Existence and local uniqueness of multi-peak solutions for the Chern-Simons-Schrödinger system
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| Format: | Preprint |
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2022
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| author | Hua, Qiaoqiao Wang, Chunhua Yang, Jing |
| author_facet | Hua, Qiaoqiao Wang, Chunhua Yang, Jing |
| contents | In the present paper, we consider the Chern-Simons-Schrödinger system \begin{equation} \left\{ \begin{aligned} &-\varepsilon^{2}Δu+V(x)u+(A_{0}+A_{1}^{2}+A_{2}^{2})u=|u|^{p-2}u,\,\,\,\,x\in \mathbb{R}^2,\\ &\partial_1 A_0 = A_2 u^2,\ \partial_{2}A_{0}=-A_{1}u^{2},\\ &\partial_{1}A_{2}-\partial_{2}A_{1}=-\frac{1}{2}|u|^{2},\ \partial_{1}A_{1}+\partial_{2}A_{2}=0,\\ \end{aligned} \right. \end{equation} where $p>2,$ $\varepsilon>0$ is a parameter and $V:\mathbb{R}^{2}\rightarrow\mathbb{R}$ is a bounded continuous function. Under some mild assumptions on $V(x)$, we show the existence and local uniqueness of positive multi-peak solutions. Our methods mainly use the finite dimensional reduction method, various local Pohozaev identities, blow-up analysis and the maximum principle. Because of the nonlocal terms involved by $A_{0},A_{1}$ and $A_{2},$ we have to obtain a series of new and technical estimates. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_17427 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Existence and local uniqueness of multi-peak solutions for the Chern-Simons-Schrödinger system Hua, Qiaoqiao Wang, Chunhua Yang, Jing Analysis of PDEs In the present paper, we consider the Chern-Simons-Schrödinger system \begin{equation} \left\{ \begin{aligned} &-\varepsilon^{2}Δu+V(x)u+(A_{0}+A_{1}^{2}+A_{2}^{2})u=|u|^{p-2}u,\,\,\,\,x\in \mathbb{R}^2,\\ &\partial_1 A_0 = A_2 u^2,\ \partial_{2}A_{0}=-A_{1}u^{2},\\ &\partial_{1}A_{2}-\partial_{2}A_{1}=-\frac{1}{2}|u|^{2},\ \partial_{1}A_{1}+\partial_{2}A_{2}=0,\\ \end{aligned} \right. \end{equation} where $p>2,$ $\varepsilon>0$ is a parameter and $V:\mathbb{R}^{2}\rightarrow\mathbb{R}$ is a bounded continuous function. Under some mild assumptions on $V(x)$, we show the existence and local uniqueness of positive multi-peak solutions. Our methods mainly use the finite dimensional reduction method, various local Pohozaev identities, blow-up analysis and the maximum principle. Because of the nonlocal terms involved by $A_{0},A_{1}$ and $A_{2},$ we have to obtain a series of new and technical estimates. |
| title | Existence and local uniqueness of multi-peak solutions for the Chern-Simons-Schrödinger system |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2210.17427 |