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2020 The fibering method approach for a non-linear Schrödinger equation coupled with the electromagnetic field
Gaetano Siciliano, Kaye Silva
Publ. Mat. 64(2): 373-390 (2020). DOI: 10.5565/PUBLMAT6422001

Abstract

We study, with respect to the parameter $q\neq0$, the following Schrödinger-Bopp-Podolsky system in $\mathbb R^{3}$ \begin{equation*} \begin{cases} -\Delta u+\omega u+q^2\phi u=|u|^{p-2}u, \\ -\Delta \phi+a^2\Delta^2 \phi = 4\pi u^2, \end{cases} \end{equation*} where $p\in(2,3]$, $\omega>0$, $a\geq0$ are fixed. We prove, by means of the fibering approach, that the system has no solutions at all for large values of $q$ and has two radial solutions for small $q$'s. We give also qualitative properties about the energy level of the solutions and a variational characterization of these extremal values of $q$. Our results recover and improve some results in [2,5].

Citation

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Gaetano Siciliano. Kaye Silva. "The fibering method approach for a non-linear Schrödinger equation coupled with the electromagnetic field." Publ. Mat. 64 (2) 373 - 390, 2020. https://doi.org/10.5565/PUBLMAT6422001

Information

Received: 2 July 2018; Revised: 23 January 2019; Published: 2020
First available in Project Euclid: 3 July 2020

zbMATH: 07236047
MathSciNet: MR4119258
Digital Object Identifier: 10.5565/PUBLMAT6422001

Subjects:
Primary: 35A02 , 35J50 , 35J91 , 35Q60

Keywords: fibering methods , Nehari manifold , Schrödinger-Poisson type system , variational methods

Rights: Copyright © 2020 Universitat Autònoma de Barcelona, Departament de Matemàtiques

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Vol.64 • No. 2 • 2020
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