In this paper, we prove the existence of weak solutions for the following nonlinear elliptic system - div A ( x, ? u ) = - a ( x ) | u | p ( x ) - 2 u - b ( x ) | u | α ( x ) | v | β ( x ) v + f ( x ) in Ω , - div B ( x, ? v ) = ? - c ( x ) | v | q ( x ) - 2 v - d ( x ) | v | β ( x ) | u | α ( x ) u + g ( x ) in Ω , u = v = 0 on ? Ω , where Ω is an open bounded domains of R N with a smooth boundary ? Ω. The existence of weak solutions is proved using the theory of monotone operators. ? 2000 Mathematics Subject Classification. Primary 35B45; Secondary 35J55. Key words and phrases. Weak solutions; nonlinear elliptic systems; p ( x )-Laplacian; monotone operators; generalized Lebesgue-Sobolev spaces.
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机译：在本文中，我们证明了以下非线性椭圆系统-div A（x，？u）=-a（x）|的弱解的存在。你p（x）-2 u-b（x）|你α（x）| v | β（x）v + f（x）单位为Ω-div B（x，？v）=？ -c（x）| v | q（x）-2 v-d（x）| v | β（x）|你α（x）u + g（x）以Ω为单位，u = v = 0 on？ Ω，其中Ω是具有平滑边界的R N的开放有界域？ Ω。利用单调算子理论证明了弱解的存在。 ？ 2000年数学学科分类。初级35B45;中学35J55。关键字和词组。解决方案薄弱;非线性椭圆系统p（x）-Laplacian;单调运算符;广义Lebesgue-Sobolev空间。
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