Classification of nonnegative solutions to static Schrödinger–Hartree–Maxwell system involving the fractional Laplacian
Abstract This paper is mainly concerned with the following semi-linear system involving the fractional Laplacian: { ( − Δ ) α 2 u ( x ) = ( 1 | ⋅ | σ ∗ v p 1 ) v p 2 ( x ) , x ∈ R n , ( − Δ ) α 2 v ( x ) = ( 1 | ⋅ | σ ∗ u q 1 ) u q 2 ( x ) , x ∈ R n , u ( x ) ≥ 0 , v ( x ) ≥ 0 , x ∈ R n , $$ \textst...
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oai:doaj.org-article:b6e46c06aabf47fda94f59c003f27b832021-11-21T12:07:14ZClassification of nonnegative solutions to static Schrödinger–Hartree–Maxwell system involving the fractional Laplacian10.1186/s13661-021-01568-91687-2770https://doaj.org/article/b6e46c06aabf47fda94f59c003f27b832021-11-01T00:00:00Zhttps://doi.org/10.1186/s13661-021-01568-9https://doaj.org/toc/1687-2770Abstract This paper is mainly concerned with the following semi-linear system involving the fractional Laplacian: { ( − Δ ) α 2 u ( x ) = ( 1 | ⋅ | σ ∗ v p 1 ) v p 2 ( x ) , x ∈ R n , ( − Δ ) α 2 v ( x ) = ( 1 | ⋅ | σ ∗ u q 1 ) u q 2 ( x ) , x ∈ R n , u ( x ) ≥ 0 , v ( x ) ≥ 0 , x ∈ R n , $$ \textstyle\begin{cases} (-\Delta )^{\frac{\alpha }{2}}u(x)= (\frac{1}{ \vert \cdot \vert ^{\sigma }} \ast v^{p_{1}} )v^{p_{2}}(x), \quad x\in \mathbb{R}^{n}, \\ (-\Delta )^{\frac{\alpha }{2}}v(x)= (\frac{1}{ \vert \cdot \vert ^{\sigma }} \ast u^{q_{1}} )u^{q_{2}}(x), \quad x\in \mathbb{R}^{n}, \\ u(x)\geq 0,\quad\quad v(x)\geq 0, \quad x\in \mathbb{R}^{n}, \end{cases} $$ where 0 < α ≤ 2 $0<\alpha \leq 2$ , n ≥ 2 $n\geq 2$ , 0 < σ < n $0<\sigma <n$ , and 0 < p 1 , q 1 ≤ 2 n − σ n − α $0< p_{1}, q_{1}\leq \frac{2n-\sigma }{n-\alpha }$ , 0 < p 2 , q 2 ≤ n + α − σ n − α $0< p_{2}, q_{2}\leq \frac{n+\alpha -\sigma }{n-\alpha }$ . Applying a variant (for nonlocal nonlinearity) of the direct method of moving spheres for fractional Laplacians, which was developed by W. Chen, Y. Li, and R. Zhang (J. Funct. Anal. 272(10):4131–4157, 2017), we derive the explicit forms for positive solution ( u , v ) $(u,v)$ in the critical case and nonexistence of positive solutions in the subcritical cases.Yunting LiYaqiong LiuYunhui YiSpringerOpenarticleFractional LaplaciansNonnegative solutionsNonlocal nonlinearitiesDirect method of moving spheresAnalysisQA299.6-433ENBoundary Value Problems, Vol 2021, Iss 1, Pp 1-23 (2021) |
| institution |
DOAJ |
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DOAJ |
| language |
EN |
| topic |
Fractional Laplacians Nonnegative solutions Nonlocal nonlinearities Direct method of moving spheres Analysis QA299.6-433 |
| spellingShingle |
Fractional Laplacians Nonnegative solutions Nonlocal nonlinearities Direct method of moving spheres Analysis QA299.6-433 Yunting Li Yaqiong Liu Yunhui Yi Classification of nonnegative solutions to static Schrödinger–Hartree–Maxwell system involving the fractional Laplacian |
