Abstract
In this work, we investigate the existence and the uniqueness of solutions for the nonlocal elliptic system involving a singular nonlinearity as follows:
{(-Delta(p))(s)u = a(x)vertical bar u vertical bar(q-2)u + 1-alpha/2-alpha-beta c(x) vertical bar u vertical bar(-alpha)vertical bar v vertical bar(1-beta), in Omega,
(-Delta(p))(s)v = b(x)vertical bar v vertical bar(q-2)v + 1-beta/2-alpha-beta c(x) vertical bar u vertical bar(1-alpha)vertical bar v vertical bar(-beta), in Omega,
u = v = 0, in R-N\Omega,
where Omega is a bounded domain in R-n with smooth boundary, 0 < alpha < 1, 0 < beta < 1, 2 - alpha - beta < p < q <= p(s)* = Np/N-sp, a, b, c. C((Omega) over bar) are non-negative weight functions with compact support in Omega, and (-Delta)(p)(s) 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.