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arXiv · 2506.09752

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

Erasmo Caponio·Pietro d'Avenia·Alessio Pomponio·Gaetano Siciliano·Lianfeng Yang
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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.

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Erasmo Caponio, Pietro d'Avenia, Alessio Pomponio, Gaetano Siciliano, Lianfeng Yang. 2025-06-11. On a nonlinear Schr\"odinger-Bopp-Podolsky system in the zero mass case: functional framework and existence. https://arxiv.org/abs/2506.09752

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