Quasilinear elliptic systems in divergence form with weak monotonicity and nonlinear physical data

We study the quasilinear elliptic system $$ -mathop{ m div}sigma(x,u,Du) =v(x)+f(x,u)+mathop{ m div}g(x,u) $$ on a bounded domain of $mathbb{R}^n$ with homogeneous Dirichlet boundary conditions. This system corresponds to a diffusion problem with a source $v$ in a moving and dissolving substance, where the motion is described by $g$ and the dissolution by $f$. We prove existence of a weak solution of this system under classical regularity, growth, and coercivity conditions for $sigma$, but with only very mild monotonicity assumptions.