Boundary Blow-up Rate of the Large Solution for an Elliptic Cooperative System

In this article we consider positive large solution of cooperative systems of the form −Δu1=λ1u1+a1u1u2q1−b1(x)u1p1+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-{\Delta}u_1=\lambda_{1}u_{1}+a_{1}u_{1}u_{2}^{q_{1}}-b_1(x)u_{1}^{p_{1}+1}$$\end{document}, −Δu2=λ2u2+a2u1q2u2−b2(x)u2p2+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-{\Delta}u_2=\lambda_{2}u_{2}+a_{2}u_{1}^{q_{2}}u_{2}-b_2(x)u_{2}^{p_{2}+1}$$\end{document} in a bounded smooth domain Ω ⊂ RN(λi ∈ R, ai, bi > 0, 0 < qi < pj, i, j ∈ {1, 2}, i ≠ j), Based on the construction of certain sup and sub-solution, we show existence, uniqueness and blow-up rate of the large solution.