Nonlinear elliptic system with singular coefficient and with diffuse measure data

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We give an existence result of the nonlinear elliptic system of the type: $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \displaystyle -div\Big (A(x,v)\nabla u\Big )=\mu &{} \ \ \text{ in }\ \Omega \\ \displaystyle -div\Big (B(x,v)\nabla v\Big )=\gamma |\nabla u|^{q_0} &{} \ \ \text{ in }\ \Omega ,\\ \end{array} \right. \end{aligned}$$where $$\Omega $$ is a bounded open subset of $$\mathbb {R}^{N},\ N\ge 2,\ \mu $$ is a diffuse measure. A(x, s) is a Caratheodory function. The function B(x, s) blows up (uniformly with respect to x) as $$s\rightarrow m^{-}$$ (with $$m>0$$) and $$\gamma $$ is a positive constant and $$q_{0}\in [1, \frac{N}{N-1}[$$. The main contribution of our work is to prove the existence of a renormalized solution.