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Nonlinear Elliptic System with Variable Exponents and Singular Coefficient and with Diffuse Measure Data
Mediterranean Journal of Mathematics ( IF 1.1 ) Pub Date : 2021-04-19 , DOI: 10.1007/s00009-021-01766-w
A. Eljazouli , H. Redwane

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(xs) is a Carathéodory function. The function B(xs) 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:

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\(\ Omega \)\({\ mathbb {R}} ^ {N},\ N \ ge 2,\ 2- \ frac {1} {N} <p(x)< N,\,\ mu \)是一个弥散量度。Ax,  s)是Carathéodory函数。功能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\(q_ {0}(x)\ in [1,\ frac {N(p(x)-1)} {N-1} [\)}

更新日期:2021-04-19
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