On a class of planar Schrödinger-Poisson system with a bounded potential well
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arXiv
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| Formato: | Preprint |
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2023
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| _version_ | 1866913402235912192 |
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| author | Du, Miao Xu, Jiaxin |
| author_facet | Du, Miao Xu, Jiaxin |
| contents | In this paper, we deal with the planar Schrödinger-Poisson system \begin{equation*}\begin{cases} -Δu + V(x) u + ϕu = b|u|^{p-2} u \ &\text{in}\ \mathbb{R}^{2},\\Δϕ= u^{2} &\text{in}\ \mathbb{R}^{2},\end{cases} \end{equation*} where $b \geq 0$, $p > 2 $ and $V \in C(\mathbb{R}^2, \mathbb{R})$ is a potential function with $\inf_{\mathbb{R}^2} V >0$. Suppose moreover that $V$ exhibits a bounded potential well in the sense that $\lim_{|x|\rightarrow \infty} V(x)$ exists and is equal to $\sup_{\mathbb{R}^2} V$. By using variational methods, we obtain the existence of ground state solutions for this system in the case where $p \geq 3$. Furthermore, we also present a minimax characterization of ground state solutions. The main feature of this work is that we do not assume any periodicity or symmetry condition on the external potential $V$, which is essential to establish the compactness condition of Cerami sequences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_07265 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On a class of planar Schrödinger-Poisson system with a bounded potential well Du, Miao Xu, Jiaxin Analysis of PDEs 35J50, 35Q40 In this paper, we deal with the planar Schrödinger-Poisson system \begin{equation*}\begin{cases} -Δu + V(x) u + ϕu = b|u|^{p-2} u \ &\text{in}\ \mathbb{R}^{2},\\Δϕ= u^{2} &\text{in}\ \mathbb{R}^{2},\end{cases} \end{equation*} where $b \geq 0$, $p > 2 $ and $V \in C(\mathbb{R}^2, \mathbb{R})$ is a potential function with $\inf_{\mathbb{R}^2} V >0$. Suppose moreover that $V$ exhibits a bounded potential well in the sense that $\lim_{|x|\rightarrow \infty} V(x)$ exists and is equal to $\sup_{\mathbb{R}^2} V$. By using variational methods, we obtain the existence of ground state solutions for this system in the case where $p \geq 3$. Furthermore, we also present a minimax characterization of ground state solutions. The main feature of this work is that we do not assume any periodicity or symmetry condition on the external potential $V$, which is essential to establish the compactness condition of Cerami sequences. |
| title | On a class of planar Schrödinger-Poisson system with a bounded potential well |
| topic | Analysis of PDEs 35J50, 35Q40 |
| url | https://arxiv.org/abs/2312.07265 |