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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2410.05271 |
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| _version_ | 1866912062431559680 |
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| author | Li, Ting Tang, Zhongwei Wang, Heming Zhang, Xiaojing |
| author_facet | Li, Ting Tang, Zhongwei Wang, Heming Zhang, Xiaojing |
| contents | In this paper, we study the following critical fractional Schrödinger equation: \begin{equation} (-Δ)^s u+V(|y'|,y'')u=K(|y'|,y'')u^{\frac{n+2s}{n-2s}},\quad u>0,\quad y =(y',y'') \in \mathbb{R}^3\times\mathbb{R}^{n-3}, \qquad(0.1)\end{equation} where $n\geq 3$, $s\in(0,1)$, $V(|y'|,y'')$ and $K(|y'|,y'')$ are two bounded nonnegative potential functions. Under the conditions that $K(r,y'')$ has a stable critical point $(r_0,y_0'')$ with $r_0>0$, $K(r_0,y_0'')>0$ and $V(r_0,y_0'')>0$, we prove that equation (0.1) has a new type of infinitely many solutions that concentrate at points lying on the top and the bottom of a cylinder. In particular, the bubble solutions can concentrate at a pair of symmetric points with respect to the origin. Our proofs make use of a modified finite-dimensional reduction method and local Pohozaev identities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_05271 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | New type of bubbling solutions to a critical fractional Schrödinger equation with double potentials Li, Ting Tang, Zhongwei Wang, Heming Zhang, Xiaojing Analysis of PDEs In this paper, we study the following critical fractional Schrödinger equation: \begin{equation} (-Δ)^s u+V(|y'|,y'')u=K(|y'|,y'')u^{\frac{n+2s}{n-2s}},\quad u>0,\quad y =(y',y'') \in \mathbb{R}^3\times\mathbb{R}^{n-3}, \qquad(0.1)\end{equation} where $n\geq 3$, $s\in(0,1)$, $V(|y'|,y'')$ and $K(|y'|,y'')$ are two bounded nonnegative potential functions. Under the conditions that $K(r,y'')$ has a stable critical point $(r_0,y_0'')$ with $r_0>0$, $K(r_0,y_0'')>0$ and $V(r_0,y_0'')>0$, we prove that equation (0.1) has a new type of infinitely many solutions that concentrate at points lying on the top and the bottom of a cylinder. In particular, the bubble solutions can concentrate at a pair of symmetric points with respect to the origin. Our proofs make use of a modified finite-dimensional reduction method and local Pohozaev identities. |
| title | New type of bubbling solutions to a critical fractional Schrödinger equation with double potentials |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2410.05271 |