Existence and boundedness of weak solutions to some vectorial Dirichlet problems

For integers N≥2, n≥3 and Ω a bounded subset of Rn, we prove existence and boundedness of a weak solution u of the following prototype of nonlinear vectorial Dirichlet problem u∈W01,2(Ω,RN)−∑i=1nDiAiν(x,Du)=−∑i=1nDi(Eiν(x)uν)+Fν(x),for any x∈Ω and ν=1,2,…,N, where uν and Fν denote the ν−th component of the vectors u and F, respectively, and the tensor A(x,ξ) satisfies suitable structural assumptions.