Quasilinear degenerated elliptic system in divergence form with mild monotonicity in weighted sobolev spaces

Abstract

We consider, for a bounded open domain $\Omega$ in ${{\mathbb {R}}}^{n}$ and a function u: $\Omega \rightarrow {{\mathbb {R}}}^{m}$, the quasilinear elliptic system

$$\begin{aligned} \text{(QES) } \left\{ \begin{array}{rcl} -\text{ div }\sigma \left( x,u\left( x\right) ,Du\left( x\right) \right) &{} = &{} f(x) \text{ in } \Omega \\ u &{} = &{} 0 \; \;\;\; \text{ on } \partial \Omega , \end{array} \right. \end{aligned}$$

(0.1)

where f belongs to the dual space $W^{-1,p'}\left( \Omega ,\omega ^{*},{{\mathbb {R}}}^{m}\right)$ of $W_{0}^{1,p}\left( \Omega ,\omega ,{{\mathbb {R}}}^{m}\right)$. We prove existence of a regularity, growth and coercivity conditions for $\sigma$, but with only very mild monotonicity.