标题： 关于零质量情形下的非线性 Schrödinger-Bopp-Podolsky 系统：函数框架与存在性 显示英文标题

标题： On a nonlinear Schrödinger-Bopp-Podolsky system in the zero mass case: functional framework and existence

摘要： 本文中，我们在$\mathbb{R}^3$中考虑如下零质量的薛定谔-玻普-波多尔斯基系统\[ \begin{cases} -\Delta u +q^2\phi u=|u|^{p-2}u\\ -\Delta \phi+a^2\Delta^2\phi=4\pi u^2 \end{cases} \]，其中 $a>0$, $q\ne 0$和 $p\in (3,6)$。 受 [Ruiz, Arch. Ration. Mech. Anal. 198 (2010)] 启发，我们引入了一个带有包含非局部项范数的索伯列夫空间$\mathcal{E}$。首先，我们提供了空间$\mathcal{E}$的一些基本性质，包括嵌入到勒贝格空间中的性质。此外，获得了玻普-波多尔斯基能量的一个一般下界。基于这些事实，通过应用摄动论证，最终证明了上述系统的弱解的存在性。 显示英文摘要

摘要： 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.