A nonlocal elliptic system with nonlinear singular terms

Abstract

In this work we deal with the nonlocal elliptic system: 0 & \\hbox{in}\\ \\Omega, \\end{cases} \\] ]]> ( S ) { ( − Δ ) s 1 u = θ v q u 1 − θ in Ω , ( − Δ ) s 2 v = q v q − 1 u θ in Ω , u = v = 0 in R N ∖ Ω , u , v > 0 in Ω , where Ω ⊂ R N is a bounded regular domain ( C 2 is sufficient), 2s_2 $ ]]> N > 2 s 2 , s 1 , s 2 ∈ ( 0 , 1 ) with s 2 ≠ s 1 , 0 < θ < 1 and q>0. This work extends previous results obtained in the local case ( s 1 = s 2 = 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 s 1 , s 2 , 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.