Minimizers of the planar Schrödinger–Newton equations

Wang, Wenbo; Zhang, Wei

doi:10.1080/17476933.2020.1816987

Cited by 3 publications

(2 citation statements)

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“…and φ p or lφ p e iθ is the unique (up to translations) optimizer, where l ∈ R \ {0}, θ ∈ [2kπ, 2(k + 1)π], k = 0, ±1, ±2, .... Similar to the paper [5] or [12], we divide it into three cases:…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Some remarks on the magnetic field operators \nabla±iA and its applications

WANG

2024

Proc. Rom. Acad. Ser. A - Math. Phys. Tech. Sci. Inf. Sci.

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In the present paper, we give some remarks on the magnetic field operators $\nabla \pm iA$. As its applications, we study the Schr\"{o}dinger equation with a magnetic field \begin{equation*} -\Delta u+|A(x)|^{2}u+iA(x)\cdot \nabla u=\mu u+|u|^{p}u,~x\in \mathbb{R}^{N}, \end{equation*} where $u$ is a complex-valued function and $\mu\in \mathbb{R}$. When $N>2$, for $2 p+2 \frac{2N}{N-2}$ or $N=2$, for $2 p+2 +\infty$, the existence and nonexistence of minimizers of the corresponding minimization problem are given via constrained variational methods. As a by-product, the above equation admits a normalized solution. We point out that the condition ${div}A(x)=0$ plays a crucial role in our study.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Some remarks on the magnetic field operators \nabla±iA and its applications

WANG

2024

Proc. Rom. Acad. Ser. A - Math. Phys. Tech. Sci. Inf. Sci.

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“…Over the past few decades, the system (1.1) has attracted considerable attention due to its physical relevance (cf. [3,4,6,7,8,18,23,27,31,32,33]). More specifically, taking no account of the nonlinear term |ψ| 2 ψ and the external potential V (x) in (1.1), Harrison, Moroz and Tod [23] established the existence of bound states for (1.1) by numerical study, after which, applying a shooting method, Choquard, Stubbe and Vuffray [6] proved the existence of a unique positive radially symmetric solution for (1.1).…”

Section: Introductionmentioning

confidence: 99%

Core Content Selection and Textbook Design for High School Mathematics

Zhang¹,

Zhou²,

Wang³

et al. 2021

School Mathematics Textbooks in China

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This paper is devoted to studying the following fractional L 2 -critical nonlinear Schrödinger equation≥ 0 is an external potential. We obtain normalized L 2 -norm solutions of the above equation by solving the associated constraint minimization problem (1.4). It shows that there is a threshold a * > 0 such that (1.4) has minimizers for 0 < a < a * , and minimizers do not exist for any a > a * . For the case of a = a * , it gives a fact that the existence and non-existence of minimizers depend strongly on the value of V (0). Especially for V (0) = 0, we prove that minimizers occur blow-up behavior and the mass of minimizers concentrates at the origin as a ր a * . Applying implicit function theorem, the uniqueness of minimizers is also proved for a > 0 small enough.

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Existence and uniqueness of constraint minimizers for the planar Schrödinger-Poisson system with logarithmic potentials

Guo

Liang

2023

Journal of Differential Equations

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Minimizers of the planar Schrödinger–Newton equations

Cited by 3 publications

References 23 publications

Some remarks on the magnetic field operators \nabla±iA and its applications

Some remarks on the magnetic field operators \nabla±iA and its applications

Core Content Selection and Textbook Design for High School Mathematics

Existence and uniqueness of constraint minimizers for the planar Schrödinger-Poisson system with logarithmic potentials

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