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

Abstract

We study, with respect to the parameter $q\ne 0$, the following Schrödinger-Bopp-Podolsky system in ${\mathbb{R}}^3$ \begin{equation*} \left\{ \begin{aligned} -&Δu+ωu+q^2ϕu=|u|^{p-2}u, \\ &-Δϕ+a^2Δ^2 ϕ= 4πu^2, \end{aligned} \right. \end{equation*} where $p\in (2,3],\omega >0,a\ge 0$ are fixed. We prove, by means of the fibering approach, that the system has no solutions at all for large values of $q^{\prime }s$, and has two radial solutions for small $q^{\prime }s$. 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.