Nonlinear Fractional Schrödinger Equations coupled by power-type nonlinearities

Abstract

In this work we study the following class of systems of coupled nonlinear fractional nonlinear Schrödinger equations, \begin{equation*} \left \{ \begin{array}{l} (-Δ)^s u_1+ λ_1 u_1= μ_1 |u_1|^{2p-2}u_1+β|u_2|^{p} |u_1|^{p-2}u_1 \quad\text{in }\mathbb{R}^N,\\[3pt] (-Δ)^s u_2 + λ_2 u_2= μ_2 |u_2|^{2p-2}u_2+β|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 $N=1,2,3$; ${\lambda }_j,{\mu }_j>0$, $j=1,2$, $\beta \in \mathbb{R}$, $p\ge 2$ and $\frac{p-1}{2p}N<s<1$. Precisely, we prove the existence of positive radial bound and ground state solutions provided the parameters $\beta ,p,{\lambda }_j,{\mu }_j$, ($j=1,2$) satisfy appropriate conditions. We also study the previous system with $m$-equations, $$(-\Delta {)}^s{u}_j+{\lambda }_j{u}_j={\mu }_j|u_j{|}^{2p-2}{u}_j+\sum _{\begin{array}{c}k=1\\ k\ne j\end{array}}^m{\beta }_{jk}|u_k{|}^p|u_j{|}^{p-2}{u}_j,u_j\in W^{s,2}({\mathbb{R}}^N);j=1,\dots ,m$$ where ${\lambda }_j,{\mu }_j>0$ for $j=1,\dots ,m\ge 3$, the coupling parameters ${\beta }_{jk}={\beta }_{kj}\in \mathbb{R}$ for $j,k=1,\dots ,m$, $j\ne k$. 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 $j,k=1,\dots ,m$, $j\ne k$).