A nonlocal elliptic system with nonlinear singular terms

In this work we deal with the nonlocal elliptic system: (S){(−Δ)s1u=θvqu1−θin Ω,(−Δ)s2v=qvq−1uθin Ω,u=v=0in RN∖Ω,u,v>0in Ω, where Ω⊂RN is a bounded regular domain (C2 is sufficient), N>2s2, s1,s2∈(0,1) with s2≠s1, 0<θ<1 and q>0. This work extends previous results obtained in the local case (s1=s2=1) see Boccardo L, Orsina L. [A variational semilinear singular system. Nonlinear Anal Theory Methods Appl. 2011;74(12):3849–3860. doi: 10.1016/j.na.2011.01.017]. Under some suitable conditions on the parameters on s1,s2,θ and q, we obtain the existence results by using approximation methods, variational techniques and the classical minimization, a nonexistence result has also been treated.