  In this article, we study the following fractional-Laplacian system with
singular nonlinearity \begin{equation*} (P_{\lambda,\mu}) \left\{
\begin{array}{lr} (-\Delta)^s u = \lambda f(x) u^{-q}+
\frac{\alpha}{\alpha+\beta}b(x) u^{\alpha-1} w^\beta\; \text{in}\;\Omega \\
(-\Delta)^s w = \mu g(x) w^{-q}+ \frac{\beta}{\alpha+\beta} b(x) u^{\alpha}
w^{\beta-1}\; \text{in}\;\Omega \\ \quad \quad u, w>0\;\text{in}\;\Omega, \quad
u = w = 0 \; \mbox{in}\; \mathbb{R}^n \setminus\Omega, \end{array} \quad
\right. \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^n$
with smooth boundary $\partial \Omega$, $n>2s$, $s\in(0,1)$, $0<q<1$,
$\alpha>1$, $\beta>1$ satisfy $2<\alpha+\beta< 2_{s}^*-1$ with
$2_{s}^*=\frac{2n}{n-2s}$, the pair of parameters $(\lambda,\mu) \in
\mathbb{R}^2\setminus\{(0,0)\}$. The weight functions $f,g:
\Omega\subset\mathbb{R}^n \to \mathbb{R}$ such that $0<f$, $g\in
L^{\frac{\alpha+\beta}{\alpha+\beta-1+q}}(\Omega)$, and $b:\Omega\subset
\mathbb{R}^n \to \mathbb{R}$ is a sign-changing function such that $b(x)\in
L^{\infty}(\Omega)$. Using variational methods, we show existence and
multiplicity of positive solutions of $(P_{\lambda,\mu})$ with respect to the
pair of parameters $(\lambda,\mu)$.
