We investigate the existence and multiplicity of solutions for nonlocal elliptic systems driven by the fractional Laplacian. Specifically, we establish the existence of two positive solutions for the following class of nonlocal elliptic systems: \(\begin{aligned} {\left\{ \begin{array}{ll} \,(-\Delta )^su +V_1(x)\hspace{1.111pt}u = \lambda |u|^{p - 2}\hspace{1.111pt}u+ \frac{\alpha }{\alpha +\beta }\,\theta |u|^{\alpha - 2}\hspace{1.111pt}u|v|^{\beta }, & \text{ in }\;\; \mathbb {R}^N,\\ \,(-\Delta )^sv +V_2(x)\hspace{1.111pt}v= \lambda |v|^{q - 2}\hspace{1.111pt}v+ \frac{\beta }{\alpha +\beta }\,\theta |u|^{\alpha }|v|^{\beta -2}\hspace{1.111pt}v, & \text{ in }\;\; \mathbb {R}^N, \end{array}\right. } \end{aligned}\) \((u, v) \in H^s(\mathbb {R}^N) \hspace{1.111pt}{\times }\hspace{1.111pt}H^s(\mathbb {R}^N)\) . Here, \(\alpha ,\beta > 1\) , \(1 \leqslant p \leqslant q< 2< \alpha + \beta < 2^*_s\) , \(\theta ,\lambda > 0\) , \(N > 2s\) , \(s \in (0,1)\) , and \(V_1, V_2:\mathbb {R}^N \rightarrow \mathbb {R}\) are continuous and positive potentials. Furthermore, we find the largest positive number \(\lambda ^* > 0\) such that the above problem admits at least two positive solutions for each \( \lambda \in (0, \lambda ^*)\) . This can be done by using the nonlinear Rayleigh quotient together with the Nehari method. The main feature here is to minimize the energy functional in Nehari manifold which allows us to prove our main results without any restriction on size of parameter \(\theta > 0\) .
