Ground states for a nonhomogeneous elliptic system involving Hardy–Sobolev critical exponents

This paper studies the following nonhomogeneous elliptic system involving Hardy–Sobolev critical exponentsΔ u + λ 1u p − 1| x | s+ μ 1u 2 * ( s 1 ) − 1| x | s 1+α γ2 * ( s 2 )u α − 1v β| x | s 2= 0inΩ ,Δ v + λ 2v p − 1| x | s+ μ 2v 2 * ( s 1 ) − 1| x | s 1+β γ2 * ( s 2 )u α v β − 1| x | s 2= 0inΩ ,u ≥ 0 , v ≥ 0inΩ , u = 0 , v = 0on∂ Ω ,whereλ 1 , λ 2 , μ 1 , μ 2 , γ > 0 , 2 < p < 2 * ( s ) : = 2 ( N − s ) N − 2 , 0 ≤ s < 2 , 0 < s 1 , s 2 < 2 , α , β > 1 , α + β = 2 * ( s 2 ) , Ω is a C 1 open bounded domain in R N containing the origin, and N ≥ 4 . The existence result of positive ground state solution is established.