A quasilinear elliptic system with natural growth terms

Abstract

In this paper, we prove existence of solutions for an elliptic system of the type $$\begin{aligned} {\left\{ \begin{array}{ll} -\mathrm{div}(a(x,z) \nabla u) = f, &{}\quad \text{ in } \varOmega \text{; } \\ -\mathrm{div}(b(x) \nabla z)+ h(x,z)|\nabla u|^2 = g, &{}\quad \text{ in } \varOmega \text{; } \\ \,u = 0 = z, &{}\quad \text{ on } \partial \varOmega \text{, } \end{array}\right. } \end{aligned}$$under various assumptions on the functions \(a(x,s)\) and \(h(x,s)\), and on the data \(f\) and \(g\) (in Lebesgue spaces).