In this paper, we establish the blow-up rate of the large positive solution of the singular boundary value problem −△u=λu−a(x)up,u|∂Ω=+∞, where Ω is a bounded smooth domain in RN. The weight function a(x) in front of the nonlinearity can vanish on the boundary of the domain Ω at different rates according to the point x0 of the boundary. The decay rate of the weight function a(x) may not be approximated by a power function of distance near the boundary ∂Ω. We combine the localization method of [J. López-Gómez, The boundary blow-up rate of large solutions, J. Differential Equations 195 (2003) 25–45] with some previous radially symmetric results of [T. Ouyang, Z. Xie, The uniqueness of blow-up solution for radially symmetric semilinear elliptic equation, Nonlinear Anal. 64 (9) (2006) 2129–2142] to prove that any large solution u(x) must satisfy limx→x0u(x)K(bx0∗(dist(x,∂Ω)))−β=1for each x0∈∂Ω, where bx0∗(r)=∫0r∫0sbx0(t)dtds,K=[β((β+1)C0−1)]1p−1,β=1p−1,C0=limr→0(∫0rbx0(t)dt)2bx0∗(r)bx0(r) and bx0(r) is the boundary normal section of a(x) at x0∈∂Ω, i.e., bx0(r)=a(x0−rnx0),r>0,r∼0, and nx0 stands for the outward unit normal vector at x0∈∂Ω.