Normalized solutions for a coupled fractional Schrödinger system in low dimensions
Abstract We consider the following coupled fractional Schrödinger system: { ( − Δ ) s u + λ 1 u = μ 1 | u | 2 p − 2 u + β | v | p | u | p − 2 u , ( − Δ ) s v + λ 2 v = μ 2 | v | 2 p − 2 v + β | u | p | v | p − 2 v in R N , $$ \textstyle\begin{cases} (-\Delta )^{s}u+\lambda _{1}u=\mu _{1} \vert u \v...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2020-10-01
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| Series: | Boundary Value Problems |
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| Online Access: | http://link.springer.com/article/10.1186/s13661-020-01463-9 |
| _version_ | 1818341424153755648 |
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| author | Meng Li Jinchun He Haoyuan Xu Meihua Yang |
| author_facet | Meng Li Jinchun He Haoyuan Xu Meihua Yang |
| author_sort | Meng Li |
| collection | DOAJ |
| description | Abstract We consider the following coupled fractional Schrödinger system: { ( − Δ ) s u + λ 1 u = μ 1 | u | 2 p − 2 u + β | v | p | u | p − 2 u , ( − Δ ) s v + λ 2 v = μ 2 | v | 2 p − 2 v + β | u | p | v | p − 2 v in R N , $$ \textstyle\begin{cases} (-\Delta )^{s}u+\lambda _{1}u=\mu _{1} \vert u \vert ^{2p-2}u+ \beta \vert v \vert ^{p} \vert u \vert ^{p-2}u, \\ (-\Delta )^{s}v+\lambda _{2}v=\mu _{2} \vert v \vert ^{2p-2}v+\beta \vert u \vert ^{p} \vert v \vert ^{p-2}v \end{cases}\displaystyle \quad \text{in } {\mathbb{R}^{N}}, $$ with 0 < s < 1 $0< s<1$ , 2 s < N ≤ 4 s $2s< N\le 4s$ and 1 + 2 s N < p < N N − 2 s $1+\frac{2s}{N}< p<\frac{N}{N-2s}$ , under the following constraint: ∫ R N | u | 2 d x = a 1 2 and ∫ R N | v | 2 d x = a 2 2 . $$ \int _{\mathbb{R}^{N}} \vert u \vert ^{2}\,dx=a_{1}^{2} \quad \text{and}\quad \int _{ \mathbb{R}^{N}} \vert v \vert ^{2}\,dx=a_{2}^{2}. $$ Assuming that the parameters μ 1 $\mu _{1}$ , μ 2 $\mu _{2}$ , a 1 $a_{1}$ , a 2 $a_{2}$ are fixed quantities, we prove the existence of normalized solution for different ranges of the coupling parameter β > 0 $\beta >0$ . |
| first_indexed | 2024-12-13T15:58:34Z |
| format | Article |
| id | doaj.art-819c79d5e8ef4ac2acd130ba9612f5e8 |
| institution | Directory Open Access Journal |
| issn | 1687-2770 |
| language | English |
| last_indexed | 2024-12-13T15:58:34Z |
| publishDate | 2020-10-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Boundary Value Problems |
| spelling | doaj.art-819c79d5e8ef4ac2acd130ba9612f5e82022-12-21T23:39:15ZengSpringerOpenBoundary Value Problems1687-27702020-10-012020112910.1186/s13661-020-01463-9Normalized solutions for a coupled fractional Schrödinger system in low dimensionsMeng Li0Jinchun He1Haoyuan Xu2Meihua Yang3School of Mathematics and Statistics, Huazhong University of Science and TechnologySchool of Mathematics and Statistics, Huazhong University of Science and TechnologySchool of Mathematics and Statistics, Huazhong University of Science and TechnologySchool of Mathematics and Statistics, Huazhong University of Science and TechnologyAbstract We consider the following coupled fractional Schrödinger system: { ( − Δ ) s u + λ 1 u = μ 1 | u | 2 p − 2 u + β | v | p | u | p − 2 u , ( − Δ ) s v + λ 2 v = μ 2 | v | 2 p − 2 v + β | u | p | v | p − 2 v in R N , $$ \textstyle\begin{cases} (-\Delta )^{s}u+\lambda _{1}u=\mu _{1} \vert u \vert ^{2p-2}u+ \beta \vert v \vert ^{p} \vert u \vert ^{p-2}u, \\ (-\Delta )^{s}v+\lambda _{2}v=\mu _{2} \vert v \vert ^{2p-2}v+\beta \vert u \vert ^{p} \vert v \vert ^{p-2}v \end{cases}\displaystyle \quad \text{in } {\mathbb{R}^{N}}, $$ with 0 < s < 1 $0< s<1$ , 2 s < N ≤ 4 s $2s< N\le 4s$ and 1 + 2 s N < p < N N − 2 s $1+\frac{2s}{N}< p<\frac{N}{N-2s}$ , under the following constraint: ∫ R N | u | 2 d x = a 1 2 and ∫ R N | v | 2 d x = a 2 2 . $$ \int _{\mathbb{R}^{N}} \vert u \vert ^{2}\,dx=a_{1}^{2} \quad \text{and}\quad \int _{ \mathbb{R}^{N}} \vert v \vert ^{2}\,dx=a_{2}^{2}. $$ Assuming that the parameters μ 1 $\mu _{1}$ , μ 2 $\mu _{2}$ , a 1 $a_{1}$ , a 2 $a_{2}$ are fixed quantities, we prove the existence of normalized solution for different ranges of the coupling parameter β > 0 $\beta >0$ .http://link.springer.com/article/10.1186/s13661-020-01463-9Fractional LaplacianSchrödinger systemPositive radial solution |
| spellingShingle | Meng Li Jinchun He Haoyuan Xu Meihua Yang Normalized solutions for a coupled fractional Schrödinger system in low dimensions Boundary Value Problems Fractional Laplacian Schrödinger system Positive radial solution |
| title | Normalized solutions for a coupled fractional Schrödinger system in low dimensions |
| title_full | Normalized solutions for a coupled fractional Schrödinger system in low dimensions |
| title_fullStr | Normalized solutions for a coupled fractional Schrödinger system in low dimensions |
| title_full_unstemmed | Normalized solutions for a coupled fractional Schrödinger system in low dimensions |
| title_short | Normalized solutions for a coupled fractional Schrödinger system in low dimensions |
| title_sort | normalized solutions for a coupled fractional schrodinger system in low dimensions |
| topic | Fractional Laplacian Schrödinger system Positive radial solution |
| url | http://link.springer.com/article/10.1186/s13661-020-01463-9 |
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