The exact asymptotic behaviour of the unique solution to a singular nonlinear Dirichlet problem

摘要

By Karamata regular varying theory, a perturbed argument and constructing comparison functions, we show the exact asymptotic behaviour of the unique solution u epsilon C-2(Omega) boolean AND C((Omega) over bar) near the boundary to a singular Dirichlet problem -Delta u = b(x)g(u) + lambda f (u), u > 0, x epsilon Omega, u vertical bar(partial derivative Omega)=0, which is independent on lambda f (u), and we also show the existence and uniqueness of solutions to the problem, where Omega is a bounded domain with smooth boundary in R-N, lambda > 0, g epsilon C-1 ((0, infinity), (0, infinity)) and there exists gamma > 1 such that lim(t -> 0)+ g'(xi t)/g'(t) = xi-(1+gamma), for all xi > 0, f epsilon C-loc(alpha) ([0, infinity), [0, infinity)), the function f (s)/s+s(0) is decreasing on (0, infinity) for some s(0) > 0, and b is nonnegative nontrivial on Omega, which may be vanishing on the boundary. (c) 2006 Elsevier Inc. All rights reserved.