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 − Δ θ = k ( θ ) | ∇ p | r + q ( x ) in Ω ; − div { ( k ( θ ) | ∇ p | r − 2 + β ( x ) | ∇ p | r 0 − 2 ) ∇ p } = 0 in Ω ; θ = θ 0 , and p = p 0 on ∂ Ω .