A nonlinear elliptic system with a transport term and singular data

We consider the nonlinear elliptic system {u∈W0NN−1(Ω):−div(M(x)∇u)+u=−div(uM(x)∇ψ)+f(x),ψ∈W01,2(Ω):−div(M(x)∇ψ)+ψ=R(u)+E(x)∇ψ,where Ω is a bounded, open subset of RN, N≥ 3; M(x) is a coercive, symmetric matrix with L∞(Ω) coefficients; f(x) and E(x) belong to some Lebesgue space, and R(s) is a continuous function such that 0≤R(s)≤|s|θ,for θ<2N.Using a duality technique, we prove existence of at least a weak solution (u,ψ). Moreover, if N=3 or N=4, we prove under stronger assumptions on f(x) and E(x) that the solution u belongs to W01,2(Ω).