UNIQUENESS AND EXISTENCE OF SOLUTIONS FOR A SINGULAR SYSTEM WITH NONLOCAL OPERATOR VIA PERTURBATION METHOD

In this work, we investigate the existence and the uniqueness of solutions for the nonlocal elliptic system involving a singular nonlinearity as follows: $ \begin{eqnarray*} \left\{\begin{array}{ll} (-\Delta_p)^su = a(x)|u|^{q-2}u +\frac{1-\alpha}{2-\alpha-\beta} c(x)|u|^{-\alpha}|v|^{1-\beta}, \quad \text{in }\Omega, \ (-\Delta_p)^s v = b(x)|v|^{q-2}v +\frac{1-\beta}{2-\alpha-\beta} c(x)|u|^{1-\alpha}|v|^{-\beta}, \quad \text{in }\Omega, \ u = v = 0 , \;\;\mbox{ in }\, \mathbb{R}^N\setminus\Omega, \end{array} \right. \end{eqnarray*} $ where $ \Omega $ is a bounded domain in $ \mathbb{R}^{n} $ with smooth boundary, $ 0<\alpha <1, $ $ 0<\beta <1, $ $ 2-\alpha -\beta <p<q\leq p_{s}^{\ast } = \frac{Np}{N-sp}, $ $ a, \, b, \, c \in C(\overline{\Omega}) $ are non-negative weight functions with compact support in $ \Omega, - $ and $ (-\Delta)^s_p $ is the fractional $ p $-laplacian operator. We use a perturbation method combine with some variationals methods in order to show the existence of a solution to the above system. We also prove the uniqueness of the solution to the system for some additional condition.