| description |
Abstract This paper is mainly concerned with the following semi-linear system involving the fractional Laplacian: { ( − Δ ) α 2 u ( x ) = ( 1 | ⋅ | σ ∗ v p 1 ) v p 2 ( x ) , x ∈ R n , ( − Δ ) α 2 v ( x ) = ( 1 | ⋅ | σ ∗ u q 1 ) u q 2 ( x ) , x ∈ R n , u ( x ) ≥ 0 , v ( x ) ≥ 0 , x ∈ R n , $$ \textstyle\begin{cases} (-\Delta )^{\frac{\alpha }{2}}u(x)= (\frac{1}{ \vert \cdot \vert ^{\sigma }} \ast v^{p_{1}} )v^{p_{2}}(x), \quad x\in \mathbb{R}^{n}, \\ (-\Delta )^{\frac{\alpha }{2}}v(x)= (\frac{1}{ \vert \cdot \vert ^{\sigma }} \ast u^{q_{1}} )u^{q_{2}}(x), \quad x\in \mathbb{R}^{n}, \\ u(x)\geq 0,\quad\quad v(x)\geq 0, \quad x\in \mathbb{R}^{n}, \end{cases} $$ where 0 < α ≤ 2 $0<\alpha \leq 2$ , n ≥ 2 $n\geq 2$ , 0 < σ < n $0<\sigma <n$ , and 0 < p 1 , q 1 ≤ 2 n − σ n − α $0< p_{1}, q_{1}\leq \frac{2n-\sigma }{n-\alpha }$ , 0 < p 2 , q 2 ≤ n + α − σ n − α $0< p_{2}, q_{2}\leq \frac{n+\alpha -\sigma }{n-\alpha }$ . Applying a variant (for nonlocal nonlinearity) of the direct method of moving spheres for fractional Laplacians, which was developed by W. Chen, Y. Li, and R. Zhang (J. Funct. Anal. 272(10):4131–4157, 2017), we derive the explicit forms for positive solution ( u , v ) $(u,v)$ in the critical case and nonexistence of positive solutions in the subcritical cases. |
| format |
article |
| author |
Yunting Li Yaqiong Liu Yunhui Yi |
| author_facet |
Yunting Li Yaqiong Liu Yunhui Yi |
| author_sort |
Yunting Li |
| title |
Classification of nonnegative solutions to static Schrödinger–Hartree–Maxwell system involving the fractional Laplacian |
| title_short |
Classification of nonnegative solutions to static Schrödinger–Hartree–Maxwell system involving the fractional Laplacian |
| title_full |
Classification of nonnegative solutions to static Schrödinger–Hartree–Maxwell system involving the fractional Laplacian |
| title_fullStr |
Classification of nonnegative solutions to static Schrödinger–Hartree–Maxwell system involving the fractional Laplacian |
| title_full_unstemmed |
Classification of nonnegative solutions to static Schrödinger–Hartree–Maxwell system involving the fractional Laplacian |
| title_sort |
classification of nonnegative solutions to static schrödinger–hartree–maxwell system involving the fractional laplacian |
| publisher |
SpringerOpen |
| publishDate |
2021 |
| url |
https://doaj.org/article/b6e46c06aabf47fda94f59c003f27b83 |
| work_keys_str_mv |
AT yuntingli classificationofnonnegativesolutionstostaticschrodingerhartreemaxwellsysteminvolvingthefractionallaplacian AT yaqiongliu classificationofnonnegativesolutionstostaticschrodingerhartreemaxwellsysteminvolvingthefractionallaplacian AT yunhuiyi classificationofnonnegativesolutionstostaticschrodingerhartreemaxwellsysteminvolvingthefractionallaplacian |
| _version_ |
1718419208398700544 |