Normalized solutions for a coupled fractional schrodinger system in low dimensions

Abstract

We consider the following coupled fractional Schrödinger system: \begin{equation*} \left\{ \begin{aligned} &(-Δ)^su+λ_1u=μ_1|u|^{2p-2}u+β|v|^p|u|^{p-2}u\\ &(-Δ)^sv+λ_2v=μ_2|v|^{2p-2}v+β|u|^p|v|^{p-2}v\\ \end{aligned} \right.\quad\text{in}~{\mathbb{R}^N}, \end{equation*} with $0<s<1$, $2s<N\le 4s$ and $1+\frac{2s}{N}<p<\frac{N}{N-2s}$, under the following constraint \begin{align*} \int_{\mathbb{R}^N}|u|^2dx=a_1^2\quad\text{and}\quad \int_{\mathbb{R}^N}|v|^2dx=a_2^2. \end{align*} Assuming that the parameters ${\mu }_1,{\mu }_2,a_1,a_2$ are fixed quantities, we prove the existence of normalized solution for different ranges of the coupling parameter $\beta >0$ .