A boundary blow-up elliptic problem with an inhomogeneous term

By a perturbation method and constructing comparison functions, we reveal how the inhomogeneous term hh affects the exact asymptotic behaviour of solutions near the boundary to the problem △u=b(x)g(u)+λh(x)△u=b(x)g(u)+λh(x), u>0u>0 in ΩΩ, u|∂Ω=∞u|∂Ω=∞, where ΩΩ is a bounded domain with smooth boundary in RNRN, λ>0λ>0, g∈C1[0,∞)g∈C1[0,∞) is increasing on [0,∞)[0,∞), g(0)=0g(0)=0, g′g′ is regularly varying at infinity with positive index ρρ, the weight bb, which is non-trivial and non-negative in ΩΩ, may be vanishing on the boundary, and the inhomogeneous term hh is non-negative in ΩΩ and may be singular on the boundary.