On a nonlinear Schr\"odinger-Bopp-Podolsky system in the zero mass case: functional framework and existence

TL;DR

This paper establishes a functional framework and proves the existence of weak solutions for a nonlinear Schrödinger-Bopp-Podolsky system in three-dimensional space, involving nonlocal terms and Sobolev space embeddings.

Contribution

It introduces a new Sobolev space with a nonlocal norm and proves the existence of solutions for the zero mass Schrödinger-Bopp-Podolsky system.

Findings

Defined a Sobolev space with nonlocal terms

Established embeddings into Lebesgue spaces

Proved existence of weak solutions

Abstract

In this paper, we consider in $\mathbb{R}^3$ the following zero mass Schr\"odinger-Bopp-Podolsky system \[ \begin{cases} -\Delta u +q^2\phi u=|u|^{p-2}u\\ -\Delta \phi+a^2\Delta^2\phi=4\pi u^2 \end{cases} \] where $a>0$, $q\ne 0$ and $p\in (3,6)$. Inspired by [Ruiz, Arch. Ration. Mech. Anal. 198 (2010)], we introduce a Sobolev space $\mathcal{E}$ endowed with a norm containing a nonlocal term. Firstly, we provide some fundamental properties for the space $\mathcal{E}$ including embeddings into Lebesgue spaces. Moreover a general lower bound for the Bopp-Podolsky energy is obtained. Based on these facts, by applying a perturbation argument, we finally prove the existence of a weak solution to the above system.