Optimal global asymptotic behaviour of the solution to a class of singular Dirichlet problems

This paper is mainly concerned with the global asymptotic behaviour of the unique solution to a class of singular Dirichlet problems -Delta u = b(x)g(u), u > 0, x is an element of Omega, u vertical bar(partial derivative Omega) = 0, where Omega is a bounded smooth domain in R-n, g is an element of C-1(0, infinity) is positive and decreasing in (0, infinity) with lim(s -> 0) + g(s) = infinity, b is an element of C-alpha(Omega) for some alpha is an element of (0, 1), which is positive in Omega, but may vanish or blow up on the boundary properly. Moreover, we reveal the asymptotic behaviour of such a solution when the parameters on b tend to the corresponding critical values.