Normalized Solutions of Nonlinear Schrödinger Equations with Potentials and Non-autonomous Nonlinearities

Yang, Zuo; Qi, Shijie; Zou, Wenming

doi:10.1007/s12220-022-00897-0

Cited by 23 publications

(7 citation statements)

References 25 publications

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“…As to the studies about the multiplicity of normalized solutions to the non-autonomous Schrödinger equation, Yang et al. [42] studied Equation (1.6) with f being L 2 -subcritical and satisfying Berestycki–Lions type conditions. Alves [1] considered the multiplicity of normalized solutions to with f being L 2 -subcritical.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Multiplicity and stability of normalized solutions to non-autonomous Schrödinger equation with mixed non-linearities

Li,

Xu,

Zhu

2023

Proceedings of the Edinburgh Mathematical Society

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This paper first studies the multiplicity of normalized solutions to the non-autonomous Schrödinger equation with mixed nonlinearities \begin{equation*} \begin{cases} -\Delta u=\lambda u+h(\epsilon x)|u|^{q-2}u+\eta |u|^{p-2}u,\quad x\in \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2\,\textrm{d}x=a^2, \end{cases} \end{equation*} where $a, \epsilon, \eta \gt 0$ , q is L2-subcritical, p is L2-supercritical, $\lambda\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier and h is a positive and continuous function. It is proved that the numbers of normalized solutions are at least the numbers of global maximum points of h when ϵ is small enough. The solutions obtained are local minimizers and probably not ground state solutions for the lack of symmetry of the potential h. Secondly, the stability of several different sets consisting of the local minimizers is analysed. Compared with the results of the corresponding autonomous equation, the appearance of the potential h increases the number of the local minimizers and the number of the stable sets. In particular, our results cover the Sobolev critical case $p=2N/(N-2)$ .

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Multiplicity and stability of normalized solutions to non-autonomous Schrödinger equation with mixed non-linearities

Li,

Xu,

Zhu

2023

Proceedings of the Edinburgh Mathematical Society

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“…On the other hand, the solvability of (1) with various nonconstant potentials and general nonlinearities is rather poorly understood so far. When ε = 1, [29,41] give the existence of solutions for L 2 -subcritical case under the assumption lim x→∞ V (x) = V ∞ ≥ V ̸ ≡ V ∞ ; [23] considered with similar potential assumption and Ambrosetti-Rabinowitz type conditions on nonlinearity in the L 2 -supercritical case; and [3] studied the L 2 -supercritical problem with a power type nonlinearity f (u) = u p−1 and a positive potential vanishing at infinity. We also note that [1] studied solutions of multibump type with periodic assumptions under a strict nondegeneracy condition.…”

Section: Introduction and Main Resultsmentioning

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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“…It is easy to verify that I(u) is a well-defined and C 1 functional on H 1 (R 3 ). Recently, there are numerous contributions flourishing within this topic, for instance, see [2,3,4,5,10,18,20,27,28,29,42,43,45].…”

Section: Introductionmentioning

confidence: 99%

Normalized solutions for Sobolev critical Schrodinger-Bopp-Podolsky systems

Li,

Chang,

Feng

2023

ejde

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We study the Sobolev critical Schrodinger-Bopp-Podolsky system $$\displaylines{ -\Delta u+\phi u=\lambda u+\mu|u|^{p-2}u+|u|^4u\quad \text{in }\mathbb{R}^3,\cr -\Delta\phi+\Delta^2\phi=4\pi u^2\quad \text{in } \mathbb{R}^3, }$$ under the mass constraint $\int_{\mathbb{R}^3}u^2\,dx=c $ for some prescribed $c>0$, where $2<p<8/3$, $\mu>0$ is a parameter, and $\lambda\in\mathbb{R}$ is a Lagrange multiplier. By developing a constraint minimizing approach, we show that the above system admits a local minimizer. Furthermore, we establish the existence of normalized ground state solutions. For more inofrmation see https://ejde.math.txstate.edu/Volumes/2023/56/abstr.html

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Normalized Solutions of Nonlinear Schrödinger Equations with Potentials and Non-autonomous Nonlinearities

Cited by 23 publications

References 25 publications

Multiplicity and stability of normalized solutions to non-autonomous Schrödinger equation with mixed non-linearities

Multiplicity and stability of normalized solutions to non-autonomous Schrödinger equation with mixed non-linearities

Core Content Selection and Textbook Design for High School Mathematics

Normalized solutions for Sobolev critical Schrodinger-Bopp-Podolsky systems

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