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: (−∆p)u = a(x)|u|q−2u+ 1−α 2−α−β c(x)|u| −α|v|1−β , in Ω, (−∆p)v = b(x)|v|q−2v + 1−β 2−α−β c(x)|u| 1−α|v|−β , in Ω, u = v = 0, in R \ Ω, where Ω is a bounded domain in R with smooth boundary, 0 < α < 1, 0 < β < 1, 2 − α − β < p < q ≤ ps = Np N−sp , a, b, c ∈ C(Ω) are non-negative weight functions with compact support in Ω, and (−∆)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.