Nonlinear Schr\"odinger equation in the Bopp-Podolsky electrodynamics: solutions in the electrostatic case

TL;DR

This paper investigates the existence and nonexistence of solutions to a nonlinear Schr"odinger-Bopp-Podolsky system in three dimensions, analyzing parameter effects and the limiting behavior as the Bopp-Podolsky parameter approaches zero.

Contribution

It provides new existence and nonexistence results for the nonlinear Schr"odinger-Bopp-Podolsky system and explores the asymptotic connection to the classical Schr"odinger-Poisson system.

Findings

Solutions exist depending on parameters q and p.

Solutions tend to classical Schr"odinger-Poisson solutions as a→0.

Nonexistence results are established for certain parameter regimes.

Abstract

We study the following nonlinear Schr\"odinger-Bopp-Podolsky system \[ \begin{cases} -\Delta u + \omega u + q^{2}\phi u = |u|^{p-2}u -\Delta \phi + a^2 \Delta^2 \phi = 4\pi u^2 \end{cases} \hbox{ in }\mathbb{R}^3 \] with $a,\omega>0$. We prove existence and nonexistence results depending on the parameters $q,p$. Moreover we also show that, in the radial case, the solutions we find tend to solutions of the classical Schr\"odinger-Poisson system as $a\to0$.