Normalized solutions for nonlinear Schrödinger systems

Bartsch, Thomas; Jeanjean, Louis

doi:10.1017/s0308210517000087

Cited by 97 publications

(92 citation statements)

References 36 publications

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“…[2,3,4,6,7,8,18,19,20,27,28,36]. In [18], Jeanjean considered the following semi-linear Schrödinger equation:…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Normalized solutions for the Chern–Simons–Schrödinger equation in R^2

Li¹,

Luo²

2017

Ann. Acad. Sci. Fenn. Math.

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Abstract. In this paper, we study the existence and multiplicity of solutions with a prescribed L 2 -norm for a class of nonlinear Chern-Simons-Schrödinger equations in RTo get such solutions we look for critical points of the energy functional, we prove a sufficient condition for the nonexistence of constrain critical points of I on S r (c) for certain c and get infinitely many minimizers of I on S r (8π). For the value p ∈ (4, +∞) considered, the functional I is unbounded from below on S r (c). By using the constrained minimization method on a suitable submanifold of S r (c), we prove that for certain c > 0, I has a critical point on S r (c). After that, we get an H 1 -bifurcation result of our problem. Moreover, by using a minimax procedure, we prove that there are infinitely many critical points of I restricted on S r (c) for any c ∈ 0,.

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“…[2,3,4,6,7,8,18,19,20,27,28,36]. In [18], Jeanjean considered the following semi-linear Schrödinger equation:…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Recently, Bartsch et al considered normalized solutions to the nonlinear Schrö-dinger systems in [2,3]. In [3], the following coupled cubic Schrödinger systems was considered:…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Normalized solutions for the Chern–Simons–Schrödinger equation in R^2

Li¹,

Luo²

2017

Ann. Acad. Sci. Fenn. Math.

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“…Remark 3.5. In [3], (3.13) was not observed and the fact that the weak limit belongs to S r (a 1 , a 2 ) was proved using Liouville's type arguments, as developed in [14], see also [7,13]. It is the use of these arguments, which induces the restriction on the dimension N in [3, Theorem 2.1].…”

Section: Lemma 33 Any Minimizing Sequence For (14) Is Up To Transmentioning

confidence: 99%

“…In [7] the existence of one minimizer had been achieved still for N = 1. The restriction on the dimension was subsequently removed in [3] where the existence of a minimizer for (1.4) was obtained in full generality in H 1 (R N ) for N = 2, 3, 4 and under some restrictions for N ≥ 5, see [3, Theorem 2.1] for a precise statement.…”

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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“…Later, under the assumptions

false(H_{1} false)

and

false(H_{2} false)

, Bartsch and De Valeriola obtained the existence of infinitely many critical points for the following functional come from :

Ē false(u false) = \frac{1}{2} \int_{{double-struckR}^{N}} false| \nabla u {false|}^{2} - \int_{{double-struckR}^{N}} F false(u false)

on the constraint

S_{r} false(c false) = ({}, u \in H_{r}^{1} false({double-struckR}^{3} false) : false| false| u false| {false|}_{2}^{2} = c)

for

c > 0

. In addition, we refer the readers to the literature about the normalized solutions for nonlinear Schrödinger systems.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Existence and multiplicity of normalized solutions for a class of Schrödinger‐Poisson equations with general nonlinearities

Xie

Chen

2019

Math Methods in App Sciences

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In this paper, we study the existence and multiplicity of standing waves with prescribed L2‐norm Schrödinger‐Poisson equations with general nonlinearities in double-struckR3: i∂tψ+Δψ−κ(|x|−1*|φ|2)ψ+f(ψ)=0, where κ>0 and f is superlinear and satisfies the monotonicity condition. To this end, we look for critical points of the following functional Eκ(u)=12∫R3|∇u|2+κ4∫R3(|x|−1*u2)u2−∫R3F(u) constrained on the L2‐spheres Sfalse(cfalse)={}u∈H1false(double-struckR3false):false|false|ufalse|false|22=c,1emc>0, where Ffalse(sfalse):=∫0sffalse(tfalse)dt. We consider the case where Eκ is unbounded from below on Sfalse(cfalse) and establish the existence of critical points of Eκ on Sfalse(cfalse) for c>0 sufficiently small and under some mild assumptions on f. In addition, we show that there are infinitely many radial critical points false{unκfalse} of Eκ on Sfalse(cfalse) when f is odd and present a convergence property of unκ as κ↘0. Our results generalize some recent results in the literature.

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Normalized solutions for nonlinear Schrödinger systems

Cited by 97 publications

References 36 publications

Normalized solutions for the Chern–Simons–Schrödinger equation in R^2

Normalized solutions for the Chern–Simons–Schrödinger equation in R^2

Existence and orbital stability of standing waves for nonlinear Schrödinger systems

Existence and multiplicity of normalized solutions for a class of Schrödinger‐Poisson equations with general nonlinearities

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