Global dynamics above the ground state energy for the combined power-type nonlinear Schrödinger equations with energy-critical growth at low frequencies

Akahori, Takafumi; Ibrahim, Slim; Kikuchi, Hiroaki; Nawa, Hayato

doi:10.1090/memo/1331

Cited by 14 publications

(33 citation statements)

References 27 publications

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“…About the uniqueness and nondegeneracy of ground states to (1.1), these properties were proved in [2] under the assumptions d ≥ 4, 1 + 4 d < p < 2 * − 1 and ω ≪ 1. We also mention that the papers [17,18] studied the uniqueness and nondegeneracy of ground states to equations in bounded domains with single power type nonlinearity whose exponent is the critical one or close to it.…”

Section: Introductionmentioning

confidence: 98%

“…When the energy of initial data is less than the ground state energy, only two scenarios can happen: finite time blow-up or scattering. For example, we refer to [1,2,20,21,28]. However, when the energy of initial data is slightly greater than the ground state energy, the dynamic is much more complicated, and the combination of finite time blow-up, scattering and nondispersion behaviors are shown in forward or backward in time.…”

Section: Introductionmentioning

confidence: 99%

“…However, when the energy of initial data is slightly greater than the ground state energy, the dynamic is much more complicated, and the combination of finite time blow-up, scattering and nondispersion behaviors are shown in forward or backward in time. We refer to [2,29,30] for more details. In studying the dynamics around the ground state, basic properties of ground state such as the uniqueness and nondegeneracy play a crucial role.…”

Section: Introductionmentioning

confidence: 99%

“…For the sake of clarity and self-content, a sketch of the proof of Proposition 1.1 will be given in Appendix A, using simpler arguments than those in [3,38]. Using a standard argument for semilinear elliptic equation (see [2,15,24]), we can derive the following properties of the ground states to (1.1): Proposition 1.2. Assume either d = 3 and 3 < p < 5 or else d ≥ 4 and 1 < p < d+2 d−2 .…”

Section: Introductionmentioning

confidence: 99%

“…In [2], for ω ≪ 1 and Φ ω ∈ G ω , we use the following rescaling corresponding to the subcritical power p:…”

Section: Introductionmentioning

confidence: 99%

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Uniqueness and nondegeneracy of ground states to nonlinear scalar field equations involving the Sobolev critical exponent in their nonlinearities for high frequencies

et al. 2019

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The study of the uniqueness and nondegeneracy of ground state solutions to semilinear elliptic equations is of great importance because of the resulting energy landscape and its implications for the various dynamics. In [2], semilinear elliptic equations with combined power-type nonlinearities involving the Sobolev critical exponent are studied. There, it is shown that if the dimension is four or higher, and the frequency is sufficiently small, then the positive radial ground state is unique and nondegenerate. In this paper, we extend these results to the case of high frequencies when the dimension is five and higher. After suitably rescaling the equation, we demonstrate that the main behavior of the solutions is given by the Sobolev critical part for which the ground states are explicit, and their degeneracy is well characterized. Our result is a key step towards the study of the different dynamics of solutions of the corresponding nonlinear Schrödinger and Klein-Gordon equations with energies above the energy of the ground state. Our restriction on the dimension is mainly due to the existence of resonances in dimension three and four.

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Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In [2], for ω ≪ 1 and Φ ω ∈ G ω , we use the following rescaling corresponding to the subcritical power p:…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Uniqueness and nondegeneracy of ground states to nonlinear scalar field equations involving the Sobolev critical exponent in their nonlinearities for high frequencies

et al. 2019

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Uniqueness of ground states for combined power-type nonlinear scalar field equations involving the Sobolev critical exponent at high frequencies in three and four dimensions

Akahori

Murata

2022

Nonlinear Differ. Equ. Appl.

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Asymptotic profiles for a nonlinear Schrödinger equation with critical combined powers nonlinearity

Moroz

2023

Math. Z.

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We study asymptotic behaviour of positive ground state solutions of the nonlinear Schrödinger equation $$\begin{aligned} -\Delta u+u=u^{2^*-1}+\lambda u^{q-1} \quad \textrm{in}\, {\mathbb {R}}^N,\qquad \qquad \qquad \qquad \qquad {(P_\lambda )} \end{aligned}$$ - Δ u + u = u 2 ∗ - 1 + λ u q - 1 in R N , ( P λ ) where $$N\ge 3$$ N ≥ 3 is an integer, $$2^{*}=\frac{2N}{N-2}$$ 2 ∗ = 2 N N - 2 is the Sobolev critical exponent, $$2<q<2^*$$ 2 < q < 2 ∗ and $$\lambda >0$$ λ > 0 is a parameter. It is known that as $$\lambda \rightarrow 0$$ λ → 0 , after a rescaling the ground state solutions of $$(P_\lambda )$$ ( P λ ) converge to a particular solution of the critical Emden-Fowler equation $$-\Delta u=u^{2^*-1}$$ - Δ u = u 2 ∗ - 1 . We establish a novel sharp asymptotic characterisation of such a rescaling, which depends in a non-trivial way on the space dimension $$N=3$$ N = 3 , $$N=4$$ N = 4 or $$N \ge 5$$ N ≥ 5 . We also discuss a connection of these results with a mass constrained problem associated to $$(P_{\lambda })$$ ( P λ ) . Unlike previous work of this type, our method is based on the Nehari-Pohožaev manifold minimization, which allows to control the $$L^{2}$$ L 2 norm of the groundstates.

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Global dynamics above the ground state energy for the combined power-type nonlinear Schrödinger equations with energy-critical growth at low frequencies

Abstract: We consider the combined power-type nonlinear Schrödinger equations with energy-critical growth, and study the solutions slightly above the ground state threshold at low frequencies, so that we obtain a so-called nine-set theory developed by Nakanishi and Schlag.

Cited by 14 publications

References 27 publications

Uniqueness and nondegeneracy of ground states to nonlinear scalar field equations involving the Sobolev critical exponent in their nonlinearities for high frequencies

Uniqueness and nondegeneracy of ground states to nonlinear scalar field equations involving the Sobolev critical exponent in their nonlinearities for high frequencies

Uniqueness of ground states for combined power-type nonlinear scalar field equations involving the Sobolev critical exponent at high frequencies in three and four dimensions

Asymptotic profiles for a nonlinear Schrödinger equation with critical combined powers nonlinearity

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