Boundary blow-up elliptic problems of Bieberbach and Rademacher type with nonlinear gradient terms

By a perturbation method and constructing comparison functions, we show the exact asymptotic behaviour of solutions near the boundary to nonlinear elliptic problems △u±|∇u|q=b(x)eu,x∈Ω,u|∂Ω=+∞, where Ω is a bounded domain with smooth boundary in RN, q≥0, b is non-negative and non-trivial in Ω, which may be vanishing on the boundary.