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 $\displaystyle\frac{p-1}{2p}N<s<1$. Precisely, we prove the existence of positive radial bound and ground state solutions provided the parameters $\beta, p, \lambda_j,\mu_j$, ($j=1,\, 2$) satisfy appropriate conditions. We also study the previous system with $m$-equations, $$ (-\Delta)^s u_j+ \lambda_j u_j =\mu_j |u_j|^{2p-2}u_j+ \sum_{\substack{k=1\\k\neq j}}^m\beta_{jk} |u_k|^p|u_j|^{p-2}u_j,\quad u_j\in W^{s,2}(\mathbb{R}^N);\: j=1,\ldots,m $$ where $\lambda_j,\, \mu_j>0$ 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 $j,k=1,\ldots,m$, $j\neq k$).