On a nonlinear Schrödinger-Bopp-Podolsky system in the zero mass case: functional framework and existence
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910071490871296 |
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| author | Caponio, Erasmo d'Avenia, Pietro Pomponio, Alessio Siciliano, Gaetano Yang, Lianfeng |
| author_facet | Caponio, Erasmo d'Avenia, Pietro Pomponio, Alessio Siciliano, Gaetano Yang, Lianfeng |
| contents | In this paper, we consider in $\mathbb{R}^3$ the following zero mass Schrödinger-Bopp-Podolsky system \[ \begin{cases} -Δu +q^2ϕu=|u|^{p-2}u\\ -Δϕ+a^2Δ^2ϕ=4πu^2 \end{cases} \] 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_09752 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a nonlinear Schrödinger-Bopp-Podolsky system in the zero mass case: functional framework and existence Caponio, Erasmo d'Avenia, Pietro Pomponio, Alessio Siciliano, Gaetano Yang, Lianfeng Analysis of PDEs 35J48, 35J50, 35Q60 In this paper, we consider in $\mathbb{R}^3$ the following zero mass Schrödinger-Bopp-Podolsky system \[ \begin{cases} -Δu +q^2ϕu=|u|^{p-2}u\\ -Δϕ+a^2Δ^2ϕ=4πu^2 \end{cases} \] 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. |
| title | On a nonlinear Schrödinger-Bopp-Podolsky system in the zero mass case: functional framework and existence |
| topic | Analysis of PDEs 35J48, 35J50, 35Q60 |
| url | https://arxiv.org/abs/2506.09752 |