Uniqueness of boundary blow-up solutions on exterior domain of RN

In this paper, we consider the existence and uniqueness of positive solutions of the degenerate logistic type elliptic equation -Delta u = a(x)u - b(x)|u|(q-1) u, x is an element of R-N \ D, u|(partial derivative D) = infinity, where N >= 2, D subset of R-N is a bounded domain with smooth boundary and a(x), b(x) are continuous functions on R-N with b(x) >= 0, b(x) not equivalent to 0. We show that under rather general conditions on a(x) and b(x) for large |x|, there exists a unique positive solution. Our results improve the corresponding ones in [W Dong, Y. Du, Unbounded principal eigenfunctions and the logistic equation on R-N, Bull. Austral. Math. Soc. 67 (2003) 413-427] and [Y. Du, L. Ma, Logistic type equations on R-N by a squeezing method involving boundary blow-up solutions, J. London Math. Soc. (2) 64 (2001) 107-124]. (C) 2006 Elsevier Inc. All rights reserved.