Existence of Weak Solutions for a Nonlinear Elliptic System

We investigate the existence of weak solutions to the following Dirichlet boundary value problem, which occurs when modeling an injection molding process with a partial slip condition on the boundary. We have Δθ=k(θ)| p|r+q(x) in Ω; div{(k(θ)| p|r 2+β(x)| p|r0 2) p}=0 in Ω; θ=θ0, and p=p0 on Ω.