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

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].