On instability of standing waves for the mass-supercritical fractional nonlinear Schrödinger equation

Dinh, Van Duong

doi:10.1007/s00033-019-1104-4

Cited by 14 publications

(5 citation statements)

References 25 publications

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“…If

μ = 0

and

q \in false(\overset{q}{true}, 2_{σ}^{*} false)

, we learn from the forthcoming Lemma 5.5 that () possesses a positive radial ground state at energy level m ( a , 0) > 0 for any a > 0. Moreover, the associated standing wave is strongly unstable since we are in a L 2 ‐supercritical (with respect to q ) regime (see Thomas et al and Dinh 14,16 ). From the variational point of view, the stabilization is reflected by the discontinuity of the ground state energy level m ( a , μ ): We have m ( a , μ ) < 0 for every μ > 0 small, while m ( a , 0) > 0.…”

Section: Introduction and Main Resultsmentioning

confidence: 98%

“…By using a sharp Gagliardo–Nirenberg‐type inequality and the profile decomposition in

H^{σ} false(ℝ^{N} false)

, Peng and Shi in their work 13 proved that the standing waves of () with

f false(false| ψ false| false) ψ = {false| ψ false|}^{q - 2} ψ

are orbitally stable when

2 < q < \overset{q}{true}

and the ground state solitary waves are strongly unstable to blowup when

q = \overset{q}{true}

. Thomas et al 14 obtained a general criterion for blow‐up of radial solutions of () with

f false(false| ψ false| false) ψ = {false| ψ false|}^{q - 2} ψ

q \geq \overset{q}{true}

while N ≥ 2; see also Dinh 15,16 . For () with combined power‐type nonlinearities, that is,

f false(ψ false) = γ false| ψ {false|}^{q - 2} ψ + μ false| ψ {false|}^{p - 2} ψ

, one can refer to Feng and Zhu 17,18 …”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Normalized solutions for the fractional Schrödinger equation with a focusing nonlocal perturbation

Luo

Yang

2021

Math Methods in App Sciences

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In this paper, we study the existence and asymptotic properties of solutions to the following fractional Schrödinger equation: (−Δ)σu=λu+|u|q−2u+μIα∗|u|p|u|p−2uinℝN under the normalized constraint ∫ℝNu2=a2, where N ≥ 2, σ ∈ (0, 1), α ∈ (0, N), q ∈ (2 + 4σ/N, 2N/N − 2σ], p ∈ [2, 1 + 2σ + α/N), a > 0, μ > 0, Iαfalse(xfalse)0.1em=0.1emfalse|xfalse|α−N and λ0.1em∈0.1emℝ appears as a Lagrange multiplier. In the Sobolev subcritical case q ∈ (2 + 4σ/N, 2N/N − 2σ), we show that the problem admits at least two solutions under suitable assumptions on α, a, and μ. In the Sobolev critical case q0.1em=0.1em2Nfalse/N−2σ, we prove that the problem possesses at least one ground state solution. Furthermore, we give some stability and asymptotic properties of the solutions. We mainly extend the results of S. Bhattarai published in 2017 on J. Differ. Equ. and B. H. Feng et al published in 2019 on J. Math. Phys. concerning the above problem from L2‐subcritical and L2‐critical setting to L2‐supercritical setting with respect to q, involving Sobolev critical case especially.

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“…If

μ = 0

and

q \in false(\overset{q}{true}, 2_{σ}^{*} false)

Section: Introduction and Main Resultsmentioning

confidence: 98%

“…By using a sharp Gagliardo–Nirenberg‐type inequality and the profile decomposition in

H^{σ} false(ℝ^{N} false)

, Peng and Shi in their work 13 proved that the standing waves of () with

f false(false| ψ false| false) ψ = {false| ψ false|}^{q - 2} ψ

are orbitally stable when

2 < q < \overset{q}{true}

and the ground state solitary waves are strongly unstable to blowup when

q = \overset{q}{true}

. Thomas et al 14 obtained a general criterion for blow‐up of radial solutions of () with

f false(false| ψ false| false) ψ = {false| ψ false|}^{q - 2} ψ

q \geq \overset{q}{true}

while N ≥ 2; see also Dinh 15,16 . For () with combined power‐type nonlinearities, that is,

f false(ψ false) = γ false| ψ {false|}^{q - 2} ψ + μ false| ψ {false|}^{p - 2} ψ

, one can refer to Feng and Zhu 17,18 …”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Normalized solutions for the fractional Schrödinger equation with a focusing nonlocal perturbation

Luo

Yang

2021

Math Methods in App Sciences

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“…s u + |u| σ u = 0 is strongly unstable by blow up arguments, where N ≥ 2, N 2N −1 < s < 1 and 4s N < σ < 2s N −2s . From the results in [15,38], our assumption in (F 4 ) is optimal in a sense.…”

Section: Introductionmentioning

confidence: 98%

“…s u + |u| σ u = 0 with N ≥ 2 and 1 2 < s < 1. An immediate complement result is [15] in which the author proves that the ground state standing wave of i∂ t u − (−∆)…”

Section: Introductionmentioning

confidence: 99%

Stable standing waves of nonlinear fractional Schrödinger equations

Zhang

2022

CPAA

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“…Compared with [32], we use a constrained minimization method instead of a mini-max pro- For any ω > 0, the existence of ground state solution u ω to problem (1.2) has been studied in [11,18,31,42,50]. Next, we analyze the connection between the couple of weak solution (u c , ω c ) to (1.2) obtained in Theorem 1.2 and u ωc .…”

Section: Introductionmentioning

confidence: 99%

Normalized ground states for the fractional nonlinear Schrödinger equations

Feng,

Ren,

Wang

2019

Preprint

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In this paper, we study the existence and instability of standing waves with a prescribed L 2 -norm for the fractional Schrödinger equationTo this end, we look for normalized solutions of the associated stationary equationFirstly, by constructing a suitable submanifold of a L 2 -sphere, we prove the existence of a normalized solution for (0.2) with least energy in the L 2 -sphere, which corresponds to a normalized ground state standing wave of (0.1). Then, we show that each normalized ground state of (0.2) coincides a ground state of (0.2) in the usual sense. Finally, we obtain the sharp threshold of global existence and blow-up for (0.1). Moreover, we can use this sharp threshold to show that all normalized ground state standing waves are strongly unstable by blow-up.

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On instability of standing waves for the mass-supercritical fractional nonlinear Schrödinger equation

Cited by 14 publications

References 25 publications

Normalized solutions for the fractional Schrödinger equation with a focusing nonlocal perturbation

Normalized solutions for the fractional Schrödinger equation with a focusing nonlocal perturbation

Stable standing waves of nonlinear fractional Schrödinger equations

Normalized ground states for the fractional nonlinear Schrödinger equations

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