Normalized solutions for a coupled fractional Schrödinger system in low dimensions

Meng Li, Jinchun He, Haoyuan Xu, Meihua Yang
2020 Boundary Value Problems  
We consider the following coupled fractional Schrödinger system: $$ \textstyle\begin{cases} (-\Delta )^{s}u+\lambda _{1}u=\mu _{1} \vert u \vert ^{2p-2}u+ \beta \vert v \vert ^{p} \vert u \vert ^{p-2}u, \\ (-\Delta )^{s}v+\lambda _{2}v=\mu _{2} \vert v \vert ^{2p-2}v+\beta \vert u \vert ^{p} \vert v \vert ^{p-2}v \end{cases}\displaystyle \quad \text{in } {\mathbb{R}^{N}}, $$ { ( − 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
more » ... | 2 p − 2 v + β | u | p | v | p − 2 v in R N , with $0< s<1$ 0 < s < 1 , $2s< N\le 4s$ 2 s < N ≤ 4 s and $1+\frac{2s}{N}< p<\frac{N}{N-2s}$ 1 + 2 s N < p < N N − 2 s , under the following constraint: $$ \int _{\mathbb{R}^{N}} \vert u \vert ^{2}\,dx=a_{1}^{2} \quad \text{and}\quad \int _{ \mathbb{R}^{N}} \vert v \vert ^{2}\,dx=a_{2}^{2}. $$ ∫ R N | u | 2 d x = a 1 2 and ∫ R N | v | 2 d x = a 2 2 . Assuming that the parameters $\mu _{1}$ μ 1 , $\mu _{2}$ μ 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 $\beta >0$ β > 0 .
doi:10.1186/s13661-020-01463-9 fatcat:hbb34665hzh5vnzevjylbmt6ay