Nonlinear Fractional Schr\"odinger Equations coupled by power-type nonlinearities

Abstract

In this work we study the following class of systems of coupled nonlinear fractional nonlinear Schr\"odinger equations, \begin{equation*} \left \{ \begin{array}{l} (-\Delta)^s u_1+ \lambda_1 u_1= \mu_1 |u_1|^{2p-2}u_1+\beta |u_2|^{p} |u_1|^{p-2}u_1 \quad\text{in }\mathbb{R}^N,\\[3pt] (-\Delta)^s u_2 + \lambda_2 u_2= \mu_2 |u_2|^{2p-2}u_2+\beta |u_1|^{p}|u_2|^{p-2}u_2 \quad\text{in }\mathbb{R}^N, \end{array} \right. \end{equation*} where $ u_1,\, u_2\in W^{s,2}(\mathbb{R}^N)$, with $ N=1,\, 2,\, 3$; $\lambda_j,\,\mu_j>0$, $j=1,2$, $\beta\in \mathbb{R}$, $p\geq 2$ and $\displaystyle\frac{p-1}{2p}N 0$ for $j=1,\ldots ,m\ge 3$, the coupling parameters $\beta_{jk}=\beta_{kj}\in \mathbb{R}$ for $j,k=1,\ldots,m$, $j\neq k$. For this system we prove similar results as for $m=2$, depending on the values of the parameters $\beta_{jk}, p, \lambda_j,\mu_j$, (for $j,k=1,\ldots,m$, $j\neq k$).