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

Abstract

We study the quasilinear elliptic system -div σ(x, u, Du) = v(x) + ƒ(x, u) + div g(x, u) on a bounded domain of ℝn 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 ƒ. We prove existence of a weak solution of this system under classical regularity, growth, and coercivity conditions for σ, but with only very mild monotonicity assumptions.