An existence theorem for weak solutions for a class of elliptic partial differential systems in general Orlicz–Sobolev spaces

I prove the existence of a weak solution for the Dirichlet problem of a class of elliptic partial differential systems −∂Aαi∂xα(x,u(x),Du(x))+Bi(x,u(x),Du(x))=0 in general Orlicz–Sobolev spaces W01LM(Ω,RN), where i=1,…,N,α=1,…,ni=1,…,N,α=1,…,n, u:Ω→RNu:Ω→RN is a vector-valued function, and the summation convention is used throughout with i,ji,j running from 1 to NN and α,βα,β running from 1 to nn.