Stable standing waves of nonlinear Schrödinger equations with a general nonlinear term

Shibata, Masataka

doi:10.1007/s00229-013-0627-9

Cited by 113 publications

(78 citation statements)

References 12 publications

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“…Nowadays it is still the object of an intense activity. We refer, in particular, to [9,10,11,16,27,28]. In these works the authors are mainly interested in the existence of ground states, namely of solutions to (P m ) which can be characterized as minimizers of I among all the solutions.…”

Section: Introductionmentioning

confidence: 99%

“…An emphasize is also given to the issue of stability of these solutions, as standing waves of (1.1). This is done either, following the strategy laid down in [9], by showing that any minimizing sequences of I on S m is precompact up to translations [11,27] or by using more analytic approaches [28]. Likely, as far as the existence of ground states and their orbital stability is concerned, the most general result is contained in [27].…”

Section: Introductionmentioning

confidence: 99%

“…Here again it is essential to know in advance that the suspected critical level is negative. Remark 1.1 If the stability of the ground states, as studied for example in [11,27,28], is by now relatively well understood, the issue of the orbital stability, or more likely orbital instability, of the other critical points of I restricted to S m is still totally open. We believe an interesting but challenging question would be to prove that the solution obtained in Theorem 1.1, which enjoys a well defined variational characterization, is orbitally unstable.…”

Section: Introductionmentioning

confidence: 99%

“…When we do not have(1.2), this number can be positive. To see this, following closely[27], we assume, in addition to (f 1) − (f 5), that…”

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confidence: 99%

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Nonradial normalized solutions for nonlinear scalar field equations

Jeanjean

2019

Nonlinearity

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We study the following nonlinear scalar field equationHere f ∈ C(R, R), m > 0 is a given constant and µ ∈ R arises as a Lagrange multiplier. In a mass subcritical case but under general assumptions on the nonlinearity f , we show the existence of one nonradial solution for any N ≥ 4, and obtain multiple (sometimes infinitely many) nonradial solutions when N = 4 or N ≥ 6. In particular, all these solutions are sign-changing. MSC: 35J60, 58E05

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…When we do not have(1.2), this number can be positive. To see this, following closely[27], we assume, in addition to (f 1) − (f 5), that…”

mentioning

confidence: 99%

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Nonradial normalized solutions for nonlinear scalar field equations

Jeanjean

2019

Nonlinearity

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“…Most are based on some homogeneity type property. In autonomous case it is also possible to use scaling arguments, see for example [4,10,26]. In the case of multiple constraints how to establish strict subadditivity conditions is much less understood.…”

Section: Introductionmentioning

confidence: 99%

Existence and orbital stability of standing waves for nonlinear Schrödinger systems

Gou

Jeanjean

2016

Nonlinear Analysis

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In this paper we investigate the existence of solutions in H 1 (R N ) × H 1 (R N ) for nonlinear Schrödinger systems of the formunder the constraints4 N for i = 1, 2 and r 1 +r 2 < 2+ 4 N . This problem is motivated by the search of standing waves for an evolution problem appearing in several physical models. Our solutions are obtained as constrained global minimizers of an associated functional. Note that in the system λ 1 and λ 2 are unknown and will correspond to the Lagrange multipliers. Our main result is the precompactness of the minimizing sequences, up to translation. Assuming the local well posedness of the associated evolution problem we then obtain the orbital stability of the standing waves associated to the set of minimizers.

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Normalized solutions of the Schrödinger equation with potential

Zhao,

Zou

2023

Mathematische Nachrichten

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In this paper, for dimension and prescribed mass , we consider the following nonlinear scalar field equation with L2 constraint: where is a Lagrange multiplier, . In particular, satisfies mass supercritical and Sobolev subcritical growth. We prove the existence results of the normalized solution and infinitely many normalized solutions to the above system under some proper assumptions on the functions by the mountain pass argument.

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Stable standing waves of nonlinear Schrödinger equations with a general nonlinear term

Cited by 113 publications

References 12 publications

Nonradial normalized solutions for nonlinear scalar field equations

Nonradial normalized solutions for nonlinear scalar field equations

Existence and orbital stability of standing waves for nonlinear Schrödinger systems

Normalized solutions of the Schrödinger equation with potential

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