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*} \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.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2018
- DOI:
- 10.48550/arXiv.1806.05260
- arXiv:
- arXiv:1806.05260
- Bibcode:
- 2018arXiv180605260S
- Keywords:
-
- Mathematics - Analysis of PDEs;
- 35A02;
- 35J50;
- 35J91;
- 35Q60