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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2501.05983 |
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| _version_ | 1866913643660050432 |
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| author | Guo, Qidong He, Rui Hua, Qiaoqiao Wang, Qingfang |
| author_facet | Guo, Qidong He, Rui Hua, Qiaoqiao Wang, Qingfang |
| contents | We study the Schrödinger-Poisson-Slater equation \begin{equation*}\left\{\begin{array}{lll}
-Δu + λu + \big(|x|^{-1} \ast |u|^{2}\big)u = V(x) u^{ p_{\varepsilon}-1 }, \, \text{ in } \mathbb{R}^{3},\\[2mm]
\int_{\mathbb{R}^3}u^2 \,dx= a,\,\, u > 0,\,\, u \in H^{1}(\mathbb{R}^{3}),
\end{array}
\right. \end{equation*} where $λ$ is a Lagrange multiplier, $V(x)$ is a real-valued potential, $a\in \mathbb{R}_{+}$ is a constant, $ p_{\varepsilon} = \frac{10}{3} \pm \varepsilon$ and $\varepsilon>0$ is a small parameter. In this paper, we prove that it is the positive critical value of the potential $V$ that affects the existence of single-peak solutions for this problem. Furthermore, we prove the local uniqueness of the solutions we construct. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_05983 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Normalized Solutions for nonlinear Schrödinger-Poisson equations involving nearly mass-critical exponents Guo, Qidong He, Rui Hua, Qiaoqiao Wang, Qingfang Analysis of PDEs We study the Schrödinger-Poisson-Slater equation \begin{equation*}\left\{\begin{array}{lll} -Δu + λu + \big(|x|^{-1} \ast |u|^{2}\big)u = V(x) u^{ p_{\varepsilon}-1 }, \, \text{ in } \mathbb{R}^{3},\\[2mm] \int_{\mathbb{R}^3}u^2 \,dx= a,\,\, u > 0,\,\, u \in H^{1}(\mathbb{R}^{3}), \end{array} \right. \end{equation*} where $λ$ is a Lagrange multiplier, $V(x)$ is a real-valued potential, $a\in \mathbb{R}_{+}$ is a constant, $ p_{\varepsilon} = \frac{10}{3} \pm \varepsilon$ and $\varepsilon>0$ is a small parameter. In this paper, we prove that it is the positive critical value of the potential $V$ that affects the existence of single-peak solutions for this problem. Furthermore, we prove the local uniqueness of the solutions we construct. |
| title | Normalized Solutions for nonlinear Schrödinger-Poisson equations involving nearly mass-critical exponents |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.05983 |