The second order expansion of solutions to a singular Dirichlet boundary value problem

In this paper, we mainly study the second order expansion of classical solutions in a neighborhood of partial derivative Omega to the singular Dirichlet problem -Delta u = b(x)g(u) + lambda a(x)f(u), u > 0, x is an element of Omega, u vertical bar partial derivative Omega = 0, where Omega is a bounded domain with smooth boundary in R-N, lambda >= 0. The weight functions b, a is an element of C-loc(alpha)(Omega) are positive in Omega and both may be vanishing or be singular on the boundary. The function g is an element of C-1 ((0, infinity), (0, infinity)) satisfies lim(t -> 0+) g(t) = infinity, and f is an element of C([0,infinity),[0,infinity)). We show that the nonlinear term lambda a(x) f(u) does not affect the second order expansion of solutions in a neighborhood of partial derivative Omega to the problem for some kinds of functions b and a. (C) 2015 Elsevier Inc. All rights reserved.