Quasilinear Degenerated Elliptic Systems with Weighted in Divergence Form with Weak Monotonicity with General Data

We consider, for a bounded open domain Ω in Rn and a function u : Ω → Rm, the quasilinear elliptic system: (1). We generalize the system (QES)(f,g) in considering a right hand side depending on the jacobian matrix Du. Here, the star in (QES)(f,g) indicates that f may depend on Du. In the right hand side, v belongs to the dual space W-1,P’(Ω, ω*, Rm), , f and g satisfy some standard continuity and growth conditions. We prove existence of a regularity, growth and coercivity conditions for σ, but with only very mild monotonicity assumptions.