On a singular elliptic system with quadratic growth in the gradient

In this paper, we deal with the following singular elliptic system:{−Δu+α|∇u|2u=pp+qa(x)|v|q|u|p−2u+f,x∈RN,−Δv+β|∇v|2v=qp+qa(x)|u|p|v|q−2v+g,x∈RN, where N≥3, α,β>N+24, p,q>1 and p+q≤N+2N−2. We show through the sub- and supersolutions method, the existence of a nonnegative solution for an approximated system. The limit of the approximated solution is a positive solution. In the case, α=β=0, p=q and f=g, we prove the uniqueness of a solution. Among others, we prove some existence and uniqueness results for some auxiliary problems by using the comparison principle, a minimization method and with the help of Nehari manifold. The proofs rely on the concentration-compactness principle.