation

On a fourth‐order nonlinear Helmholtz equation

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