The fibering method approach for a non-linear Schr\"odinger equation coupled with the electromagnetic field

TL;DR

This paper applies the fibering method to analyze a nonlinear Schrödinger system coupled with electromagnetic fields, establishing existence and non-existence results for solutions depending on the parameter q, and providing qualitative insights.

Contribution

The study introduces the fibering approach to this coupled system, proving new existence and non-existence results and characterizing extremal parameter values, improving prior findings.

Findings

No solutions for large q values.

Two radial solutions for small q values.

Qualitative energy properties of solutions.

Abstract

We study, with respect to the parameter $q\neq0$, the following Schr\"odinger-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.