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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| Accès en ligne: | https://arxiv.org/abs/2307.02272 |
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| _version_ | 1866916429980237824 |
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| author | Du, Fan Hua, Qiaoqiao Wang, Chunhua |
| author_facet | Du, Fan Hua, Qiaoqiao Wang, Chunhua |
| contents | We consider the following critical fractional Schrödinger equation \begin{equation*} (-Δ)^s u+V(|y'|,y'')u = u^{2_s^*-1},\quad u>0,\quad y =(y',y'') \in \mathbb{R}^3\times\mathbb{R}^{N-3}, \end{equation*} where $N\geq 3,s\in(0,1)$, $2_s^*=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent and $V(|y'|,y'')$ is a bounded non-negative function in $\mathbb{R}^3\times\mathbb{R}^{N-3}$. If $r^{2s}V(r,y'')$ has a stable critical point $(r_0,y_0'')$ with $r_0>0$ and $V(r_0,y_0'')>0$, by using a finite-dimensional reduction method and various local Pohozaev identities, we prove that the problem above has a new type of infinitely many solutions which concentrate at points lying on the top and the bottom of a cylinder. And the concentration points of the bubble solutions include saddle points of the function $r^{2s}V(r,y'')$. We have to overcome some difficulties caused by the non-localness of the fractional Laplacian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_02272 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A new type of bubble solutions for a critical fractional Schrödinger equation Du, Fan Hua, Qiaoqiao Wang, Chunhua Analysis of PDEs 35B40, 35B45, 35J40 We consider the following critical fractional Schrödinger equation \begin{equation*} (-Δ)^s u+V(|y'|,y'')u = u^{2_s^*-1},\quad u>0,\quad y =(y',y'') \in \mathbb{R}^3\times\mathbb{R}^{N-3}, \end{equation*} where $N\geq 3,s\in(0,1)$, $2_s^*=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent and $V(|y'|,y'')$ is a bounded non-negative function in $\mathbb{R}^3\times\mathbb{R}^{N-3}$. If $r^{2s}V(r,y'')$ has a stable critical point $(r_0,y_0'')$ with $r_0>0$ and $V(r_0,y_0'')>0$, by using a finite-dimensional reduction method and various local Pohozaev identities, we prove that the problem above has a new type of infinitely many solutions which concentrate at points lying on the top and the bottom of a cylinder. And the concentration points of the bubble solutions include saddle points of the function $r^{2s}V(r,y'')$. We have to overcome some difficulties caused by the non-localness of the fractional Laplacian. |
| title | A new type of bubble solutions for a critical fractional Schrödinger equation |
| topic | Analysis of PDEs 35B40, 35B45, 35J40 |
| url | https://arxiv.org/abs/2307.02272 |