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

Abstract

I prove the existence of a weak solution for the Dirichlet problem of a class of elliptic partial differential systems − ∂ A α i ∂ x α ( x , u ( x ) , D u ( x ) ) + B i ( x , u ( x ) , D u ( x ) ) = 0 in general Orlicz–Sobolev spaces W 0 1 L M ( Ω , R N ) , where i = 1 , … , N , α = 1 , … , n , u : Ω → R N is a vector-valued function, and the summation convention is used throughout with i , j running from 1 to N and α , β running from 1 to n .