Nonlinear Fractional Schrödinger Equations coupled by power--type nonlinearities

Abstract

In this work, we study the following class of systems of coupled nonlinear fractional Schrödinger 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 $ \frac{p-1}{2p}N < s < 1$. Precisely, we prove the existence of positive radial bound and ground state solutions provided the parameters $p, \beta, \lambda_j,\mu_j$, ($j = 1,\, 2$) satisfy appropriate conditions. We also study the previous system with $m$ equations, $$ (-\Delta)^s u_j + \lambda_j u_j = \mu_j |u_j|^{2p-2}u_j + \sum_{\substack{k = 1\\k\neq j}}^m\beta_{jk} |u_k|^p|u_j|^{p-2}u_j,\quad u_j\in W^{s,2}(\mathbb{R}^N) $$ where $ j = 1,\ldots,m\ge 3$, $\lambda_j,\, \mu_j > 0$, 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 $p, \beta_{jk}, \lambda_j,\mu_j$, (for $j,k = 1,\ldots,m$, $j\neq k$).