Multiplicity results of fractional-Laplace system with sign-changing and singular nonlinearity

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 $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^n$ with smooth boundary $\mathrm{\partial }\mathrm{\Omega }$, $n>2s$, $s\in (0,1)$, $0<q<1$, $\alpha >1$, $\beta >1$ satisfy $2<\alpha +\beta <2_s^{\ast }-1$ with $2_s^{\ast }=\frac{2n}{n-2s}$, the pair of parameters $(\lambda,\mu) \in $\mathbb{R}$^2\setminus{(0,0)}$\mathrm{.}Theweightfunctions$f,g: \Omega\subset$\mathbb{R}$^n \to $\mathbb{R}$$suchthat$0<f$,$g\in L^{\frac{\alpha+\beta}{\alpha+\beta-1+q}}(\Omega)$,and$b:\Omega\subset $\mathbb{R}$^n \to $\mathbb{R}$$isasign-changingfunctionsuchthat$b(x)\in L^{\infty}(\Omega)$. Using variational methods, we show existence and multiplicity of positive solutions of $(P_{\lambda ,\mu })$ with respect to the pair of parameters $(\lambda ,\mu )$.