The fibering method approach for a non-linear Schrödinger equation coupled with the electromagnetic field

Pub Date : 2018-06-13 DOI:10.5565/publmat6422001
G. Siciliano, K. Silva
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引用次数: 28

Abstract

We study, with respect to the parameter $q\neq0$, the following Schrodinger-Bopp-Podolsky system in $\mathbb R^{3}$ \begin{equation*} \left\{ \begin{aligned} -&\Delta u+\omega u+q^2\phi u=|u|^{p-2}u, \\ &-\Delta \phi+a^2\Delta^2 \phi = 4\pi u^2, \end{aligned} \right. \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 $q's$, 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 extremals values of $q$. Our results recover and improve some results in the literature.
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非线性电磁场耦合Schrödinger方程的纤维化方法
对于参数$q\neq0$,我们研究如下的$\mathbb R^{3}$\begin{equation*} \left\{ \begin{aligned} -&\Delta u+\omega u+q^2\phi u=|u|^{p-2}u, \\ &-\Delta \phi+a^2\Delta^2 \phi = 4\pi u^2, \end{aligned} \right. \end{equation*}中的薛定谔-波普-波多尔斯基系统,其中$p\in(2,3], \omega>0, a\geq0$是固定的。利用纤维化方法证明了系统在较大的$q's$值下无解,在较小的$q's$值下有两个径向解。我们还给出了解的能级的定性性质和$q$的这些极值的变分表征。我们的结果恢复和改进了文献中的一些结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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