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

中文翻译：

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具有指数和奇异系数和扩散测度的非线性椭圆系统。

在本文中，我们研究了以下类型的非线性椭圆系统的存在结果：

$$ \ begin {aligned} \ left \ {\ begin {array} {ll} \ displaystyle -div \ Big（A（x，v）\ left | \ nabla u \ right | ^ {p（x）-2} \ nabla u \ Big）+ \ left | 你\右| ^ {p（x）-2} u = \ mu＆{} \ \ \ text {in} \ \ Omega \\ \ displaystyle -div \ Big（B（x，v）\ left | \ nabla v \ right | ^ {p（x）-2} \ nabla v \ Big）+ \ left | v \ right | ^ {p（x）-2} v = \ gamma | \ nabla u | ^ {q_ {0}（x）}＆{} \ \ \ text {in} \ \ Omega，\\ \ end {array} \正确的。\ end {aligned} $$

其中\（\ Omega \）是\（{\ mathbb {R}} ^ {N}，\ N \ ge 2，\ 2- \ frac {1} {N} <p（x）< N，\，\ mu \）是一个弥散量度。A（x，  s）是Carathéodory函数。功能乙（X，  š）鼓起（均匀地相对于X）作为\（S \ RIGHTARROW米^ { - } \） （与\（M> 0 \） ）和\（\伽马\）是正常数和\（q_ {0}（x）\ in [1，\ frac {N（p（x）-1）} {N-1} [\）}。