Uniqueness and blow-up rate of large solutions for elliptic equation -Δu = λu - b(x)h(u)

In this paper, we establish the blow-up rate of the large positive solution of the singular boundary value problem {-Delta u = lambda u - b(x)h(u) in Omega, u = +infinity on partial derivative Omega, where Omega is a smooth bounded domain in R(N). The weight function b(x) is a non-negative continuous function in the domain. h(u) is locally Lipschitz continuous and h(u)lu is increasing on (0, infinity) and h(u) similar to Hu(p) for sufficiently large u with H > 0 and p > 1. Naturally, the blow-up rate of the problem equals its blow-up rate for the very special, but important, case when h(u) = Hu(p). We distinguish two cases: (1) Omega is a ball domain and b is a radially symmetric function on the domain in Theorem 1.1; (II) Omega is a smooth bounded domain and b satisfies some local condition on each boundary normal section assumed in Theorem 1.2. The blow-up rate is explicitly determined by functions b and h. In case (I), the singular boundary value problem has a unique solution u satisfying lim(d(x)-> 0) u(x)/KH(-beta)(b*(parallel to x - x(0)parallel to))(-beta) = 1 where d(x) = dist(x, partial derivative Omega), b*(r) and K are defined by b*(r) = (r)integral(R) (s)integral(R) b(t) dt ds, K=[beta((beta+1)C(0)-1)](1/p-1), beta:= 1/p-1. In case (II), the blow-up rates of the solutions to the boundary value problem are established and the uniqueness is proved. (C) 2009 Elsevier Inc. All rights reserved.