In this paper, we study the Dirichlet problem of elliptic systems -Δu=g(u)inΩ,u=ϱμon∂Ω,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{aligned} &{}-\Delta \mathbf {u}=\mathbf {g}(\mathbf {u})&{}\quad &{}\text{ in } \ \Omega ,\\ &{}\mathbf {u}=\mathbf {\varrho } \varvec{\mu }&{}\quad &{}\text{ on } \ \partial \Omega , \end{aligned} \end{array}\right. } \end{aligned}$$\end{document}where ϱ≥0\documentclass[12pt]{minimal}
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\begin{document}$$\varrho \ge 0$$\end{document}, Ω\documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document} is an open bounded C2\documentclass[12pt]{minimal}
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\begin{document}$$C^2$$\end{document} domain in RN\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^N$$\end{document} with N≥2\documentclass[12pt]{minimal}
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\begin{document}$$N\ge 2$$\end{document}, and u\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {u}$$\end{document}, g(u)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {g}(\mathbf {u})$$\end{document}, μ\documentclass[12pt]{minimal}
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\begin{document}$$\varvec{\mu }$$\end{document} are nonnegative vector-valued functions. We obtain the existence of weak positive solutions for the systems. In the special case g(u)=|u|p-1u\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {g}(\mathbf {u})=|\mathbf {u}|^{p-1}\mathbf {u}$$\end{document} with p>1\documentclass[12pt]{minimal}
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\begin{document}$$p>1$$\end{document}, we shall give a better description about the positive solutions including the priori estimate, regularity, existence and nonexistence.
