Nonlocal fractional system involving the fractional p, q-Laplacians and singular potentials

In this paper, we will focus on following nonlocal quasilinear elliptic system with singular nonlinearities: $$\begin{aligned} (S) \left\{ \begin{array}{ll} (-\Delta )_{p}^{s_1} u =\dfrac{1}{v^{\alpha _1}}+v^{\beta _1}&{} \text { in }\Omega , \\ (-\Delta )_{q}^{s_2} u =\dfrac{1}{u^{\alpha _2}}+u^{\beta _2}&{} \text { in }\Omega , \\ u,v=0 &{} \text { in } ({I\!\!R}^N\setminus \Omega ) , \\ u,v>0 &{} \text { in } \Omega , \end{array} \right. \end{aligned}$$ where $$\Omega \subset {I\!\!R}^N$$ be a smooth bounded domain, $$s_1,\,s_2\in (0,1)$$ , $$\alpha _1$$ , $$\alpha _2$$ , $$\beta _1$$ , $$\beta _2$$ are suitable positive constants, $$(-\Delta )_{p}^{s_1}$$ and $$(-\Delta )_{q}^{s_2}$$ are the fractional $$p-\text {Laplacian}$$ and $$q-\text {Laplacian}$$ operators. Using approximating arguments, Rabinowitz bifurcation Theorem, and fractional Hardy inequality, we are able to show the existence of positive solution to the above system.