Nonlinear Elliptic System with Variable Exponents and Singular Coefficient and with Diffuse Measure Data

In this paper, we investigate an existence result of the nonlinear elliptic system of the type: $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -div\Big (A(x,v)\left| \nabla u\right| ^{p(x)-2}\nabla u\Big ) + \left| u\right| ^{p(x)-2} u =\mu &{}\ \ \text{ in }\ \Omega \\ \displaystyle -div\Big (B(x,v)\left| \nabla v\right| ^{p(x)-2}\nabla v\Big ) + \left| v\right| ^{p(x)-2} v =\gamma |\nabla u|^{q_{0}(x)} &{}\ \ \text{ in }\ \Omega ,\\ \end{array} \right. \end{aligned}$$ where $$\Omega $$ is a bounded open subset of $${\mathbb {R}}^{N},\ N\ge 2,\ 2-\frac{1}{N}<p(x)<N,\, \mu $$ 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 $$q_{0}(x)\in [1, \frac{N(p(x)-1)}{N-1}[$$ .