Existence of solution for a singular elliptic system with convection terms

In this paper we use the dual approach introduced by Colin and Jeanjean (2004) and Liu et al. (2003) combined with a Rabinowitz's result, Galerkin's method and an approximation argument to show the existence of solution for the following quasilinear Schrödinger elliptic system with both singular and convection terms −Δz−Δ(z2)z=μ1wθ1z−γ1+zα1+|∇w|η1inΩ,−Δw−Δ(w2)w=μ2zθ2w−γ2+wα2+|∇z|η2inΩ,z,w>0 inΩ,z=w=0 on ∂Ω,where Ω is a bounded domain of RN(N≥3) with smooth boundary, μi,θi,γi,αi,ηi>0,i=1,2 are real parameters.