Singular nonlocal elliptic systems via nonlinear Rayleigh quotient

In the present work, we establish the existence of two positive solutions for singular nonlocal elliptic systems. More precisely, we consider the following nonlocal elliptic problem: $$\left\{\begin{array}{lll} (-\Delta)^su +V_1(x)u = \lambda\frac{a(x)}{u^p} + \frac{\alpha}{\alpha+\beta}\theta |u|^{\alpha - 2}u|v|^{\beta}, \,\,\, \mbox{in} \,\,\, \mathbb{R}^N,\\ (-\Delta)^sv +V_2(x)v= \lambda \frac{b(x)}{v^q}+ \frac{\beta}{\alpha+\beta}\theta |u|^{\alpha}|v|^{\beta-2}v, \,\,\, \mbox{in} \,\,\, \mathbb{R}^N, \end{array}\right. \;\;\;(u, v) \in H^s(\mathbb{R}^N) \times H^s(\mathbb{R}^N),$$ where $ 0<p \leq q < 1<\;\alpha, \beta \;,\;2<\alpha + \beta < 2^*_s$, $\theta > 0, \lambda > 0, N > 2s$, and $s \in (0,1)$. The potentials $V_1, V_2: \mathbb{R}^N \to \mathbb{R}$ are continuous functions which are bounded from below. Under our assumptions, we prove that there exists the largest positive number $\lambda^* > 0$ such that our main problem admits at least two positive solutions for each $\lambda \in (0, \lambda^*)$. Here we apply the nonlinear Rayleigh quotient together with the Nehari method. The main feature is to minimize the energy functional in Nehari set which allows us to prove our results without any restriction on the size of parameter $\theta > 0$. Moreover, we shall consider the multiplicity of solutions for the case $\lambda = \lambda^*$ where degenerated points are allowed.