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

Abstract

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 Carathéodory 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 $q_{0}(x)\in [1, \frac{N(p(x)-1)}{N-1}[$.